Research ArticleFalknerndashSkan Equation with Heat Transfer A New StochasticNumerical Approach

Imran Khan1 Hakeem Ullah 1 Hussain AlSalman 2 Mehreen Fiza1 Saeed Islam1

Asif Zahoor Raja3 Mohammad Shoaib4 and Abdu H Gumaei 5

1Department of Mathematics Abdul Wali Khan University Mardan 23200 Khyber Pakhtunkhwa (KP) Pakistan2Department of Computer Science College of Computer and Information Sciences King Saud UniversityRiyadh 11543 Saudi Arabia3Future Technology Research Center National Yunlin University of Science and Technology 123 University Road Section 3Douliou Yunlin 64002 Taiwan4Department of Mathematics COMSATS University Islamabad Attock Campus Attock 43600 Pakistan5Department of Computer Science Faculty of Applied Sciences Taiz University Taiz 6803 Yemen

CorrespondenceshouldbeaddressedtoHakeemUllahhakeemullah1gmailcomHussainAlSalmanhalsalmanksuedusa andAbdu H Gumaei abdugumaeigmailcom

Received 6 June 2021 Revised 10 July 2021 Accepted 2 August 2021 Published 30 August 2021

Academic Editor Mohammad Yaghoub Abdollahzadeh Jamalabadi

Copyright copy 2021 Imran Khan et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study a new computingmodel is developed using the strength of feedforward neural networks with the LevenbergndashMarquardtmethod- (NN-BLMM-) based backpropagation technique It is used to find a solution for the nonlinear system obtained from thegoverning equations of FalknerndashSkan with heat transfer (FSE-HT) Moreover the partial differential equations (PDEs) for theunsteady squeezing flow of heat andmass transfer of the viscous fluid are converted into ordinary differential equations (ODEs) withthe help of similarity transformation A dataset for the proposed NN-BLMM-based model is generated in different scenarios by avariation of various embedding parameters Deborah number (β) and Prandtl number (Pr) -e training (TR) testing (TS) andvalidation (VD) of the NN-BLMMmodel are evaluated in the generated scenarios to compare the obtained results with the referenceresults For the fluidic system convergence analysis a number of metrics such as themean square error (MSE) error histogram (EH)and regression (RG) plots are utilized for measuring the effectiveness and performance of the NN-BLMM infrastructure model -eexperiments showed that comparisons between the results of the proposed model and the reference results match in terms ofconvergence up to E-05 to E-10-is proves the validity of theNN-BLMMmodel Furthermore the results demonstrated that there isan increase in the velocity profile and a decrease in the thickness of the thermal boundary layer by increasing the Deborah numberAlso the thickness of the thermal boundary layer is decreased by increasing the Prandtl number

1 Introduction

-e boundary layer flow of an incompressible liquid througha stretching sheet is commonly used in many industrial andengineering processes -e field has been attracting re-searchers in the past few years -e boundary layer flow hasmajor applications in industries such as aerodynamic ex-traction of polymer paper from debris thermal wrappingcooling plate with no cooling tuber boundary layer next tothe liquid film in the condensation phase and glass fiber

development [1ndash3] By immersing them in quiescent liquidsmany metal processes need to cool continuous fibers -emechanical features of the final product depend only on thedrawing costs and the process temperature Sakiadis [4 5]experimented with new work in this area and many re-searchers in the field have investigated the flow of theboundary layer into the ongoing stretching sheet at an in-creasing speed A closed-form solution for the viscous flowof an incompressible fluid was obtained by Crane [6] Guptaand Gupta [7] studied the same effects as given in [6] by

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 3921481 17 pageshttpsdoiorg10115520213921481

β2

Figure 1 -e geometry of the problem

Figure 2 -e structure of a single neural network model

Input

Hidden Output

OutputW

b1

10 1

1

+W

b+

Figure 3 Graphical representation of NN-BLMM input hidden and output layers

Table 1 Defining the physical parameters of interest for the scenarios and cases of FSE-HT

Scenarios CasesPhysical quantity

β Pr

11 2 052 4 053 6 05

21 4 052 4 103 4 15

2 Mathematical Problems in Engineering

β2

(a)

f primePrime(ξ) + f(ξ)fPrime(ξ) + β(1ndash(f prime(ξ))2) = 0

f (0) = 0 f prime(0) = 0 f prime(infin) = 1θ (0) = 1 θ(infin) = 0

θPrime(ξ) + Pr f (ξ)θprime(ξ) = 0

(b)

(c)

Input

Hidden Output

OutputW

b1

10 1

1

+W

b+

(d)

1

05

0

-05

-1

-15

-20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(e)

Analysis of assessmentMean square error based fitnessAbsolute error analysisState transition analysisError histograms

iii

iiiivv Regression studies

(f )

Figure 4 -e workflow of the proposed NN-BLMM for processing the FSE-HT (a) -e geometry of the problem (b) Formulation of theproblem (c) Intelligent computing neuron model integrated to develop the proposed network (d) Results (e) Comparative analysis

Mathematical Problems in Engineering 3

0 50 100 150 200 250 300330 Epochs

Best Validation Performance is 24702e-09 at epoch 324

10-2

10-4

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(a)

0 100 200 300 400 500 600 700 900 10008001000 Epochs

Best Validation Performance is 13224e-10 at epoch 1000

100

10-5

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(b)

0 50 100 150 200 250 350300366 Epochs

Best Validation Performance is 12138e-08 at epoch 366100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(c)

0 10 20 30 40 50 60 7073 Epochs

Best Validation Performance is 27001e-05 at epoch 73100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(d)

0 20 40 60 80 100 140120149 Epochs

Best Validation Performance is 50276e-07 at epoch 143100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(e)

0 10050 150 200249 Epochs

Best Validation Performance is 10658-09 at epoch 249100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(f )

Figure 5-eMSE results of the proposed NN-BLMM to solve the FSE-HT (a)-eMSE results in case 1 for scenario 1 (b)-eMSE resultsin case 2 for scenario 1 (c)-eMSE results in case 3 for scenario 1 (d)-eMSE results in case 1 for scenario 2 (e)-eMSE results in case 2for scenario 2 (f ) -e MSE results in case 3 for scenario 2

4 Mathematical Problems in Engineering

considering the fluid electrically conducting and porousmedium with suctioninjection by Anderson [8] Ariel [9]investigated the combined effect of viscoelasticity andmagnetic field on the Cranersquos problem Since a stretchingsheet can occur in a variety of ways the flow through the

stretching sheet does not always need to be of two sizes If theextension is radial it can be three A flat surface of threestretches and the same width was examined by Wang [10]Brady and Acrivos [11] monitored the flow inside thechannel or tube and the Wang flow outside the performing

0 50 100 150 200 250 300336 Epochs

Gradient = 99298e-08 at epoch 336

Mu = 1e-08 at epoch 336

Validation Checks = 0 at epoch 336

100

10-5

10-10

100

10-5

10-10

1

0

-1

grad

ient

mu

val f

ail

(a)

