application of three unsupervised neural network models to singular nonlinear bvp of transformed 2d...

ORIGINAL ARTICLE

Application of three unsupervised neural network modelsto singular nonlinear BVP of transformed 2D Bratu equation

Muhammad Asif Zahoor Raja • Raza Samar •

Mohammad Mehdi Rashidi

Received: 29 June 2013 / Accepted: 24 May 2014

� Springer-Verlag London 2014

Abstract In this paper, numerical techniques are developed

for solving two-dimensional Bratu equations using different

neural network models optimized with the sequential quadratic

programming technique. The original two-dimensional problem

is transformed into an equivalent singular, nonlinear boundary

value problem of ordinary differential equations. Three neural

network models are developed for the transformed problem

based on unsupervised error using log-sigmoid, radial basis and

tan-sigmoid functions. Optimal weights for each model are

trained with the help of the sequential quadratic programming

algorithm. Three test cases of the equation are solved using the

proposed schemes. Statistical analysis based on a large number

of independent runs is carried out to validate the models in terms

of accuracy, convergence and computational complexity.

Keywords Two-dimensional Bratu equations � Neural

networks � Sequential quadratic programming � Boundary

value problems � Nonlinear singular system

1 Introduction

Artificial neural networks (ANNs) are well known for their

inbuilt strength of universal function approximation capa-

bilities. Therefore, ANNs have been applied broadly to find

the solution of dynamical systems based on ordinary and

partial differential equations by many researchers [1–4].

The recent applications in which neural networks models

are incorporated are nonlinear Van der Pol oscillators for

stiff and non-stiff cases [5, 6], optimal control problems

[7], fluid mechanic problems based on Jeffery-Hamel flow

equations for both convergent and divergent channels in

the presence of high magnetic field [8, 9], the nonlinear

Schrodinger equations [10], first Painleve transcendent

possess strong nonlinearity [11, 12], Troesch’s boundary

values problems (BVPs) arising in the study of plasma

physics problems [13, 14], one-dimensional Bratu’s equa-

tions arising in fuel ignition model of combustion theory

[15, 16] and nonlinear singular systems based on Emden–

Fowler equations [17]. The real strength of the neural

networks can be seen by formulating the mathematical

model of complex linear and nonlinear fractional order

systems represented with differential equations [18, 19].

For instance, mathematical models of Riccati fractional

differential equations and Bagley-Torvik equation involv-

ing fractional derivative term are constructed with the help

of fractional ANNs architecture [20, 21]. Recently, the

computational intelligence approach based on global

search hybrid with efficient local search algorithms are

applied for solving transformed form of 2-dimensional

(2D) Bratu’s problems and its variants [22, 23]. This study

presents the extension of previous works regarding explo-

ration and exploitation of various artificial neural network

models in order to develop an accurate, reliable, effective

and efficient platform for solving the different cases of

M. A. Z. Raja (&)

Department of Electrical Engineering, COMSATS Institute of

Information Technology, Attock, Pakistan

e-mail: [email protected];

[email protected]

R. Samar

Mohammad Ali Jinnah University, Islamabad, Pakistan

e-mail: [email protected]

M. M. Rashidi

Mechanical Engineering Department, University of Michigan-

Shanghai Jiao Tong University Joint Institute, Shanghai Jiao

Tong University, Shanghai, Peoples Republic of China

e-mail: [email protected]

M. M. Rashidi

Mechanical Engineering Department, Engineering Faculty of

Bu-Ali Sina University, Hamedan, Iran

123

Neural Comput & Applic

DOI 10.1007/s00521-014-1641-x

transformed 2D Bratu’s equation. The training of design

parameters of each neural network is carried out using

efficient sequential quadratic programming techniques.

Sequential quadratic programming (SQP) method is a

very powerful technique for solving constrained nonlinear

optimization problems; details can be seen in [24–26]. The

efficiency, accuracy and successful application of the SQP

technique over a wide range of benchmark problems prove

its effectiveness and applicability. Nocedal and Wright [26]

have given an extensive introduction and overview of

mathematical programming methods for large-scale opti-

mization problems. A number of recent applications of the

SQP method include optimal power flow problems [27],

filter design for harmonic suppression [28], economic dis-

patch problems [29], cutting-stock problem with rotatable

polygons [30], optimal topology design problems of

mechanical structures [31], and non-convex, non-smooth

constrained optimization [32], etc.

In the present study, three mathematical models are

developed for singular nonlinear boundary value problem

(BVP) of transform 2D Bratu equation using log-sigmoid,

radial basis and tan-sigmoid transfer functions in neural

networks methodology, and training of weights of the

networks is performed with SQP techniques. Three test

cases of the BVP are evaluated with these models to verify

and validate the applicability. The statistical analysis based

on large numbers of simulation for all three designed

models is performed to determine the accuracy, conver-

gence and effectiveness. Comparative studies of proposed

schemes are also provided with the exact solutions.

Organization of the paper is as fallows; in the next

section, the governing equation of system model based on

transformed 2D Bratu problem is introduced. In Sect. 3,

design methodologies in terms of development of unsu-

pervised neural network models of the problem and pro-

cedure adapted for training of the networks are presented.

In Sect. 4, simulations and results of the proposed models

on three variants of the problem are given. In Sect. 5, a

thorough comparative analysis based on results of statisti-

cal analyses is provided. Finally, research findings along

with recommendations for future work are presented in

conclusion section.

2 System model

The governing equation of boundary value problem of

nonlinear singular system is presented here from Bratu

problem of higher dimension along with its brief history,

importance and applications.

The general form of the n-dimensional nonlinear Bratu

equation is given as [22, 23]:

Duþ keu ¼ 0; in X

u ¼ 0; on X

(ð1Þ

Here, D represents the Laplacian operator, u is the solution

of the equation, and k is a real number. Usually, the domain

X is taken to be the unit interval [0, 1] in <, or the unit

square [0, 1] � [0, 1] in <2, or the unit cube [0, 1] � [0, 1] �[0, 1] in <3.

In case X = B is the unit ball in <n, n [ 1, the

n-dimensional Bratu equation (1) is transformed into an

equivalent singular, nonlinear boundary value problem

(BVP) of ordinary differential equations given as [22, 23,

33, 34]:

u00ðrÞ þ n� 1

ru0ðrÞ þ keuðrÞ ¼ 0; 0� r� 1

uð1Þ ¼ u0ð0Þ ¼ 0

8<: ð2Þ

A detailed discussion of this transformation is given in

[35]. In this study, intension is to explore the problem for

the case n = 2, i.e., the 2-dimensional Bratu problem,

defined by the equivalent BVP as:

u00ðrÞ þ 1

ru0ðrÞ þ keuðrÞ ¼ 0; 0� r� 1

uð1Þ ¼ u0ð0Þ ¼ 0

8<: ð3Þ

The Bratu Problem given in (3) has zero, one and two

solutions for k[ kc, k = kc and k\ kc, respectively,

where kc is the critical value of k. It was reported in [34]

that the critical value of k for this equation is given as

kc = 2.

The origin of the Bratu Problem (1–3) lies in the work

on solid fuel ignition models in the field of thermal com-

bustion theory. Bratu equations are expressed in the form

of nonlinear eigenvalue problems of partial differential

equations. The Bratu problem [36] has a long history; a few

well-known generalizations are the ‘‘Liouville-Gelfand’’ or

‘‘Liouville-Gelgand-Bratu’’ problem in appreciation of the

great scientists Liouville and Gelfand [37]. The introduc-

tion, history and significance of the Bratu problem are

nicely summarized in [38]. The problem comes up in many

practical applications of applied science and engineering,

including chemical reactor theory, nanotechnology, radia-

tive heat transfer and the Chandrasekhar model for the

expansion of the universe [37–41].