1000 Epochs0 100 200 300 400 500 600 700 800 900 1000

Gradient = 10553e-07 at epoch 1000

Mu = 1e-08 at epoch 1000

Validation Checks = 0 at epoch 1000

1010

100

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(b)

0 50 100 150 200 250 300 350366 Epochs

Gradient = 99259e-08 at epoch 366

Mu = 1e-08 at epoch 366

Validation Checks = 0 at epoch 366

100

10-5

10-10

100

10-5

10-10

4

2

0

grad

ient

mu

val f

ail

(c)

73 Epochs0 10 20 30 40 50 60 70

Gradient = 96455e-08 at epoch 73

Mu = 1e-08 at epoch 73

Validation Checks = 0 at epoch 73

100

10-5

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(d)

0 20 40 60 80 100 120 140149 Epochs

Gradient = 84492e-07 at epoch 149

Mu = 1e-08 at epoch 149

Validation Checks = 6 at epoch 149

100

10-5

10-10

100

10-5

10-10

10

5

0

grad

ient

mu

val f

ail

(e)

249 Epochs0 50 100 150 200

Gradient = 97359e-08 at epoch 249

Mu = 1e-09 at epoch 249

Validation Checks = 0 at epoch 249

100

10-5

10-10

100

10-5

10-10

4

2

0

val f

ail

mu

grad

ient

(f )

Figure 6 -e dynamic results of transition states for the NN-BLMM to solve the FSE-HT (a) -e transition state results in case 1 for scenario 1(b)-e transition state results in case 2 for scenario 1 (c)-e transition state results in case 3 for scenario 1 (d)-e transition state results in case 1 forscenario 2 (e) -e transition state results in case 2 for scenario 2 (f) -e transition state results in case 3 for scenario 2

Mathematical Problems in Engineering 5

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

Targets - Outputs

06

05

04

03

02

01

0O

utpu

t and

Tar

get

1050

-5

Erro

r

times10-3

Function Fit for Output Element 1

Figure 7 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

5

4

3

2

1

0

20

-2-4

Out

put a

nd T

arge

tEr

ror Targets - Outputs

times10-5

Function Fit for Output Element 1

Figure 8 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 25 3

Input

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

0010

001-002

Erro

r Targets - Outputs

Function Fit for Output Element 1

Figure 9 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 1 for FSE-HT

6 Mathematical Problems in Engineering

0 05 1 15 2 25 3

Input

10

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

05

04

03

02

01

0

-01O

utpu

t and

Tar

get

Figure 10 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT

0 05 1 15 2 25 3

Input

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Targets - Outputs

Figure 11 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT

Function Fit for Output Element 1

210

-1

Erro

r

times10-3

0 05 1 15 2 25 3

Input

04504

03503

02502

01501

0050

Out

put a

nd T

arge

t

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

Figure 12 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT

Mathematical Problems in Engineering 7

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

β2

(a)

f primePrime(ξ) + f(ξ)fPrime(ξ) + β(1ndash(f prime(ξ))2) = 0

f (0) = 0 f prime(0) = 0 f prime(infin) = 1θ (0) = 1 θ(infin) = 0

θPrime(ξ) + Pr f (ξ)θprime(ξ) = 0

(b)

(c)

Input

Hidden Output

OutputW

b1

10 1

1

+W

b+

(d)

1

05

0

-05

-1

-15

-20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(e)

Analysis of assessmentMean square error based fitnessAbsolute error analysisState transition analysisError histograms

iii

iiiivv Regression studies

(f )

Figure 4 -e workflow of the proposed NN-BLMM for processing the FSE-HT (a) -e geometry of the problem (b) Formulation of theproblem (c) Intelligent computing neuron model integrated to develop the proposed network (d) Results (e) Comparative analysis

Mathematical Problems in Engineering 3

0 50 100 150 200 250 300330 Epochs

Best Validation Performance is 24702e-09 at epoch 324

10-2

10-4

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(a)

0 100 200 300 400 500 600 700 900 10008001000 Epochs

Best Validation Performance is 13224e-10 at epoch 1000

100

10-5

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(b)

0 50 100 150 200 250 350300366 Epochs

Best Validation Performance is 12138e-08 at epoch 366100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(c)

0 10 20 30 40 50 60 7073 Epochs

Best Validation Performance is 27001e-05 at epoch 73100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(d)

0 20 40 60 80 100 140120149 Epochs

Best Validation Performance is 50276e-07 at epoch 143100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(e)

0 10050 150 200249 Epochs

Best Validation Performance is 10658-09 at epoch 249100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(f )

Figure 5-eMSE results of the proposed NN-BLMM to solve the FSE-HT (a)-eMSE results in case 1 for scenario 1 (b)-eMSE resultsin case 2 for scenario 1 (c)-eMSE results in case 3 for scenario 1 (d)-eMSE results in case 1 for scenario 2 (e)-eMSE results in case 2for scenario 2 (f ) -e MSE results in case 3 for scenario 2

4 Mathematical Problems in Engineering

considering the fluid electrically conducting and porousmedium with suctioninjection by Anderson [8] Ariel [9]investigated the combined effect of viscoelasticity andmagnetic field on the Cranersquos problem Since a stretchingsheet can occur in a variety of ways the flow through the

stretching sheet does not always need to be of two sizes If theextension is radial it can be three A flat surface of threestretches and the same width was examined by Wang [10]Brady and Acrivos [11] monitored the flow inside thechannel or tube and the Wang flow outside the performing

0 50 100 150 200 250 300336 Epochs

Gradient = 99298e-08 at epoch 336

Mu = 1e-08 at epoch 336

Validation Checks = 0 at epoch 336

100

10-5

10-10

100

10-5

10-10

1

0

-1

grad

ient

mu

val f

ail

(a)

1000 Epochs0 100 200 300 400 500 600 700 800 900 1000

Gradient = 10553e-07 at epoch 1000

Mu = 1e-08 at epoch 1000

Validation Checks = 0 at epoch 1000

1010

100

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(b)

0 50 100 150 200 250 300 350366 Epochs

Gradient = 99259e-08 at epoch 366

Mu = 1e-08 at epoch 366

Validation Checks = 0 at epoch 366

100

10-5

10-10

100

10-5

10-10

4

2

0

grad

ient

mu

val f

ail

(c)

73 Epochs0 10 20 30 40 50 60 70

Gradient = 96455e-08 at epoch 73

Mu = 1e-08 at epoch 73

Validation Checks = 0 at epoch 73

100

10-5

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(d)

0 20 40 60 80 100 120 140149 Epochs

Gradient = 84492e-07 at epoch 149

Mu = 1e-08 at epoch 149

Validation Checks = 6 at epoch 149

100

10-5

10-10

100

10-5

10-10

10

5

0

grad

ient

mu

val f

ail

(e)

249 Epochs0 50 100 150 200

Gradient = 97359e-08 at epoch 249

Mu = 1e-09 at epoch 249

Validation Checks = 0 at epoch 249

100

10-5

10-10

100

10-5

10-10

4

2

0

val f

ail

mu

grad

ient

(f )