3 Design methodology

In this section, a detailed description of the proposed

methodology is presented to solve the BVP of Bratu-type

equations. Neural networks, mathematical models, fitness


123

function formulation and procedure for training the weights

are given here.

3.1 Neural Network Mathematical Modeling

Neural network models of the Bratu equation are devel-

oped using the following continuous mapping for the

solution u(r) along with the nth-order derivative dnu/drn,

respectively, as follows [41, 42]:

uðrÞ ¼Xm

i¼1

aif ðwir þ biÞ

dnuðrÞdrn

¼Xm

i¼1

ai

dn

drnf ðwir þ biÞ;

8>>>><>>>>:

ð4Þ

where m is the number of neurons and f is the activation

function; a, w, and b are real-valued bounded adaptive

parameters, i.e., weights (W) given as:

W ¼ ða1; a2; . . .; am;w1;w2; . . .;wm; b1; b2; . . .; bmÞ

Three different mathematical models are constructed using

log-sigmoid fLS, radial basis fRB and tan-sigmoid fTS

transfer functions for the hidden layers of the network.

These transfer functions are given as follows:

fLSðxÞ ¼1

1þ e�x

fRBðxÞ ¼ e�x2

fTSðxÞ ¼2

1þ e�2x� 1

8>>>>><>>>>>:

ð5Þ

Differential equation neural networks (DENN) using fLS

(DENN-LS), fRB (DENN-RB) and fTS (DENN-TS) func-

tions have been developed to approximate solutions uLS,

uRB and uLS, along with their first- and second-order

derivatives using equations (3) and (4) as [22, 23]:

uLSðrÞ ¼Xm

i¼1

ai

1

1þ e�ðwirþbiÞ

� �

u0LSðrÞ ¼Xm

i¼1

aiwi

e�ðwirþbiÞ

ð1þ e�ðwirþbiÞÞ2

!

u00LSðrÞ ¼Xm

i¼1

aiw2i

2e�2ðwirþbiÞ

ð1þ e�ðwirþbiÞÞ3� e�ðwirþbiÞ

ð1þ e�ðwirþbiÞÞ2

!

8>>>>>>>>>><>>>>>>>>>>:

ð6Þ

uRBðrÞ ¼Xm

i¼1

ai e�ðwirþbiÞ2� �

u0RBðrÞ ¼Xm

i¼1

aiwi �2ðwir þ biÞe�ðwirþbiÞ2� �

u00RBðrÞ ¼Xm

i¼1

aiw2i �2e�ðwirþbiÞ2 þ 4ðwir þ biÞ2e�ðwirþbiÞ2� �

8>>>>>>>>><>>>>>>>>>:

ð7Þ

uTSðrÞ ¼Xm

i¼1

ai

2

1þ e�2ðwirþbiÞ� 1

� �

u0TSðrÞ ¼Xm

i¼1

aiwi

�4e�2ðwirþbiÞ

ð1þ e�2ðwirþbiÞÞ2

!

u00TSðrÞ ¼Xm

i¼1

2aiw2i

8e�4ðwirþbiÞ

ð1þ e�2ðwixþbiÞÞ3� 4e�2ðwirþbiÞ

ð1þ e�2ðwirþbiÞÞ2

!

8>>>>>>>>>><>>>>>>>>>>:

ð8Þ

The neural networks given by the set of Eqs. (6), (7) and

(8) can arbitrarily construct models for the nonlinear Bratu

Equation; the general form of the network architecture is

shown in Fig. 1.

3.2 Fitness functions

The fitness function for the Bratu BVPs is formulated by

defining an unsupervised error function e as the sum of the

mean squared errors [22, 23]:

e ¼ e1 þ e2 ð9Þ

The error function e1 is formulated as:

e1 ¼1

N þ 1

XN

m¼0

u00m þ1

rm

u0m þ keum

� �2

; r 2 ð0; 1Þ

N ¼ 1=h; ym ¼ yðrmÞ; rm ¼ mh

8><>:

ð10Þ

The interval r 2 ð0; 1Þ is divided into N? steps r 2ðr0 ¼ 0; r1; r2; . . .; rN ¼ 1Þ with a step size of h; uðrÞ,u0ðrÞ and u00ðrÞ are the neural networks as represented by

the set of equations (6), (7) and (8).

The error e2 is associated with the boundary conditions

and is given as:

e2 ¼1

2u1ð Þ2þ u00

� �2� �

ð11Þ

It is evident that with optimal weights W for any of

the three neural network models such that the func-

tions e1 and e2 approach zero, the value of the fitness

function e will also approach zero. The proposed

neural networks will in this case approximate the

solution y(r) of the BVP with uðrÞ as given in (6), (7)

and (8).

3.3 Learning procedure

In this section, the methodology for training the weights of

the network using sequential quadratic programming (SQP)

algorithm is presented.

In the standard SQP technique, one has to find a vector

W, i.e., weights, for optimization of a problem-specific

fitness function e(W) as:


123

minW

e Wð Þ ð12Þ

subject to K equality and inequality constraints gi(W) for

i = 1,2,…,K. The problem is solved by transforming into a

quadratic programming (QP) subproblem with the help of

Lagrangian function as:

L W ; kð Þ ¼ e Wð Þ þXMi¼1

ligi Wð Þ ð13Þ

where l is the Lagrangian multiplier. The basic SQP

algorithm iteratively updates weights Wk?1 from the pre-

vious weights Wk as:

Wkþ1 ¼ Wk þ akdk ð14Þ

where ak is the step length parameter and dk is the feasible

search direction to minimize the problem-specific fitness

function. The optimization subproblem is given as:

mind2<n

1

2dT Hkd þre Wð ÞTd

rgi Wkð ÞT d þ gi Wkð Þ ¼ 0 i ¼ 1; 2;

rgi Wkð ÞT d þ gi Wkð Þ� 0 i ¼ ke þ 1; ke þ 2;

8>>><>>>:

ð15Þ

Here, Hk is the positive definite approximation of the

Hessian matrix of the Lagrangian function L, r represents

the gradient operator and <n is the dimension of the search

space. The implementation of the SQP technique involves

the following three major steps: (1) updating the Hessian

matrix of the Lagrangian function using quasi-Newton

methods, (2) formulation of the QP problem and (3) line

search and merit function calculation. Further details for

the interested reader about the SQP technique can be found

in [24–26].

In the present study, we train the weights of the neural

network models for the Bratu equation through the SQP

method. The Matlab� function ‘‘fmincon’’ with parameter

settings given in Table 1 is employed for the optimization.

The generalized flowchart of the proposed scheme is given

in

Fig. 2: the procedural steps are described below:

Step 1: Tool initialization: Set the Matlab ‘‘optimset’’

parameters as given in Table 1.

Step 2: Program initialization: Randomly generate

bounded real values to construct a candidate solution

with as many members as the number of unknown

weights. The vector of weights for the ith iteration cycle

is written as

Wi ¼ a1; a2. . .am;w1;w2. . .wm; b1; b2. . .bmð Þjiwhere m is the number of neurons. The same initial

weight vector is used for all three neural network models

of the equation.

Step 3: Fitness evaluation: Invoke the Matlab� function

‘‘fmincon’’ and provide the following:

• Current W and ‘‘optimset’’ parameters,

• Fitness function e as given in equations (9), (10) and

(11)

• Termination criterion and other settings as given in

Table 1.

Fig. 1 Neural network

architecture for transformed

BVP of the Bratu equation


123

Step 4: Storage: Store the initial weight vector, the final

optimized weight vector, the fitness value e and the

execution time for this run of the algorithm.

Step 5: Statistical analysis: Repeat steps 2 to 4 with a

sufficiently large number of independent runs by varying

initial weights for a reliable statistical analysis of the

results.

3.4 Simulation and results

The proposed neural network models are applied to three

different cases of BVPs by taking different values of k: 0.5,

1 and 2. Results of numerical simulation are presented here

along with comparison with exact solutions for each case.

3.5 Bratu Problem for k = 0.5

The BVP of the Bratu equation for this case is given as:

u00ðrÞ þ 1

ru0ðrÞ þ 0:5euðrÞ ¼ 0; 0� r� 1

uð1Þ ¼ u0ð0Þ ¼ 0

8<: ð16Þ

The exact solution to the problem is given as:

uðrÞ ¼ log16 7þ 4

ffiffiffi3p� �

7þ 4ffiffiffi3pþ r2

� �2

!ð17Þ

The three proposed neural network models are applied

to this equation with 10 neurons each. This results in 30

unknown weight parameters contained in the vector W. The

training set is taken from inputs r e (0, 1) with a step size of

0.1. Fitness function e as given in Eq. (9) is written as:

e ¼ 1

11

X10

m¼0

u00m þ1

rm

u0m þ 0:5eum

� �2

þ 1

2u1ð Þ2þ u00

� �2� �

ð18Þ

Our desire is to find optimal weights for each

model, which optimize the fitness function given

above. The training of weights of the networks is

carried out using the SQP algorithm with parameter

settings as given in Table 1, and results are shown

graphically in Fig. 3. A particular set of weights

learned by the algorithm for the three networks

(DENN-LS, DENN-RB and DENN-TS) results in

approximate solutions uLS, uRB and uTS yielding

fitness values of 1.8789 9 10-12, 1,5242 9 10-13,

and 7.2558 9 10-14, respectively. These are given

as:

Start

Initialize ProgramParameters.

Fitness Evaluation

Termination Criterionfulfilled?

Store Final Weights,Fitness & Execution

Time

Yes

Step Increments inWeights usig SQP

No

Stop

fmincon

optim

set

Initial WeightsBounds,Declaration

Fig. 2 Generic flow diagram for optimization using the SQP method

Table 1 Parameter settings for function ‘‘fmincon’’

Parameters Settings/

values

Parameters Settings/values

‘Algorithm’ ‘SQP’ ‘TolCon’ 10-24

Hessian BFGS ‘MaxIter’ 800

‘TolX’ 10-15 ‘TolConSQP’ 10-10

‘FinDiffType’ ‘Central’ w, a, b initial value Random e[? 1.5,

-1.5]

‘MaxFunEvals’ 400,000 w, a, b bounds ±10

‘TolFun’ 10-24 Rest of the settings As default


123

(a) DENN-LS-SQP (b) DENN-RB-SQP (c) DENN-TS-SQP

Fig. 3 Graphical view of learning of weights by proposed neural network models for Bratu problem in case of k = 0.5

Table 2 Comparison of the results for l = 0.5

r u(r) u(r) |u(r) 2 u(r)|

Exact DENN-LS DENN-RB DENN-TS DENN-LS DENN-RB DENN-TS

0.0 0.1386729284 0.1386735014 0.1386729328 0.1386729406 5.73E-07 4.43E-09 1.22E-08

0.1 0.1372375082 0.1372379757 0.1372375065 0.1372375151 4.67E-07 1.68E-09 6.85E-09

0.2 0.1329374187 0.1329377111 0.1329374065 0.1329374133 2.92E-07 1.22E-08 5.44E-09

0.3 0.1257910845 0.1257912653 0.1257910725 0.1257910760 1.81E-07 1.20E-08 8.48E-09

0.4 0.1158289224 0.1158290662 0.1158289149 0.1158289180 1.44E-07 7.47E-09 4.34E-09

0.5 0.1030929130 0.1030930511 0.1030929033 0.1030929090 1.38E-07 9.78E-09 4.01E-09

0.6 0.0876360203 0.0876361406 0.0876360037 0.0876360111 1.20E-07 1.66E-08 9.28E-09

0.7 0.0695214731 0.0695215536 0.0695214559 0.0695214611 8.05E-08 1.72E-08 1.20E-08

0.8 0.0488219305 0.0488219729 0.0488219184 0.0488219211 4.24E-08 1.22E-08 9.40E-09

0.9 0.0256185525 0.0256185813 0.0256185372 0.0256185415 2.88E-08 1.54E-08 1.10E-08

1.0 0.0000000000 -0.0000000074 -0.0000000002 0.0000000007 7.37E-09 1.60E-10 7.46E-10

uLS rð Þ ¼

�0:8816

1þ e� 0:8800r�0:8988ð Þ þ�0:3681

1þ e� �0:1914r�1:22208ð Þ þ�3:7801

1þ e� �0:5820r�1:3073ð Þ þ1:3923

1þ e� 0:0063rþ0:9265ð Þ þ0:7143

1þ e� �0:4158r�0:9652ð Þ

þ �2:2292

1þ e� 0:7107r�2:7649ð Þ þ�0:1155

1þ e� �0:6407r�1:2859ð Þ þ0:5791

1þ e0:3690rþ1:4739Þ þ0:0018

1þ e� 0:17701rþ0:8431ð Þ þ�1:0864

1þ e� 0:6719r�1:3288ð Þ

0BB@

1CCAð19Þ

uRB rð Þ ¼� 0:02285e� �1:0604r�1:9673ð Þ2 � 0:1599e� �0:6274r�0:7541ð Þ2 þ 0:2012e� �0:5568r�0:6690ð Þ2 � 0:0372e� þ0:5735rþ1:13032ð Þ2

� 0:5914e� þ0:2933r�1:6718ð Þ2 � 0:9632e� �0:3401r�1:4795ð Þ2 � 0:3773e� �0:3758rþ1:3868ð Þ2 þ 1:3571e� þ0:2140r�0:1290ð Þ2

� 1:2920e� �0:02880rþ0:2517ð Þ2 þ 1:0847e� þ0:1546rþ1:3227ð Þ2

0BB@

1CCAð20Þ


123

The Eqs. (19)–(21) are given in appendix Table 8 with

14 decimal point of accuracy in order to reproduced results

without round off errors. The solutions uLSðrÞ, uRBðrÞ and

uTSðrÞto the problem determined using Eqs. (19), (20) and

(21) for points r e (0, 1) with a step size of 0.1 are given in

Table 2 along with the exact solution (17), while the

comparison of the results is shown graphically in Fig. 4.

Results of proposed schemes based on each model are

overlapping the exact solution. Therefore, the values of

absolute error (AE) juðrÞ � uðrÞj are calculated and given

in Table 2 for the same inputs. The mean value of the

absolute error (MAE) for solutions with DENN-LS,

DENN-RB and DENN-TS models are 1.8863 9 10-07,

9.9134 9 10-09, 7.6151 9 10-09, respectively.