Figure 6 -e dynamic results of transition states for the NN-BLMM to solve the FSE-HT (a) -e transition state results in case 1 for scenario 1(b)-e transition state results in case 2 for scenario 1 (c)-e transition state results in case 3 for scenario 1 (d)-e transition state results in case 1 forscenario 2 (e) -e transition state results in case 2 for scenario 2 (f) -e transition state results in case 3 for scenario 2

Mathematical Problems in Engineering 5

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

Targets - Outputs

06

05

04

03

02

01

0O

utpu

t and

Tar

get

1050

-5

Erro

r

times10-3

Function Fit for Output Element 1

Figure 7 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

5

4

3

2

1

0

20

-2-4

Out

put a

nd T

arge

tEr

ror Targets - Outputs

times10-5

Function Fit for Output Element 1

Figure 8 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 25 3

Input

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

0010

001-002

Erro

r Targets - Outputs

Function Fit for Output Element 1

Figure 9 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 1 for FSE-HT

6 Mathematical Problems in Engineering

0 05 1 15 2 25 3

Input

10

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

05

04

03

02

01

0

-01O

utpu

t and

Tar

get

Figure 10 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT

0 05 1 15 2 25 3

Input

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Targets - Outputs

Figure 11 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT

Function Fit for Output Element 1

210

-1

Erro

r

times10-3

0 05 1 15 2 25 3

Input

04504

03503

02502

01501

0050

Out

put a

nd T

arge

t

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

Figure 12 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT

Mathematical Problems in Engineering 7

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

0 50 100 150 200 250 300330 Epochs

Best Validation Performance is 24702e-09 at epoch 324

10-2

10-4

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(a)

0 100 200 300 400 500 600 700 900 10008001000 Epochs

Best Validation Performance is 13224e-10 at epoch 1000

100

10-5

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(b)

0 50 100 150 200 250 350300366 Epochs

Best Validation Performance is 12138e-08 at epoch 366100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(c)

0 10 20 30 40 50 60 7073 Epochs

Best Validation Performance is 27001e-05 at epoch 73100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(d)

0 20 40 60 80 100 140120149 Epochs

Best Validation Performance is 50276e-07 at epoch 143100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(e)

0 10050 150 200249 Epochs

Best Validation Performance is 10658-09 at epoch 249100

10-4

10-2

10-6

10-8

10-10

Mea

n Sq

uare

d Er

ror (

mse

)

TrainValidation

TestBest

(f )

Figure 5-eMSE results of the proposed NN-BLMM to solve the FSE-HT (a)-eMSE results in case 1 for scenario 1 (b)-eMSE resultsin case 2 for scenario 1 (c)-eMSE results in case 3 for scenario 1 (d)-eMSE results in case 1 for scenario 2 (e)-eMSE results in case 2for scenario 2 (f ) -e MSE results in case 3 for scenario 2

4 Mathematical Problems in Engineering

considering the fluid electrically conducting and porousmedium with suctioninjection by Anderson [8] Ariel [9]investigated the combined effect of viscoelasticity andmagnetic field on the Cranersquos problem Since a stretchingsheet can occur in a variety of ways the flow through the

stretching sheet does not always need to be of two sizes If theextension is radial it can be three A flat surface of threestretches and the same width was examined by Wang [10]Brady and Acrivos [11] monitored the flow inside thechannel or tube and the Wang flow outside the performing

0 50 100 150 200 250 300336 Epochs

Gradient = 99298e-08 at epoch 336

Mu = 1e-08 at epoch 336

Validation Checks = 0 at epoch 336

100

10-5

10-10

100

10-5

10-10

1

0

-1

grad

ient

mu

val f

ail

(a)

1000 Epochs0 100 200 300 400 500 600 700 800 900 1000

Gradient = 10553e-07 at epoch 1000

Mu = 1e-08 at epoch 1000

Validation Checks = 0 at epoch 1000

1010

100

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(b)

0 50 100 150 200 250 300 350366 Epochs

Gradient = 99259e-08 at epoch 366

Mu = 1e-08 at epoch 366

Validation Checks = 0 at epoch 366

100

10-5

10-10

100

10-5

10-10

4

2

0

grad

ient

mu

val f

ail

(c)

73 Epochs0 10 20 30 40 50 60 70

Gradient = 96455e-08 at epoch 73

Mu = 1e-08 at epoch 73

Validation Checks = 0 at epoch 73

100

10-5

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(d)

0 20 40 60 80 100 120 140149 Epochs

Gradient = 84492e-07 at epoch 149

Mu = 1e-08 at epoch 149

Validation Checks = 6 at epoch 149

100

10-5

10-10

100

10-5

10-10

10

5

0

grad

ient

mu

val f

ail

(e)

249 Epochs0 50 100 150 200

Gradient = 97359e-08 at epoch 249

Mu = 1e-09 at epoch 249

Validation Checks = 0 at epoch 249

100

10-5

10-10

100

10-5

10-10

4

2

0

val f

ail

mu

grad

ient

(f )

Figure 6 -e dynamic results of transition states for the NN-BLMM to solve the FSE-HT (a) -e transition state results in case 1 for scenario 1(b)-e transition state results in case 2 for scenario 1 (c)-e transition state results in case 3 for scenario 1 (d)-e transition state results in case 1 forscenario 2 (e) -e transition state results in case 2 for scenario 2 (f) -e transition state results in case 3 for scenario 2

Mathematical Problems in Engineering 5

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

Targets - Outputs

06

05

04

03

02

01

0O

utpu

t and

Tar

get

1050

-5

Erro

r

times10-3

Function Fit for Output Element 1

Figure 7 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

5

4

3

2

1

0

20

-2-4

Out

put a

nd T

arge

tEr

ror Targets - Outputs

times10-5

Function Fit for Output Element 1

Figure 8 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 25 3

Input

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

0010

001-002

Erro

r Targets - Outputs

Function Fit for Output Element 1

Figure 9 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 1 for FSE-HT

6 Mathematical Problems in Engineering

0 05 1 15 2 25 3

Input

10

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

05

04

03

02

01

0

-01O

utpu

t and

Tar

get

Figure 10 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT

0 05 1 15 2 25 3

Input

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Targets - Outputs

Figure 11 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT

Function Fit for Output Element 1

210

-1

Erro

r

times10-3

0 05 1 15 2 25 3

Input

04504

03503

02502

01501

0050

Out

put a

nd T

arge

t

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

Figure 12 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT

Mathematical Problems in Engineering 7

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

considering the fluid electrically conducting and porousmedium with suctioninjection by Anderson [8] Ariel [9]investigated the combined effect of viscoelasticity andmagnetic field on the Cranersquos problem Since a stretchingsheet can occur in a variety of ways the flow through the

stretching sheet does not always need to be of two sizes If theextension is radial it can be three A flat surface of threestretches and the same width was examined by Wang [10]Brady and Acrivos [11] monitored the flow inside thechannel or tube and the Wang flow outside the performing