3.6 Bratu Problem for k = 1.0

The 1-dimensional transformed BVP (3) for this case is

given as:

Solu

tion

Solu

tion

Solu

tion

Inputs ‘r’ Inputs ‘r’ Inputs ‘r’

(a) (b) (c)

Fig. 4 Comparison of solutions of proposed neural network models for Bratu Problem in case of k = 0.5


Fig. 5 Graphical view of learning of weights by proposed neural network models for Bratu Problem in case of k = 1.0

uTS rð Þ

¼�0:7171þ �0:10653� 2

1þ e�2 0:1128rþ1:2264ð Þ þ0:0142� 2

1þ e�2 �0:7505rþ0:6065ð Þ þ0:3273� 2

1þ e�2 �0:2906rþ0:12620ð Þ þ�0:4327� 2

1þ e�2 0:1925rþ0:1044ð Þ þ0:1920� 2

1þ e�2 0:0003r�0:0665ð Þ

þ 0:9262� 2

1þ e�2 �0:4049rþ1:12584ð Þ þ�0:4607� 2

1þ e�2 0:1374rþ1:18332ð Þ þ0:5177� 2

1þ e�2 0:7250rþ1:3431ð Þ þ0:6463� 2

1þ e�2 0:5074rþ0:3602ð Þ þ�0:9067� 2

1þ e�2 0:2223rþ1:4157ð Þ

0BB@

1CCAð21Þ


123

u00ðrÞ þ 1

ru0ðrÞ þ euðrÞ ¼ 0; 0� r� 1

uð1Þ ¼ u0ð0Þ ¼ 0

8<: ð22Þ


uðrÞ ¼ log8ð3þ 2

ffiffiffi2pÞ

ð3þ 2ffiffiffi2pþ r2Þ2

!ð23Þ

This problem is solved on the same lines as adopted for

the k = 0.5 case; however, the fitness function e formu-

lated in this case is given as:

e ¼ 1

11

X10

m¼0

u00m þ1

rm

u0m þ eum

� �2

þ 1

2u1ð Þ2þ u00

� �2� �

ð24Þ

The optimization of fitness function (24) is carried out

by SQP, and results are shown for one particular case in

Fig. 5 for each model. Approximate solutions uLS, uRB and

uTS derived using one set of weights trained by the SQP

algorithm for DENN-LS, DENN-RB and DENN-TS

yielding fitness values of 2.0282 9 10-12,

3.6211 9 10-11, and 2.5345 9 10-13, respectively, are

given as:

Table 3 Comparison of the results for k = 1.0

r u(r) u(r) |u(r) - u(r)|


0.0 0.3166943676 0.3166944887 0.3166947421 0.3166942771 1.21E-07 3.74E-07 9.05E-08

0.1 0.3132658505 0.3132659305 0.3132660495 0.3132658020 8.00E-08 1.99E-07 4.85E-08

0.2 0.3030154228 0.3030154260 0.3030153749 0.3030153985 3.13E-09 4.79E-08 2.44E-08

0.3 0.2860472653 0.2860472479 0.2860473052 0.2860472600 1.74E-08 3.99E-08 5.26E-09

0.4 0.2625311275 0.2625311260 0.2625312327 0.2625311300 1.41E-09 1.05E-07 2.59E-09

0.5 0.2326967839 0.2326967765 0.2326967806 0.2326967894 7.33E-09 3.23E-09 5.56E-09

0.6 0.1968268057 0.1968267675 0.1968266924 0.1968268224 3.82E-08 1.13E-07 1.67E-08

0.7 0.1552481067 0.1552480553 0.1552480119 0.1552481329 5.14E-08 9.48E-08 2.62E-08

0.8 0.1083227634 0.1083227243 0.1083227255 0.1083227875 3.92E-08 3.80E-08 2.41E-08

0.9 0.0564386025 0.0564385518 0.0564385199 0.0564386336 5.07E-08 8.25E-08 3.11E-08

1.0 0.0000000000 -0.0000000002 -0.0000000024 0.0000000000 1.98E-10 2.41E-09 4.08E-11

Solu

tion

Solu

tion

Solu

tion


(a) (b) (c)


uLS rð Þ ¼

�0:0646

1þ e� 1:1734rþ0:3883ð Þ þ�1:5629

1þ e� 0:9678r�0:5230ð Þ þ�1:3194

1þ e� �1:1150r�1:5778ð Þ þ�0:7698

1þ e� 0:2773rþ0:8088ð Þ þ�1:7646

1þ e� �1:3237r�1:6337ð Þ

þ 0:7410

1þ e� �0:1712r�0:4849ð Þ þ�0:5746

1þ e� 0:5227r�0:5860ð Þ þ�0:4441

1þ e �1:5901r�0:1701ð Þ þ2:2962

1þ e� �0:9253rþ2:2512ð Þ þ0:0508

1þ e� �0:3305rþ0:5438ð Þ

0BB@

1CCAð25Þ


123

The Eqs. (25)–(27) are given in appendix Table 9 with 14

decimal point of accuracy. Approximate solutions uLSðrÞ,uRBðrÞ and uTSðrÞ determined using Eqs. (25), (26) and (27)

for points r e (0, 1) with step size of 0.1 are given in Table 3

along with the exact solution (23); the comparison of the

results is given graphically in Fig. 6. Values of absolute error

(AE) juðrÞ � uðrÞj are also calculated and provided in the

table for the same inputs. Values of MAE for DENN-LS,

DENN-RB and DENN-TS are 3.7283 9 10-08,

1.0007 9 10-07 and 2.4995 9 10-09, respectively.

3.7 Bratu Problem for k = 2

The 1-dimensional transformed Bratu equation (3) for this

case is given as:

u00ðrÞ þ 1

ru0ðrÞ þ 2:0euðrÞ ¼ 0; 0� r� 1

uð1Þ ¼ u0ð0Þ ¼ 0

8<: ð28Þ


uðrÞ ¼ log4

ð1þ r2Þ2

!ð29Þ

This problem is solved using the same procedure as

outlined for earlier cases; however, the fitness function eformulated for this case is:

e ¼ 1

11

X10

m¼0

u00m þ1

rm

u0m þ 2eum

� �2

þ 1

2u1ð Þ2þ u00

� �2� �

ð30Þ


Fig. 7 Graphical view of learning of weights by proposed neural network models for Bratu Problem in case of k = 2.0

uRB rð Þ ¼� 0:0184e� þ2:7104rþ2:3622ð Þ2 þ 0:4238e� þ0:8189rþ1:9677ð Þ2 � 1:5370e� �0:3798rþ1:6333ð Þ2 � 3:2135e� þ0:0004r�0:0255ð Þ2

þ 2:3098e� �0:3428rþ1:18746ð Þ2 þ 0:8914e� þ2:1768rþ3:1580ð Þ2 þ 1:0923e� þ0:0956r�2:3492ð Þ2 þ 0:4692e� þ0:3647rþ2:2750ð Þ2

þ 3:1085e� �0:3834r�0:1284ð Þ2 � 1:2627e� þ0:3855rþ2:4861ð Þ2

0BB@

1CCA

ð26Þ

uTS rð Þ

¼2:5369þ �0:4351� 2

1þ e�2 0:0687rþ0:6055ð Þ þ�1:03551� 2

1þ e�2 0:5150r�0:5924ð Þ þ�0:6076� 2

1þ e�2 �0:7748r�0:7136ð Þ þ0:5125� 2

1þ e�2 �0:3200rþ1:1825ð Þ þ�0:0152� 2

1þ e�2 �2:4710r�1:6287ð Þ

þ 1:3806 � 2

1þ e�2 �0:4520rþ1:8799ð Þ þ0:4969� 2

1þ e�2 0:6093rþ0:3564ð Þ þ�1:7782� 2

1þ e�2 0:0903rþ1:7928ð Þ þ�0:8245� 2

1þ e�2 0:1182rþ0:7882ð Þ þ�0:2306� 2

1þ e�2 �0:0786rþ0:3272ð Þ

0BB@

1CCAð27Þ


123

Table 4 Comparison of the results for k = 2.0

r u(r) u(r) |u(r) - u(r)|


0.0 1.3862943611 1.3861911057 1.3869312052 1.3861166258 1.03E-04 6.37E-04 1.78E-04

0.1 1.3663936994 1.3662856611 1.3670001865 1.3662122225 1.08E-04 6.06E-04 1.81E-04

0.2 1.3078529348 1.3077471338 1.3084139896 1.3076799865 1.06E-04 5.61E-04 1.73E-04

0.3 1.2139389686 1.2138415625 1.2144413947 1.2137801428 9.74E-05 5.02E-04 1.59E-04

0.4 1.0894543509 1.0893680651 1.0898855026 1.0893140613 8.63E-05 4.31E-04 1.40E-04

0.5 0.9400072585 0.9399342487 0.9403607613 0.9398901687 7.30E-05 3.54E-04 1.17E-04

0.6 0.7713249616 0.7712664665 0.7715988179 0.7712320828 5.85E-05 2.74E-04 9.29E-05

0.7 0.5887421212 0.5886981446 0.5889374361 0.5886725869 4.40E-05 1.95E-04 6.95E-05

0.8 0.3969018774 0.3968718823 0.3970225184 0.3968555124 3.00E-05 1.21E-04 4.64E-05

0.9 0.1996406706 0.1996239595 0.1996920994 0.1996158362 1.67E-05 5.14E-05 2.48E-05

1.0 0.0000000000 -0.0000043808 -0.0000115280 -0.0000019775 4.38E-06 1.15E-05 1.98E-06

Solu

tion

Solu

tion

Solu

tion


(a) (b) (c)


uLS rð Þ

¼

�3:0827

1þ e� �2:0504r�2:5228ð Þ þ3:4322

1þ e� �1:0312rþ2:0662ð Þ þ�3:8023

1þ e� 1:2749r�0:2961ð Þ þ3:9229

1þ e� �1:6024r�0:5304ð Þ þ�0:3039

1þ e� 1:7645r�3:1048ð Þ

þ �3:0545

1þ e� �2:2386r�1:8462ð Þ þ2:5177

1þ e� �1:8467r�2:2580ð Þ þ�2:4280

1þ e� �2:6025r�2:6242ð Þ þ�0:6238

1þ e� �2:5401r�2:8177ð Þ þ�2:4722

1þ e� �3:0170r�0:6066ð Þ

0BB@

1CCAð31Þ

uRB rð Þ ¼� 0:3452e� �1:9554r�3:19882ð Þ2 � 1:4843e� �1:3289r�1:0876ð Þ2 � 4:0338e� þ1:4470rþ4:1032ð Þ2 � 3:3053e� þ0:3534r�1:4515ð Þ2

þ 2:4967e� �0:6189r�0:3251ð Þ2 � 0:2265e� þ0:7441r�2:1671ð Þ2 � 1:8484e� �0:5632r�3:9159ð Þ2 � 0:7009e� �2:8146r�2:6688ð Þ2