0 50 100 150 200 250 300336 Epochs

Gradient = 99298e-08 at epoch 336

Mu = 1e-08 at epoch 336

Validation Checks = 0 at epoch 336

100

10-5

10-10

100

10-5

10-10

1

0

-1

grad

ient

mu

val f

ail

(a)

1000 Epochs0 100 200 300 400 500 600 700 800 900 1000

Gradient = 10553e-07 at epoch 1000

Mu = 1e-08 at epoch 1000

Validation Checks = 0 at epoch 1000

1010

100

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(b)

0 50 100 150 200 250 300 350366 Epochs

Gradient = 99259e-08 at epoch 366

Mu = 1e-08 at epoch 366

Validation Checks = 0 at epoch 366

100

10-5

10-10

100

10-5

10-10

4

2

0

grad

ient

mu

val f

ail

(c)

73 Epochs0 10 20 30 40 50 60 70

Gradient = 96455e-08 at epoch 73

Mu = 1e-08 at epoch 73

Validation Checks = 0 at epoch 73

100

10-5

10-10

100

10-5

10-10

1

0

-1

val f

ail

mu

grad

ient

(d)

0 20 40 60 80 100 120 140149 Epochs

Gradient = 84492e-07 at epoch 149

Mu = 1e-08 at epoch 149

Validation Checks = 6 at epoch 149

100

10-5

10-10

100

10-5

10-10

10

5

0

grad

ient

mu

val f

ail

(e)

249 Epochs0 50 100 150 200

Gradient = 97359e-08 at epoch 249

Mu = 1e-09 at epoch 249

Validation Checks = 0 at epoch 249

100

10-5

10-10

100

10-5

10-10

4

2

0

val f

ail

mu

grad

ient

(f )

Figure 6 -e dynamic results of transition states for the NN-BLMM to solve the FSE-HT (a) -e transition state results in case 1 for scenario 1(b)-e transition state results in case 2 for scenario 1 (c)-e transition state results in case 3 for scenario 1 (d)-e transition state results in case 1 forscenario 2 (e) -e transition state results in case 2 for scenario 2 (f) -e transition state results in case 3 for scenario 2

Mathematical Problems in Engineering 5

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

Targets - Outputs

06

05

04

03

02

01

0O

utpu

t and

Tar

get

1050

-5

Erro

r

times10-3

Function Fit for Output Element 1

Figure 7 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

5

4

3

2

1

0

20

-2-4

Out

put a

nd T

arge

tEr

ror Targets - Outputs

times10-5

Function Fit for Output Element 1

Figure 8 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 25 3

Input

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

0010

001-002

Erro

r Targets - Outputs

Function Fit for Output Element 1

Figure 9 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 1 for FSE-HT

6 Mathematical Problems in Engineering

0 05 1 15 2 25 3

Input

10

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

05

04

03

02

01

0

-01O

utpu

t and

Tar

get

Figure 10 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT

0 05 1 15 2 25 3

Input

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Targets - Outputs

Figure 11 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT

Function Fit for Output Element 1

210

-1

Erro

r

times10-3

0 05 1 15 2 25 3

Input

04504

03503

02502

01501

0050

Out

put a

nd T

arge

t

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

Figure 12 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT

Mathematical Problems in Engineering 7

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

Targets - Outputs

06

05

04

03

02

01

0O

utpu

t and

Tar

get

1050

-5

Erro

r

times10-3

Function Fit for Output Element 1

Figure 7 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 425 53 35 45

Input

5

4

3

2

1

0

20

-2-4

Out

put a

nd T

arge

tEr

ror Targets - Outputs

times10-5

Function Fit for Output Element 1

Figure 8 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 1 for FSE-HT

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

0 05 1 15 2 25 3

Input

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

0010

001-002

Erro

r Targets - Outputs

Function Fit for Output Element 1

Figure 9 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 1 for FSE-HT

6 Mathematical Problems in Engineering

0 05 1 15 2 25 3

Input

10

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

05

04

03

02

01

0

-01O

utpu

t and

Tar

get

Figure 10 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT

0 05 1 15 2 25 3

Input

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Targets - Outputs

Figure 11 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT

Function Fit for Output Element 1

210

-1

Erro

r

times10-3

0 05 1 15 2 25 3

Input

04504

03503

02502

01501

0050

Out

put a

nd T

arge

t

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

Figure 12 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT

Mathematical Problems in Engineering 7

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

0 05 1 15 2 25 3

Input

10

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

05

04

03

02

01

0

-01O

utpu

t and

Tar

get

Figure 10 -e results of the NN-BLMM compared with the reference solution in case 1 of scenario 2 for FSE-HT

0 05 1 15 2 25 3

Input

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

06

05

04

03

02

01

0

Out

put a

nd T

arge

t

5

0

-5

Erro

r

times10-3

Function Fit for Output Element 1

Targets - Outputs

Figure 11 -e results of the NN-BLMM compared with the reference solution in case 2 of scenario 2 for FSE-HT

Function Fit for Output Element 1

210

-1

Erro

r

times10-3

0 05 1 15 2 25 3

Input

04504

03503

02502

01501

0050

Out

put a

nd T

arge

t

Training TargetsTraining OutputsValidation TargetsValidation OutputsTest TargetsTest OutputsErrorsFit

Targets - Outputs

Figure 12 -e results of the NN-BLMM compared with the reference solution in case 3 of scenario 2 for FSE-HT

Mathematical Problems in Engineering 7

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

tube [12] Wang [13] and Usha and Sridharan [14] tested theunstable SS Ariel [15] used HPM and expanded HPM toobtain a solution for analysis in axisymmetric flow across theflat layer A noniterative solution for MHD flow was pro-vided by Ariel et al [16] -e problem of a third-ordernonlinear two-point boundary value with no exact solutions

is commonly known as the FalknerndashSkan equation (FSE)Due to the importance of the boundary layer theory the FSEis considered widely in the last forty years Due to thenonlinear nature of the FSE having no exact solutionsavailable in the literature the scientists have tried the an-alytical and numerical approaches Hartree [17] obtained a

-35

e-06

000

0311

000

0625

000

0939

000

1253

000

1567

000

1881

000

2195

000

2509

000

2824

000

3138

000

3452

000

3766

000

408

000

4394

000

4708

000

5022

000

5337

000

5651

000

5965

Error = Targets - Outputs

Error Histogram with 20 Bins

50

40

30

20

10

0

Inst

ance

sTrainingValidationTestZero Error

(a)

-2e-

05-1

9e-

05-1

7e-

05-1

5e-

05-1

4e-

05-1

2e-

05-1

1e-

05-9

2e-

06-7

6e-

06

-45

e-06

-29

e-06

-13

e-06

237

e-07

18e

-06

337

e-06

494

e-06

65e

-06

807

e-06

963

e-06

-6e-

06

Error = Targets - Outputs

Error Histogram with 20 Bins

7

6

5

4

3

2

1

0

Inst

ance

s

TrainingValidationTestZero Error

(b)

-00

1703

-00

1587

-00

1471

-00

1355

-00

1239

-00

1123

-00

1007

-00

0891

-00

0775

-00

0659

-00

0543

-00

0428

-00

0312

-00

0196

-00

008

000

0364

000

1524

000

2683

000

3843

000

5003

Error = Targets - Outputs

Error Histogram with 20 Bins30

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(c)

-00

0011

000

0296

000

0697

000

1099

000

1501

000

1902

000

2304

000

2705

000

3107

000

3508

000

391

000

4311

000

4713

000

5114

000

5516

000

5918

000

6319

000

6721

000

7122

000

7524

Error = Targets - Outputs

Error Histogram with 20 Bins

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

(d)

-00

0401

-00

0375

-00

0349

-00

0323

-00

0297

-00

0271

-00

0245

-00

0219

-00

0194

-00

0168

-00

0142

-00

0116

-00

009

-00

0064

-00

0038

-00

0012

000

0135

000

0394

000

0652

000

0911

Error = Targets - Outputs

Error Histogram with 20 Bins18

16

14

10

12

8

6

4

2

0

Inst

ance

s

TrainingValidationTestZero Error

(e)

-45

e-05

256

e-05

96e

-05

000

0166

000

0237

000

0307

000

0378

000

0448

000

0518

000

0589

000

0659

000

073

000

080

0008

70

0009

410

0010

110

0010

820

0011

520

0012

220

0012

93

Error = Targets - Outputs

25

20

15

10

5

0

Inst

ance

s

TrainingValidationTestZero Error

Error Histogram with 20 Bins

(f )

Figure 13 Studying the error histograms for the NN-BLMM results in cases 1ndash3 of scenarios 1 and 2 for FSE-HT (a) Error histogram forscenario 1 in case 1 (b) Error histogram for scenario 1 in case 2 (c) Error histogram for scenario 1 in case 3 (d) Error histogram for scenario2 in case 1 (e) Error histogram for scenario 2 in case 2 (f ) Error histogram for scenario 2 in case 3

8 Mathematical Problems in Engineering

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

solution of the FSE numerically Smith and Cebeci andKeller [18 19] solved the FSE by the shooting methodMeksyn [20] solved the FSE by analytic approximationAsaithambi [21ndash23] found the FSE solution by finite dif-ferences and Salama studied the FSE by solving it with ahigher-order method [24] Zhang and Chen used the iter-ative method for handling the FSE [25] Rosales-Vera andValencia used the Fourier transform for the solution of theFSE [26] Liao [27] applied the homotopy analysis method(HAM) to solve the FSE recently Recently Ullah et almodified the optimal homotopy asymptotic method(OHAM) for the FSE [28] -e important special case of theFSE is known as the Blasius equation Rosales and Valencia[29] solved this problem using the Fourier series methodBoyd [30] established the solution of the FSE using thenumerical method In the literature survey the FSE is solvedby analytical and numerical methods Both these methodshave some disadvantages as analytical techniques mostlyrequired the initial guess and small parameter assumption

and may affect the result Also the numerical methodsrequired linearization and discretization techniques whichcan affect the accuracy -erefore a new stochastic methodis proposed to solve the FSE-HT and the results arecompared with the literature validating the method

-e stochastic methods are relatively less used for thesolution of the fluid system -e use of stochastic methodscan be seen in thermodynamics astrophysics offline cir-cuits atomic physics MHD plasma physics fluid dynamicselectromagnetics nanotechnology bioinformatics electric-ity energy finance and random matrix theory [31ndash36] -estochastic method would be used for important and reputedfluidic models [37ndash44]

According to the literature survey no research has ap-plied artificial intelligent techniques through the NN-BLMMto solve the FSE-HT -e purpose of this study is to use theNN-BLMM for the solution of FSE-HT

-e critical aspect of the proposed computing paradigmis given as follows

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

01 02 03 04 05Target

DataFitY = T

DataFitY = T

01 02 03 04 05Target

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

54

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

81

e-05

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

054 05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

019

Training R=1 Validation R=1

Test R=099999 All R=099998

Figure 14 Regression illustration of NN-BLMM results in case 1 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 9

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

(i) A new application based on artificial intelligence-based computing using neural network-basedbackpropagation with LevenbergndashMarquardt isimplemented for studying and solving the FSE-HT

(ii) -e dataset of the NN-BLMM model is generatedand produced for variations of Deborah and Prandtlparameters through the OHAM

(iii) -e governing equations are transformed from a set ofPDEs into ODEs by using a similarity transformation

(iv) -e processing of the NN-BLMM means TR TSand VD of the FSE-HTmodel for different scenariosto obtain the approximate solution and comparisonwith reference results

(v) -e convergence analysis based on the MSE EHand RG plots is employed to ensure the perfor-mance of NN-BLMM for the detailed analysis of theboundary layer flow model

-e mathematical modeling of the FSE-HT is explained inSection 2 -e analysis of FSE-HT is also given in Section 3Section 4 presents the numerical results and graphical repre-sentations of the NN-BLMMmodel for the FSE-HTwith morediscussions and comparisons Finally Section 5 gives concludingremarks of the proposed study for the FSE-HT

2 Problem Formulation

We consider two-dimensional laminar viscous flows over asemi-infinite flat plate under the boundary layer approximationas given in Figure 1 -e governing equations are [28]

zu

zx+

zv

zy 0 (1)

uzu

zx+ v

zv

zy Us

dUs

dx+ v

z2u

z2y

(2)

20 1 3 4 5Target

DataFitY = T

DataFitY = T

1 2 3 4Target

21 3 4Target

DataFitY = T

DataFitY = T

10 2 3 4Target

5

5

4

3

2

1

0

Out

put ~

= 1lowast

Targ

et +

22

e-08 4

3

2

1Out

put ~

= 1lowast

Targ

et +

97

e-06

4

3

2

1Out

put ~

= 1lowast

Targ

et +

18

e-05 4

3

2

1Out

put ~

= 1lowast

Targ

et +

27

e-06

0

5

Training R=1 Validation R=1

Test R=1 All R=1

Figure 15 Regression illustration of NN-BLMM results in case 2 of scenario 1 for FSE-HT

10 Mathematical Problems in Engineering

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

uzT

zx+ v

zT

zy a

z2T

zy2 (3)

where u and v are velocity components in x- and y-di-rections Us(x) is the free stream velocity ldquoardquo is the thermaldiffusivity and υ is the kinematic viscosity

In the two-dimensional case the incompressible boundarylayer flowover awedge of angle βπ is shown in Figure 1-e freestream velocity of the form Ue(x) cxr can be computedusing the similarity transformation as follows

u(x y) Usfprime(ξ)

ξ y

(r + 1)C

2

1113970

x(mminus1)2

θ(ξ) T minus Tinfin

Tw minus Tinfin

(4)