þ 1:3287e� �0:6810r�3:2489ð Þ2 � 1:9045e� þ0:6532rþ4:3523ð Þ2

0BB@

1CCAð32Þ


123

Ta

ble

5R

esu

lts

of

stat

isti

cal

anal

ysi

sfo

rth

eB

ratu

Pro

ble

ms

kr

|u(r

)-

u(r

)|

DE

NN

-LS

DE

NN

-RB

DE

NN

-TS

MIN

ME

AN

ST

DM

INM

EA

NS

TD

MIN

ME

AN

ST

D

0.5

0.0

5.7

39

10

-07

6.1

69

10

-03

4.3

39

10

-02

4.4

39

10

-09

3.0

59

10

-02

1.2

39

10

-01

1.2

29

10

-08

9.2

29

10

-06

2.1

39

10

-05

0.2

2.1

69

10

-07

5.5

69

10

-03

3.9

19

10

-02

1.1

29

10

-08

2.3

39

10

-02

7.0

39

10

-02

5.4

49

10

-09

4.1

89

10

-06

9.1

79

10

-06

0.4

1.4

49

10

-07

4.7

29

10

-03

3.3

29

10

-02

7.4

79

10

-09

1.8

99

10

-02

5.0

99

10

-02

4.3

49

10

-09

2.4

49

10

-06

5.1

79

10

-06

0.6

3.0

79

10

-08

3.9

29

10

-03

2.7

69

10

-02

1.6

69

10

-08

1.4

79

10

-02

3.7

49

10

-02

9.2

89

10

-09

1.6

19

10

-06

2.7

19

10

-06

0.8

3.1

39

10

-08

3.0

59

10

-03

2.1

49

10

-02

8.9

79

10

-09

1.0

39

10

-02

2.6

79

10

-02

2.2

59

10

-09

6.6

79

10

-07

1.2

99

10

-06

1.0

2.2

39

10

-11

2.0

39

10

-03

1.4

39

10

-02

1.9

29

10

-09

5.6

29

10

-03

2.1

59

10

-02

6.3

19

10

-11

8.3

89

10

-08

2.7

99

10

-07

1.0

0.0

1.2

19

10

-07

1.4

39

10

-02

1.0

09

10

-01

2.8

29

10

-07

8.2

49

10

-02

2.1

89

10

-01

6.3

89

10

-08

1.0

99

10

-05

1.2

59

10

-05

0.2

3.1

39

10

-09

1.3

29

10

-02

9.2

89

10

-02

4.7

99

10

-08

6.5

09

10

-02

1.4

39

10

-01

2.4

49

10

-08

5.2

79

10

-06

6.3

09

10

-06

0.4

1.4

19

10

-09

1.1

59

10

-02

8.1

19

10

-02

1.0

59

10

-07

5.3

29

10

-02

1.1

29

10

-01

2.5

99

10

-09

3.4

69

10

-06

4.1

39

10

-06

0.6

2.4

79

10

-08

9.7

99

10

-03

6.8

99

10

-02

5.4

19

10

-09

4.0

69

10

-02

8.5

49

10

-02

1.6

79

10

-08

2.2

99

10

-06

3.0

79

10

-06

0.8

1.1

69

10

-09

7.8

49

10

-03

5.5

19

10

-02

2.5

59

10

-09

2.6

79

10

-02

5.9

99

10

-02

1.1

49

10

-08

1.0

49

10

-06

1.4

99

10

-06

1.0

2.6

79

10

-11

5.6

19

10

-03

3.9

49

10

-02

1.9

29

10

-09

1.3

39

10

-02

4.2

99

10

-02

2.7

49

10

-11

2.4

69

10

-07

7.0

79

10

-07

2.0

0.0

1.0

39

10

-04

3.2

79

10

-02

2.1

59

10

-01

6.3

79

10

-04

4.4

59

10

-01

7.6

19

10

-01

1.7

89

10

-04

8.5

09

10

-02

7.7

69

10

-01

0.2

1.0

69

10

-04

3.0

49

10

-02

2.0

19

10

-01

5.6

19

10

-04

3.9

49

10

-01

6.2

39

10

-01

1.7

39

10

-04

4.8

49

10

-02

4.1

59

10

-01

0.4

8.6

39

10

-05

2.5

49

10

-02

1.7

39

10

-01

4.3

19

10

-04

3.2

39

10

-01

5.0

69

10

-01

1.4

09

10

-04

3.1

99

10

-02

2.6

59

10

-01

0.6

5.8

59

10

-05

1.9

19

10

-02

1.3

89

10

-01

2.7

49

10

-04

2.3

39

10

-01

3.6

89

10

-01

9.2

99

10

-05

1.9

49

10

-02

1.5

99

10

-01

0.8

3.0

09

10

-05

1.2

49

10

-02

9.9

19

10

-02

1.2

19

10

-04

1.3

79

10

-01

2.2

59

10

-01

4.6

49

10

-05

8.9

19

10

-03

7.2

69

10

-02

1.0

1.1

39

10

-07

5.9

49

10

-03

5.9

19

10

-02

1.9

29

10

-05

4.7

29

10

-02

1.1

49

10

-01

1.0

39

10

-07

1.2

99

10

-05

3.0

29

10

-05


123

Fit

ness

Val

ues

No. of independent run of algorithm

(a) Bratu Problem for = 0.5

Fit

ness

Val

ues


(b) Bratu Problem for = 1.0

Fit

ness

Val

ues


(c) Bratu Problem for = 2.0

Mea

n A

bsol

ute

erro

r


(d) Bratu Problem for = 0.5

Mea

n A

bsol

ute

erro

r


(e) Bratu Problem for = 1.0

Mea

n A

bsol

ute

erro

r


(f) Bratu Problem for = 2.0

Fig. 9 The Fitness and Mean absolute error values for 100 independent runs of each model

Table 6 Results of

convergence analysis for the

neural networks models

k Model % runs with MAE % runs with fitness

10-03 10-04 10-05 10-06 10-07 10-04 10-06 10-08 10-10 10-12

0.5 DENN-LS 098 098 098 089 004 098 098 098 075 001

DENN-RB 084 083 081 061 013 087 085 083 056 006

DENN-TS 100 100 099 091 017 100 100 100 076 009

1.0 DENN-LS 098 098 097 052 009 098 098 097 041 009

DENN-RB 078 078 076 046 007 081 079 078 042 000

DENN-TS 100 100 100 077 010 100 100 100 072 005

2.0 DENN-LS 035 003 000 000 000 099 099 085 009 000

DENN-RB 021 002 000 000 000 072 070 061 003 000

DENN-TS 065 006 000 000 000 100 099 095 028 002

uTS rð Þ

¼�1:05866þ �0:3224� 2

1þ e�2 �0:4704rþ0:7714ð Þ þ1:7804� 2

1þ e�2 �0:7934rþ0:4323ð Þ þ�0:6046� 2

1þ e�2 0:9799r�0:4569ð Þ þ�1:3524� 2

1þ e�2 0:0849rþ0:6872ð Þ þ�0:8231� 2

1þ e�2 1:1464r�1:8889ð Þ

þ �0:0182� 2

1þ e�2 1:0020r�0:3070ð Þ þ�1:0478� 2

1þ e�2 �1:6413r�0:3964ð Þ þ1:5584� 2

1þ e�2 2:3195rþ2:1263ð Þ þ1:6900� 2

1þ e�2 0:1248r�0:9377ð Þ þ0:1985� 2

1þ e�2 �0:1325r�0:2194ð Þ

0BB@

1CCA

ð33Þ


123

The optimal values for the fitness function (30) are

learned with SQP, and results are shown graphically

for one particular case in Fig. 7. Solutions uLS, uRB

and uTS with one particular set of weights learned by

the SQP algorithm for DENN-LS, DENN-RB and

DENN-TS yielding fitness values of 1.9428 9 10-10,

4.4574 9 10-10, and 2.3517 9 10-09, respectively, are

given as:

The Eqs. (31)–(33) are also given in appendix

Table 10 in detail format. Approximate solutions

uLSðrÞ, uRBðrÞ and uTSðrÞ to the problem obtained using

Eqs. (31), (32) and (33) for points r e (0, 1) with a step

size of 0.1 are given in Table 4, along with the exact

solution (29), while the graphical comparison of the

results is given in Fig. 8. The values of absolute error

(AE) juðrÞ � uðrÞj are determined and provided in the

table for the same inputs. The values of MAE for

solutions with DENN-LS, DENN-RB and DENN-TS

are 6.6123 9 10-05, 3.4039 9 10-04 and

1.0763 9 10-04, respectively.

4 Statistical analysis

Reliable inferences for the neural network models can only

be made on the basis of a large number of independent runs

and not just a single successful execution of the algorithm.

In this section, results of the proposed neural networks to

solve the Bratu equation based on 100 independent runs are

presented and compared in terms of accuracy, convergence

and computational complexity.

The accuracy of the models is examined using statistical

parameters, i.e., mean, standard deviation (STD) and

minimum (MIN) values of the absolute error (AE) from the

exact solution. Results are calculated for DENN-LS,

DENN-RB and DENN-TS models on the basis of one

hundred independent runs; these are provided in Table 5

for inputs r e (0, 1) with a step size of 0.2. All the three

cases for the Bratu problem are considered, i.e., k = 0.5, 1

and 2. It is observed that with increase in the value of k(i.e., as k approaches kc = 2), the accuracy of the models

decreases. There is no significant difference in the results

of MIN values for all the three models; the values of mean

of the AE are mostly the best for DENN-TS model. The

lowest values of STD are also observed for DENN-TS.

The values of fitness e, as given in equations (18), (24)

and (28), are determined for the hundred independent runs

of each model. These results along with MAE values are

plotted against the number of independents runs of each

model in Fig. 9. Results are plotted on a semilogarithmic

scale in order to elaborate the small differences in the

values. We also plot runs according to ascending order of

fitness values in Fig. 9. It is found that for the case k = kc,

the accuracy in terms of MAE values decreases as com-

pared to the other cases for all three neural network mod-

els. It can be seen from the figure that the better values of

MAE are obtained for smaller fitness values, and vice

versa. The results obtained using DENN-TS consistently

provide the lowest values of MAE and the fitness function.

DENN-RB model could not provide sufficient accuracy for

some runs, and this trend increases with increase in the

value of k. Results of DENN-RB model also mostly do not

provide sufficient precision.