Using equation (4) in equations (1)ndash(3) the Fal-knerndashSkan equation [28] can be calculated as

fprimeprimeprime(ξ) + f(ξ)fPrime(ξ) + β 1 minus fprime(ξ)( 11138572

1113872 1113873 0

θPrime(ξ) + Prf(ξ)θprime(ξ) 0(5)

with boundary conditions

f(0) 0

fprime(0) 0

fprime(infin) 1

θ(0) 1

θ(infin) 0

(6)

where Pr is the Prandtl number

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

01 02 03 0504Target

DataFitY = T

01 02 03 04 05Target

DataFitY = T

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

38

e-07

05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0065

05

04

03

02

01Out

put ~

= 0

97lowastTa

rget

+ 0

013

05

04

03

02

01Out

put ~

= 0

99lowastTa

rget

+ 0

003

7

Training R=1 Validation R=1

Test R=099888 All R=099979

Figure 16 Regression illustration of NN-BLMM results in case 3 of scenario 1 for FSE-HT

Mathematical Problems in Engineering 11

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

3 Solution Procedure

-e solution procedure is performed by implementing theproposed NN-BLMM model using nftool in the MATLABarea -e structure of the NN-BLMM is given in Figure 2and its data process is presented in Figure 3 Essentially theNN-BLMM is used for three conditions with different pa-rameters as set out in Table 1 -e design of the NN-BLMMis based on ten numbers of neurons as shown in Figure 4 Areference database of 201 input grids has been selectedwithin the interval [0 5] 80 of the data is used randomly intraining while 10 is used in the testing and validationprocess Training data are used for the construction of alimited solution built on the MSE results Verification dataare used for demonstrating the behavior of the NN whereasthe test data are used for showing the performance ofrandom inputs Details about the scenarios and cases aregiven in Table 1

-e NN-BLMM-based computing paradigm with onehidden layer structure of neural networks is exposed inFigure 2

-e input data are operated in the single-layer neuralnetwork model with the help of the NN-BLMM to obtain theoutput result as represented in Figure 3

4 Analysis of Results

-e workflow of the proposed NN-BLMM model for pro-cessing the FSE-HT is shown in Figure 4 Besides the resultsof the NN-BLMM for cases 1ndash3 of all scenarios in workingwith PF and state are displayed in Figures 5 and 6 separately-e results of FT are shown in Figures 7ndash12 -e EH resultsare given in Figure 13 whereas RA results are presented inFigures 14ndash19 on three FSE-HT cases Figure 5 shows theMSE for scenarios 1 and 2 at the best PF when the TR VDand TTdata are run Figure 6 shows the error gradient in the

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

010 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

33

e-05

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

2 04

03

02

01

Out

put ~

= 1lowast

Targ

et +

-00

0022

04

03

02

01Out

put ~

= 1lowast

Targ

et +

-00

024

0

Training R=1 Validation R=1

Test R=099778 All R=1

Figure 17 Regression illustration of NN-BLMM results in case 1 of scenario 2 for FSE-HT

12 Mathematical Problems in Engineering

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

direction of magnitude calculated during the TR of the NN-BLMM for all scenarios at each epoch and its results arecompare with numerical method results for valdiating theproposed method -e absolute errors are further explainedwith the help of Figures 7ndash13 -e accuracy of the NN-BLMM is verified by checking the results for the regressionshowing the accuracy of the NN-BLMM as given inFigures 14ndash19 In addition the convergence control pa-rameters in terms of executed epochs MSE PF time ofexecution and BP measures are presented in Table 2 InFigures 5(a)ndash5(f) MSE assembly of TR VD and TS pro-cedures is presented for cases 1 and 3 of all scenarios (FSE-HT) One can see that the best PF has been achieved at 2241000 366 72 143 and 248 epochs with an MSE of about24702times10minus8 13234times10minus10 12138times10minus8 27061times 10minus550276times10minus7 and 10658times10minus9 respectively Appropriatevalues for GD and step Mu size of BP are (99228 times10-0810553 times10ndash07 99259 times10ndash08 96455times10-8 84492times10-

7 and 27359times10-8) and (10-08 10-8 10-08 10-08 10-09)as shown in Figures 6(a)ndash6(f) -e results determine thecorrect and convergent PF of the NN-BLMM for each case-e maximum error achieved in the TS PF and VL by theproposed NN-BLMM is less than 01times 10-03 09times10-0501times 10-01 05times10-03 3times10-03 and 03times10-03 as given inFigures 7ndash12 Error variability is also assessed with EH forevery input point and the results are given in Figures 13(a)ndash13(f) -e maximum error achieved is less than minus35Eminus 6237Eminus 7 364Eminus 4 minus11Eminus 4 minus12Eminus 4 and 256Eminus 6 in allcases (FSE-HT) -e investigation over RG is performedusing the correlation studies Moreover the RG results aregiven in Figures 14ndash19 and the correlation R values in-variably revolve around unity of the NN-BLMM perfor-mance results for FSE-HT Adopted cases are given inTable 2 -e NN-BLMM performance is approximately10minus8 to 10minus10 and 10minus7 to 10minus10 in cases 1 and 3 of allscenarios with 1ndash3 cases of FSE-HT -ese results show

010 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

04

03

02

01

0

Out

put ~

= 1lowast

Targ

et +

31

e-07

04

03

02

01Out

put ~

= 0

98lowastTa

rget

+ 0

01 04

03

02

01Out

put ~

= 1lowast

Targ

et +

-8e-

07

04

03

02

01Out

put ~

= 1lowast

Targ

et +

00

0048

Training R=1 Validation R=1

Test R=099999 All R=1

g

Figure 18 Regression illustration of NN-BLMM results in case 2 of scenario 2 of FSE-HT

Mathematical Problems in Engineering 13

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

the stability of PF for the NN-BLMM in each case of FSE-HT equations -e velocity distribution along with thecomparison with numerical results is plotted in Figures 20(a)and 21(a) -e results of the NN-BLMM have been

compared with RK-4 numerical solutions in each casetherefore to achieve precision gauges AE is determinedfrom the reference solutions and the results are shown inFigures 20(b) and 21(b) for case studies 1ndash3 respectively

01 02 03 04Target

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

36

e-07

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

028

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

0025

04

035

03

025

02

015

01

005

Out

put ~

= 1lowast

Targ

et +

-00

001

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

01 02 03 04Target

DataFitY = T

Training R=1 Validation R=1

Test R=1 All R=1

Figure 19 Regression illustration of NN-BLMM results in case 3 of scenario 2 for FSE-HT

Table 2 -e NN-BLMM results of scenarios 1ndash3 for FSE-HT

Scenario 1 casesMean square error (MSE)

Performance Gradient Mu Epoch TimeTraining Validation Testing

1 777491Eminus 10 678795Eminus 05 151380Eminus 05 770Eminus 10 761Eminus 06 100Eminus 08 504 lt12 674192Eminus 09 210370Eminus 05 152281Eminus 07 674Eminus 09 126Eminus 05 100Eminus 07 1000 lt13 521240Eminus 08 320661Eminus 06 171187Eminus 05 485Eminus 08 184Eminus 04 100Eminus 07 172 lt1