The reliability of the proposed models is investigated

based on percent convergent runs, i.e., runs fulfilling a

criteria of pre-defined values of fitness e and MAE. Results

of this convergence analysis for one hundred independent

runs fulfilling the criteria of fitness and MAE are given in

Table 6. It is observed that the best convergence results are

obtained for the DENN-TS model optimized with SQP. A

decrease in convergence is observed with increase in level

Table 7 Comparative analysis of the performance and execution time

k Present results Reported results

Method GMAE MF MET (Seconds) Method MET

(Seconds) ValuesMean STD Values Values Values STD

0.5 DENN-LS 3.0170 9 10-06 4.2333 9 10-06 9.1075 9 10-10 4.1036 9 10-09 21.65 7.81 GA 161.87

DENN-RB 2.3668 9 10-05 1.6787 9 10-04 3.7242 9 10-09 2.5880 9 10-08 19.70 6.73 IPA 109.59

DENN-TS 2.8633 9 10-06 6.0301 9 10-06 5.7949 9 10-10 2.2724 9 10-09 21.56 7.90 GA-IPA 266.99

1.0 DENN-LS 6.3047 9 10-06 7.5995 9 10-06 3.3727 9 10-09 1.3196 9 10-08 21.87 6.81 GA 153.32

DENN-RB 1.0141 9 10-05 3.7138 9 10-05 1.1058 9 10-09 1.6631 9 10-09 19.37 7.19 IPA 146.72

DENN-TS 3.6951 9 10-06 4.4190 9 10-06 1.0442 9 10-09 2.0368 9 10-09 21.50 7.98 GA-IPA 260.98

2.0 DENN-LS 3.0224 9 10-03 1.4954 9 10-03 5.7798 9 10-08 3.0208 9 10-07 18.71 6.65 GA 166.93

DENN-RB 3.1414 9 10-03 1.6783 9 10-03 2.2841 9 10-09 1.6046 9 10-09 20.52 5.42 IPA 124.73

DENN-TS 2.6278 9 10-03 1.4537 9 10-03 1.6101 9 10-09 2.5062 9 10-09 20.57 7.49 GA-IPA 288.84


123

of stiffness in criteria for all the three models, but still

DENN-TS yields the best results.

The proposed models are further analyzed based on two

performance measures: the global mean absolute error

(GMAE) and the mean fitness (MF). These are defined as:

GMAE ¼ 1

R

XR

r¼1

1

P

XP

j¼1

jui � uri j;

MF ¼ 1

R

XR

r¼1

er

ð34Þ

where P and R represent the total number of inputs and

independent runs, respectively, ui and uir are the exact and

proposed solutions of the rth independent run for the ith

input, and er is the fitness value for the rth run of the

algorithm. In our simulations, the inputs are taken from r e[0, 1] with a step size of 0.1, i.e., P = 11, and R = 100.

Values of GMAE and MF along with STD are determined

for DENN-LS, DENN-RB and DENN-TS models opti-

mized with the SQP method and results presented in

Table 7 for all the three problems. Values of MF are found

to be around 10-08 to 10-10 for the three models for each

case of the problem, but generally lowest for DENN-LS.

Values of GMAE are of the order of 10-05 to 10-06 for

cases of k = 0.5 and 1.0, while this is of the order of 10-03

for k = kc = 2. The best results are obtained with DENN-

TS model for all cases of the boundary value problem.

The computational complexity of the schemes is ana-

lyzed based on the computing time taken for optimization

of weights for each scheme. The analysis is performed

based on 100 independent runs of each model, and results

are provided in terms of mean execution time (MET) and

its STD in Table 7. The reported values of MET are also

listed in Table 7 for genetic algorithms (GAs), interior-

point method and hybrid approach GA-IPA [22]. It is found

that there is no significant difference in the values of MET

for DENN-LS, DENN-RB and DENN-TS models opti-

mized with the SQP technique, while these values are 10

times better from the hybrid approach GA-IPA and around

5 times superior from GA and IPA. The computations are

carried out on a Dell Workstation 390, with

Intel(R) Core(TM) 2 CPU [email protected] GHz, 2.00 GB

RAM, and running MATLAB version 2011a.

5 Conclusions

Feed-forward artificial neural network models based on

log-sigmoid, radial basis and tan-sigmoid transfer functions

with weights optimized using sequential quadratic pro-

gramming methods can effectively solve the 2-dimensional

Bratu Problem, once it has been transformed into an

equivalent 1-dimensional boundary value problem.

Comparison with exact solutions shows that the pro-

posed results give an absolute error in the range of 10-07 to

10-10, 10-07 to 10-11 and 10-04 to 10-06 for the cases

k = 0.5, 1.0 and 2.0, respectively. With increase in the

value of k, a decrease in the precision of results is

observed. In general, the DENN-TS model gave the most

accurate results.

The reliability and effectiveness of the proposed com-

puting models were validated by a large number of inde-

pendent runs and their statistical analysis. The proposed

models based on DENN-LS, DENN-RB and DENN-TS

provided accurate and convergent results for 70, 98 and

100 % of the independent runs, respectively, for all the

three BVPs of the Bratu equation. The performance

parameters of global mean absolute error and mean fitness

also indicate the supremacy of the DENN-TS model over

other schemes.

The mean execution time for all the three models is

around 20 s, which shows that there is hardly any differ-

ence in terms of computational complexity between the

models.

The DENN-TS trained with the SQP method is the most

effective solver in terms of accuracy and convergence, i.e.,

it yields the lowest values for AE, MAE, GMAE, fitness

achieved and mean fitness, for all the three cases of the

transformed BVP.

Appendix

The derived solutions by each neural networks model in

case of all three BVPs of Bratu-type equations are given in

Tables 8, 9 and 10, respectively. The solutions are pre-

sented with 14 decimal points for the unknown weights to

exactly reproduce the results presented in the body of

manuscript and to avoid rounding of error problem.


123

Table 8 Derive solutions by neural network models in case of problem 1

Model Proposed solutions uðrÞ

DENN-

LS

�0:881621485660081

1þ e� 0:880050447224235r�0:898845301777749ð Þ þ�0:368172277088185

1þ e� �0:19146073533071r�1:222085486359630ð Þ þ�3:7801695916735

1þ e� �0:58208204163615r�1:307356930332550ð Þ

þ 1:39233553786581

1þ e� 0:006397855283667rþ0:926570468361007ð Þ þ0:714358491124986

1þ e� �0:415842335201208r�0:965278025492269ð Þ þ�2:2292083620281

1þ e� 0:710706777029788r�2:76491788736733ð Þ

þ �0:115553502503515

1þ e� �0:640712019255941r�1:28595522183011ð Þ þ0:579154241993379

1þ e0:369074467012103rþ1:47395456097188Þ þ0:001802551342664

1þ e� 0:177017061284790rþ0:843114806464935ð Þ

þ �1:08646736101306

1þ e� 0:671901926965287r�1:32881441806551ð Þ

0BBBBBBBBBB@

1CCCCCCCCCCA

DENN-

RB�0:022851199507785e� �1:060430196337180r�1:967315103560540ð Þ2 � 0:159965370093374e� �0:627409711092437r�0:754157743819931ð Þ2

þ0:201271010547020e� �0:556839938126852r�0:669069618919350ð Þ2 � 0:037232882803983e� þ0:573551097567133rþ1:130322054120400ð Þ2

�0:591405175035391e� þ0:293345858135356r�1:671809025552220ð Þ2 � 0:963236822427233e� �0:340130511071845r�1:479540269700920ð Þ2

�0:377394341615607e� �0:375816868330149rþ1:386844575227460ð Þ2 þ 1:357179842444930e� þ0:214074813191106r�0:129094113663476ð Þ2

�1:292004785668580e� �0:028809057624117rþ0:251746801766521ð Þ2 þ 1:084729263583430e� þ0:154671493949521rþ1:322720311227970ð Þ2