Scenario 2 cases Mean square error Performance Gradient Mu Epoch TimeTraining Validation Testing1 123663Eminus 07 374133Eminus 07 601470Eminus 05 996Eminus 08 255Eminus 05 100Eminus 07 28 lt12 328408Eminus 10 351972Eminus 08 146030Eminus 09 328Eminus 10 995Eminus 08 100Eminus 09 934 lt13 262694Eminus 10 425158Eminus 08 336674Eminus 07 263Eminus 10 588Eminus 07 100Eminus 09 1000 lt1

14 Mathematical Problems in Engineering

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

One can note that AE is almost 10ndash02 to 10ndash04 and 10ndash03 to10ndash06 in all scenarios respectively All of these numericaland graphical diagrams ensure the precise flexible androbust functionality of the NN-BLMM for FSE-HT -enondimensional velocity profiles and temperature profilesare given in Figures 20(a) and 21(a) respectively -evariation of physical parameters β and Pr is given inFigures 20(a) and 21(a) whereas the absolute error for eachvariation of β and Pr is given in Figures 20(b) and 21(b) -e

solution obtained by the NN-BLMM is verified by thenumerical method It has been observed that an increase inthe Deborah number makes an increase in the velocityprofile and a decrease in the thermal boundary layerthickness Similarly by increasing the Prandtl number thethermal boundary layer thickness is decreased -e resultsobtained by the NN-BLMM are compared with the resultsavailable in the literature which proves the accuracy andvalidation of the method as given in Table 3

0 1 2 3 4 5

14

12

10

8

6

4

2

0

ndash2

β = 2β = 4

β = 6Numerical

(a)

10ndash1

10ndash2

10ndash3

10ndash4

10ndash5

10ndash6

10ndash70 1 2 3 4 5

β = 2β = 4β = 6

(b)

Figure 20 -e results of the proposed NN-BLMM compared with reference results regarding scenario 1 of FSE-HT (a) Variation of β(b) Analysis of AE

1

05

0

ndash05

ndash1

ndash15

ndash20 1 2 3 4 5

Pr = 05Pr = 10

Pr = 15Numerical

(a)

10ndash7

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

0 1 2 3 4 5

Pr = 05Pr = 10Pr = 15

(b)

Figure 21 Comparison between the results of the NN-BLMM and reference results for scenario 2 of FSE-HT (a) Pr variation (b) Analysis of AE

Mathematical Problems in Engineering 15

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

5 Conclusions

In this paper we conclude that the NN-BLMM containsless computational work and does not require lineari-zation NN-BLMM is simply applicable -e NN-BLMMhas better PF as compared to other numerical methodsand it minimizes the absolute error Moreover the NN-BLMM is independent of the assumption of a smallparameter and it does not require the initial guess -eNN-BLMM converges faster when compared to othermethods -e correctness of the NN-BLMM is authen-ticated by MSE EH RG AE FT PF TS and TR -eNN-BLMM uses 80 10 and 10 of the reference dataas TR TS and VL By increasing the Deborah numberthe velocity profile increases whereas the thickness of thethermal boundary layer is decreased by increasing theDeborah number -e thickness of the thermal boundarylayer is decreased by increasing the Prandtl number

Abbreviations

β Deborah numberANNs Artificial neural networksυ Kinematic viscosityNN Neural networkBL Boundary layerk -ermal conductivityPr Prandtl numberBVP Boundary value problemEh EpochMSE Mean square errorρ Fluid densityμ Dynamic viscosityPF PerformanceTT TestingTR TrainingVD ValidationEB Error binEH Error histogramGD GradientRG RegressionAE Absolute errorEp EpochMSE Mean square error

Data Availability

All the relevant data are included within the article

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this research article

Acknowledgments

-is research was supported by the Researchers SupportingProject number (RSP-2021244) King Saud UniversityRiyadh Saudi Arabia

References

[1] T Alten S Oh and H Gegrl Metal Forming Fundamentalsand Applications American Society of Metals GeaugaCounty OH USA 1979

[2] E G Fisher Extrusion of Plastics Wiley New York NY USA1976

[3] Z Tadmor and I Klein Engineering Principles of PlasticatingExtrusion Polymer Science and Engineering Series VanNostrand Reinhold New York NY USA 1970

[4] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensionaland axisymmetric flowrdquo AIChE Journal vol 7 no 1pp 26ndash28 1961

[5] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces II -e boundary layer on a continuous flat surfacerdquoAIChE Journal vol 7 no 2 pp 221ndash225 1961

[6] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift furangewandte Mathematik und Physik ZAMP vol 21 no 4pp 645ndash647 1970

[7] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction or blowingrdquo Canadian Journalof Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[8] H I Anderson ldquoMHD flow of a viscoelastic fluid past astretching surfacerdquoActa Mechanica vol 95 pp 227ndash230 1992

[9] P D Ariel ldquoMHD flow of a viscoelastic fluid past a stretchingsheet with suctionrdquo Acta Mechanica vol 105 no 1-4pp 49ndash56 1994

[10] C Y Wang ldquo-e three-dimensional flow due to a stretchingflat surfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash19171984

[11] J F Brady and A Acrivos ldquoSteady flow in a channel or tubewith an accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of FluidMechanics vol 112 no 1 pp 127ndash150 1981

[12] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physicsof Fluids vol 31 no 3 pp 466ndash468 1988

[13] C Y Wang ldquoLiquid film on an unsteady stretching surfacerdquoQuarterly of Applied Mathematics vol 48 no 4 pp 601ndash6101990

Table 3 -e results of the initial slope fPrime(0) achieved at different values of β

β Hartree [17] Asaithambi [22] Asaithambi [23] Salama [24] Zhang andChen [25]

Rosales-Vera andValencia [26]

OHAM[28]

Stochasticmethod

2 1687 1687222 1687218 1687218 1687218 1687218 1687218 16872181 1233 123589 1232588 1232588 1232587 1232587 1232587 123258705 0927 0927682 0927680 0927680 0927680 0927680 0927680 0927680minus01 0319 0319270 0319270 0319270 0319270 0319270 0319270 0319270

16 Mathematical Problems in Engineering

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17

[14] R Usha and R Sridharan ldquo-e axisymmetric motion of aliquid film on an unsteady stretching surfacerdquo Journal ofFluids Engineering vol 117 no 1 pp 81ndash85 1995

[15] P D Ariel ldquoComputation of MHD flow due to movingboundaryrdquo Technical report MCS-2004-01 Department ofMathematical Sciences Trinity Western University LangleyCanada 2004

[16] P D Ariel T Hayat and S Asghar ldquoHomotopy perturbationmethod and axisymmetric flow over a stretching sheetrdquo In-ternational Journal of Nonlinear Sciences and NumericalStimulation vol 7 no 4 pp 399ndash406 2006

[17] D R Hartree ldquoOn an equation occurring in Falkner-Skanrsquosapproximate treatment of the equations boundary layerrdquoProceedings of the Cambridge Philosophical Society vol 33no 2 pp 233ndash239 1921

[18] A M O Smith ldquoImproved solution of the Falkner-Skanequation boundary layer equationrdquo Journal of the Aeronau-tical Sciences vol 10 1954

[19] T Cebeci and H B Keller ldquoShooting and parallel shootingmethods for solving the Falkner-Skan boundary-layer equa-tionrdquo Journal of Computational Physics vol 7 no 2pp 289ndash300 1971

[20] D Meksyn NewMethods in Laminar Boundary Layer DeoryPergamon Press New York NY USA 1961

[21] A Asaithambi ldquoNumerical solution of the Falkner-Skanequation using piecewise linear functionsrdquo Applied Mathe-matics and Computation vol 159 no 1 pp 267ndash273 2004

[22] A Asaithambi ldquoA finite-difference method for the Falkner-Skan equationrdquo Applied Mathematics and Computationvol 92 no 2-3 pp 135ndash141 1998

[23] N S Asaithambi ldquoA numerical method for the solution of theFalkner-Skan Equationrdquo Applied Mathematics and Compu-tation vol 81 no 2-3 pp 259ndash264 1997

[24] A A Salama ldquoHigher-order method for solving freeboundary-value problemsrdquo Numerical Heat Transfer Part BFundamentals vol 45 no 4 pp 385ndash394 2004

[25] J Zhang and B Chen ldquoAn iterative method for solving theFalkner-Skan equationrdquo Applied Mathematics and Compu-tation vol 210 no 1 pp 215ndash222 2009

[26] M Rosales-Vera and A Valencia ldquoSolutions of Falkner-Skanequation with heat transfer by Fourier seriesrdquo InternationalCommunications in Heat and Mass Transfer vol 37 no 7pp 761ndash765 2010

[27] S-J Liao ldquoA uniformly valid analytic solution of two-di-mensional viscous flow over a semi-infinite flat platerdquo Journalof Fluid Mechanics vol 385 pp 101ndash128 1999

[28] H Ullah S Islam M Idrees and M Arif ldquoSolution ofboundary layer flow with heat transfer by OHAMrdquo Abstractand Applied Analysis vol 2013 Article ID 324869 10 pages2013

[29] M Rosales and A Valencia ldquoA note on solution of Blasiusequation by Fourier seriesrdquo Advances and Applications inFluid Mechanics vol 6 no 1 pp 33ndash38 2009

[30] J P Boyd ldquo-e Blasius function in the complex planerdquoExperimental Mathematics vol 8 no 4 pp 381ndash394 1999

[31] Z Sabir M A Z Raja M Umar and M Shoaib ldquoDesign ofneuro-swarming-based heuristics to solve the third-ordernonlinear multi-singular EmdenndashFowler equationrdquo De Eu-ropean Physical Journal Plus vol 135 no 6 pp 1ndash17 2020

[32] Z Sabir H A Wahab M Umar M G Sakar andM A Z Raja ldquoNovel design of Morlet wavelet neural networkfor solving second order Lane-Emden equationrdquo Mathe-matics and Computers in Simulation vol 172 pp 1ndash14 2020

[33] A Mehmood A Zameer M S Aslam and M A RajaldquoDesign of nature-inspired heuristic paradigm for systems innonlinear electrical circuitsrdquo Neural Computing and Appli-cations vol 32 no 11 pp 1ndash17 2020

[34] S I Ahmad F Faisal M Shoaib and M A Z Raja ldquoA newheuristic computational solver for nonlinear singular -o-masndashFermi system using evolutionary optimized cubicsplinesrdquo De European Physical Journal Plus vol 135 no 1p 55 2020

[35] S Lodhi M A Manzar and M A Z Raja ldquoFractional neuralnetwork models for nonlinear Riccati systemsrdquo NeuralComputing and Applications vol 31 no 1 pp 359ndash378 2019

[36] M A Z Raja M A Manzar S M Shah and Y ChenldquoIntegrated intelligence of fractional neural networks andsequential quadratic programming for BagleyndashTorvik systemsarising in fluid mechanicsrdquo Journal of Computational andNonlinear Dynamics vol 15 no 5 2020

[37] Kh Hosseinzadeh M R Mardani S Salehi M PaikarM Waqas and D D Ganji ldquoEntropy generation of three-dimensional Bodewadt flow of water and hexanol base fluidsuspended by Fe3O4 and MoS2 hybrid nanoparticlesrdquo Pra-mana vol 95 no 2

[38] K Hosseinzadeh S Roghani A R Mogharrebi A AsadiaM Waqasc and D D Ganji ldquoInvestigation of cross-fluid flowcontaining motile gyrotactic microorganisms and nano-particles over a three-dimensional cylinderrdquo AlexandriaEngineering Journal vol 59 no 5 pp 3297ndash3307 2020

[39] K Hosseinzadeh S Salehi M R Mardani F Y MahmoudiM Waqas and D D Ganji ldquoInvestigation of nano-Bio-convective fluid motile microorganism and nanoparticle flowby considering MHD and thermal radiationrdquo Informatics inMedicine Unlocked vol 21 Article ID 1000462 2020

[40] Y-Q Song H Waqas K Al-Khaled et al ldquoBioconvectionanalysis for Sutterby nanofluid over an axially stretchedcylinder with melting heat transfer and variable thermalfeatures a Marangoni and solutal modelrdquo Alexandria Engi-neering Journal vol 60 no 5 pp 4663ndash4675 2021

[41] K Hosseinzadeh S Roghani A Mogharrebi and D D GanjildquoOptimization of hybrid nanoparticles with mixture fluid flowin an octagonal porous medium by effect of radiation andmagnetic fieldrdquo Journal of Dermal Analysis and Calorimetryvol 143 no 103 pp 10ndash1007 2021

[42] K Hosseinzadeh A Asadi A Mogharrebi and D D GanjildquoInvestigation of mixture fluid suspended by hybrid nano-particles over vertical cylinder by considering shape factoreffectrdquo Journal of Dermal Analysis and Calorimetry vol 143no 19ndash20 2021

[43] M Gholinia K Hosseinzadeh and D D Ganji ldquoInvestigationof different base fluids suspend by CNTs hybrid nanoparticleover a vertical circular cylinder with sinusoidal radiusrdquo CaseStudies in Dermal Engineering vol 21 Article ID 1006662020

[44] K Hosseinzadeh E Montazer M B Shafii and D D GanjildquoHeat transfer hybrid nanofluid (1-butanolMoS2-Fe3O4)through a wavy porous cavity and its optimizationrdquo Inter-national Journal of Numerical Methods for Heat amp Fluid Flowvol 31 no 5 pp 1547ndash1567 2021

Mathematical Problems in Engineering 17