0BBBBBBBBB@

1CCCCCCCCCA

DENN-

TS� 0:717176067903709þ �0:106531337437251� 2

1þ e�2 0:112851379117880rþ1:226485837521980ð Þ þ0:014215444268231� 2

1þ e�2 �0:750511660266111rþ0:606533945754926ð Þ

þ 0:327310444921969� 2

1þ e�2 �0:290625696793821rþ0:126202201682872ð Þ þ�0:432729977887967� 2

1þ e�2 0:192534380468348rþ0:104408530788519ð Þ þ0:192037504450171� 2

1þ e�2 0:00030500708988r�0:066557133802384ð Þ

þ 0:926267908397325� 2

1þ e�2 �0:404964597022221rþ1:1258424485338ð Þ þ�0:460742765339709� 2

1þ e�2 0:137423107208874rþ1:18332964159146ð Þ þ0:517740656396020� 2

1þ e�2 0:725089544176983rþ1:34311313780279ð Þ

þ 0:646327032040363� 2

1þ e�2 0:507419906625338rþ0:360259411889679ð Þ þ�0:906718841905442� 2

1þ e�2 0:222340622767519rþ1:41577352577436ð Þ

0BBBBBBBBBB@

1CCCCCCCCCCA

Table 9 Derive solutions by neural network models in case of Problem 2

Model Proposed Solutions uðrÞ

DENN-

LS

�0:064655172789555

1þ e� 1:17349225317105rþ0:388374369025659ð Þ þ�1:56298908520775

1þ e� 0:967814073271136r�0:523069441465472ð Þ þ�1:3194443898739

1þ e� �1:11506444137262r�1:57789576175984ð Þ

þ �0:769811094890991

1þ e� 0:277372107205929rþ0:808892376819103ð Þ þ�1:76466302146984

1þ e� �1:32375097247728r�1:63372618094345ð Þ þ0:741044453237249

1þ e� �0:171259803867778r�0:484955138240345ð Þ

þ �0:574613395398207

1þ e� 0:522702854283237r�0:586069447244827ð Þ þ�0:444150680963213

1þ e �1:59019127846778r�0:170130428969297ð Þ þ2:29624249379669

1þ e� �0:925315098372165rþ2:25120128184431ð Þ

þ 0:0508482260071

1þ e� �0:330558100367822rþ0:543864917256738ð Þ

0BBBBBBBBBB@

1CCCCCCCCCCA

DENN-

RB� 0:018432963298386e� þ2:710425827922060rþ2:362204476334150ð Þ2 þ 0:423869870436273e� þ0:818923185467605rþ1:967737961050590ð Þ2

� 1:537091444450670e� �0:379864980224851rþ1:633372892377530ð Þ2 � 3:213573741286320e� þ0:000424480199338r�0:025562578271033ð Þ2

þ 2:309821869932590e� �0:342845259449014rþ1:187461137915420ð Þ2 þ 0:891461992769485e� þ2:176895443896920rþ3:158055169543970ð Þ2

þ 1:092377779743590e� þ0:095625353789992r�2:349283273422530ð Þ2 þ 0:469229304843676e� þ0:364708626329813rþ2:275093719311820ð Þ2

þ 3:108582125556570e� �0:383452696280033r�0:128424998104074ð Þ2 � 1:262775925668710e� þ0:385582078484859rþ2:486183852389300ð Þ2

0BBBBBBBBB@

1CCCCCCCCCA

DENN-

TS2:5369556008777þ �0:435172735276397� 2

1þ e�2 0:068754414053185rþ0:605556966814159ð Þ þ�1:035516998756210� 2

1þ e�2 0:515083116183790r�0:592486994815203ð Þ

þ �0:607617156040548� 2

1þ e�2 �0:774893665980428r�0:713657700905530ð Þ þ0:512521758474198� 2

1þ e�2 �0:320092232914347rþ1:182569743253850ð Þ þ�0:015277731318143� 2

1þ e�2 �2:471036394020450r�1:628763736836710ð Þ

þ 1:380665996146060� 2

1þ e�2 �0:452092873587792rþ1:879910453503700ð Þ þ0:496925060666271� 2

1þ e�2 0:609381204538157rþ0:356404794625870ð Þ þ�1:778289284333950� 2

1þ e�2 0:090356709623937rþ1:792880071492960ð Þ

þ �0:824517852175721� 2

1þ e�2 0:118279395523136rþ0:788270204420893ð Þ þ�0:230676658263262� 2

1þ e�2 �0:078643302456727rþ0:327222876189896ð Þ

0BBBBBBBBBB@

1CCCCCCCCCCA


123

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Table 10 Derive solutions by neural network models in case of Problem 3

Model Proposed Solutions uðrÞ

DENN-

LS

�3:0827824329014

1þ e� �2:0504948317817r�2:52288674225653ð Þ þ3:43223827474807

1þ e� �1:03126606579591rþ2:066248569508660ð Þ þ�3:80235024525787

1þ e� 1:27494201798072r�0:296161515729911ð Þ

þ 3:92291954455834

1þ e� �1:60242434758667r�0:530405368600276ð Þ þ�0:303984191061255

1þ e� 1:76451488404892r�3:10484466972079ð Þ þ�3:05456875599926

1þ e� �2:23860127595284r�1:84624274403238ð Þ

þ 2:51770649427315

1þ e� �1:84678076426944r�2:25807161260334ð Þ þ�2:42804252598274

1þ e� �2:602553390377610r�2:624205896198740ð Þ þ�0:62384172060593

1þ e� �2:54015922086322r�2:81778437760697ð Þ

þ �2:47226910793316

1þ e� �3:01707300622596r�0:60664046837583ð Þ

0BBBBBBBBBB@

1CCCCCCCCCCA

DENN-

RB�0:345217809586040e� �1:955499408658030r�3:198821842815730ð Þ2 � 1:484398130606580e� �1:328971537655880r�1:087639896233320ð Þ2

�4:033883444468790e� þ1:447072641837670rþ4:103289249195690ð Þ2 � 3:305306095691340e� þ0:353483608506659r�1:451580712780780ð Þ2

þ2:496721775160310e� �0:618918704703375r�0:325164070635999ð Þ2 � 0:226522992475956e� þ0:744130803134757r�2:167170120586640ð Þ2

�1:848430706960440e� �0:563293350175233r�3:915901314089210ð Þ2 � 0:700986421159916e� �2:814697067388060r�2:668816369496790ð Þ2

þ1:328775798569840e� �0:681043536162712r�3:248913702568520ð Þ2 � 1:904589137297240e� þ0:653286608231759rþ4:352351738319350ð Þ2

0BBBBBBBBB@

1CCCCCCCCCA

DENN-

TS� 1:05866446203872þ �0:322457052493663� 2

1þ e�2 �0:470405861427398rþ0:771462314687142ð Þ þ1:780468693434490� 2

1þ e�2 �0:793419550309743rþ0:432356225153400ð Þ

þ �0:604675123481939� 2

1þ e�2 0:979977684434639r�0:456900429750872ð Þ þ�1:352472331476150� 2

1þ e�2 0:084913385156943rþ0:687244003839265ð Þ þ�0:823102595386064� 2

1þ e�2 1:146459901508450r�1:888925233909550ð Þ

þ �0:018259980929906� 2

1þ e�2 1:002009023801600r�0:307069326760630ð Þ þ�1:047804133033830� 2

1þ e�2 �1:641377540027920r�0:396403953261298ð Þ þ1:558417960379020� 2

1þ e�2 2:319565138751870rþ2:126364573606770ð Þ

þ 1:690042781878130� 2

1þ e�2 0:124879558762519r�0:937711465294909ð Þ þ0:198506243148627� 2

1þ e�2 �0:132579473104704r�0:219478334768398ð Þ

0BBBBBBBBBB@

1CCCCCCCCCCA


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http://dx.doi.org/10.1007/s00521-012-1170-4

http://dx.doi.org/10.1007/s00521-012-1170-4

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http://faculty.oxy.edu/ron/research/bratu/bratu.pdf

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