treating transcendental functions in partial differential...

Research ArticleTreating Transcendental Functions in PartialDifferential Equations Using the Variational IterationMethod with Bernstein Polynomials

Ahmed Farooq Qasim and Almutasim Abdulmuhsin Hamed

College of Computer Sciences and Mathematics University of Mosul Iraq

Correspondence should be addressed to Ahmed Farooq Qasim ahmednumericalyahoocom

Received 25 November 2018 Revised 3 January 2019 Accepted 11 February 2019 Published 3 March 2019

Academic Editor Shyam L Kalla

Copyright copy 2019 Ahmed FarooqQasim and AlmutasimAbdulmuhsin HamedThis is an open access article distributed under theCreative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium providedthe original work is properly cited

We aim through this paper to present an improved variational iteration method (VIM) based on Bernstein polynomials (BP)approximations to be used with transcendental functions The key benefits gained from this modification are to reach stable andfairly accurate results and at the same time to expand the unknown functionrsquos domain in partial differential equations (PDEs)The proposed approach introduces the Bernstein polynomials in the transcendental functions of nonlinear PDEs A number ofexamples were included in order to expound the methodrsquos capacity and reliability From the results we conclude that the VIMwithBP is a powerful mathematical tool that can be applied to solve nonlinear PDEs

1 Introduction

It is common knowledge that a large number of phenomenaare in essence nonlinear and therefore can be modeledusing nonlinear differential equations [1] However it isnot possible to find exact solutions to nonlinear differentialequationsTherefore the approach to tackle this shortcomingwas to apply different numerical methods suffice to reachapproximate solutions including the methods of Adomiandecomposition [2 3] homotopy perturbation [4ndash6] differ-ential transform [7ndash9] and variational iteration [10 11]

He [12] introduced the VIM in 1998 and since thenit was applied by many scientists as an approach to solvemathematical and physical problems for its capability tosimplify a difficult problem into a readily solvable problemTheVIM is considered a capable tool that can be used to solvefunctional equations Commonly used numerical approachessuch as finite difference or characteristics methods requiretedious computations These methods are often less accuratein their results because of their round-off error Solutions forSchrodinger equations based on the widely used analyticalmethods are very constrained and can be only applied

in certain scenarios while they are inapplicable to obtainsolutions for a large number of equations that model reallife scenarios The VIM is a modified Lagrange multipliermethod with promising results in solving numerous classesof nonlinear problems with approximations that converge tothe correct solution within a short number of iterations

Bernstein polynomialsrsquo role is prominent in countlessmathematics applications as they are utilized in solvingdifferential equations in addition to their contribution to theapproximation theory Bezier curves which are Bernsteinpolynomials restricted to the interval x isin [a b] played amoreprominent role with the development of computer graphics[13]

A novel iterative approach based on the VIM is applied todifferential equations by [14] Olayiwola Mo [15] applied it tosolve the Convection-Diffusion equation and for the class offractional Convection-Diffusion equation [16] The modifiedvariational iteration method for sine-Gordon equation issuggested in [17] using Chebyshev polynomials and Hersquospolynomials [18]

In this paper we adjusted the VIM to utilize the Bernsteinpolynomials so as to approximate the correction functionsrsquo

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2019 Article ID 2872867 8 pageshttpsdoiorg10115520192872867

2 International Journal of Mathematics and Mathematical Sciences

nonlinear terms The solutions obtained using our modi-fied method are comparable to VIM In our approach weapplied the Bernstein polynomial approximations to non-linear functions in the correction functions before iteratingthe numerical solution of PDEs The main objective is toachieve reliable accurate PDEs solutions if using VIM leadsto unstable solutions The feasibility of our approach isexpounded through a number of examples

The rest of the paper is divided into five sections Section 2introduces brief ideas of VIM Section 3 presents one- andtwo-dimensional Bernstein polynomials In Section 4 ananalysis of the convergence is presented Section 5 elucidatesthree examples that expound the feasibility of using VIMwith the Bernstein polynomials Section 6 is dedicated for theconclusions

2 VIM Basics

The concept of VIM is based on the general Lagrangersquosmultiplier method The key advantage of this method isreaching a mathematical problem solution by linearizationassumption to be used as an initial approximation thatpromptly converges to an exact solution [19]

The following nonlinear differential equation elucidatesthe basic concept of the VIM as follows

L (y) +N (y) = g (x) (1)

where L and N are the linear and nonlinear operatorsrespectively while g(x) is an inhomogeneous term VIM isused to formalize the following correction function

119910119899+1 (119909 119905)= 119910119899 (119909 119905)+ int1199050120582 (120591) [119871 (119910119899 (119909 119905)) + 119873 (119910119899 (119909 119905)) minus 119892 (119909)] 119889120591

(2)

where 120582(120591) is a general Lagrangian multiplier which can bedetermined through the variational theory and integration byparts yn represents a restricted variation (ie 120575 yn = 0) andthe subscript n symbolizes the nth-order approximation

The Lagrange multiplier 120582(120591) is optimally determinedthrough integration by parts The consecutive approxima-tions yn+1 n ge 0 of the solution yn(x t) are determinedusingthe Lagrange multiplier from the first step using any selectivefunction y0 Thus the solution will be as follows

119910 (119909 119905) = lim119899997888rarrinfin

119910119899 (3)

3 Bernstein Polynomials

Polynomials are considered easily defined calculated differ-entiated and integrated mathematical tools The Bernsteinbased polynomials are tested to approximate the functionsas they result in improved approximations to a function witha few terms This feature has led to their widespread use inappliedmathematics physics and computer based geometric

designs combined with other methods like Galerkin andcollocation to solve both differential and integral equations

The nth degree Bernstein Polynomials form a completebasis over [0 1] formalized as

119861119894119899 (119909) = (119899119894) 119909119894 (1 minus 119909)119899minus119894 0 le 119894 le 119899 (4)

in which ( ni ) = ni(n minus 1) represent the binomialcoefficients

Based on [13] we use the generalized Bernstein polyno-mials of nth degree which form a complete basis over [a b] asfollows

119861119894119899 (119909) = 1(119887 minus 119886)119899 (

119899119894) (119909 minus 119886)

119894 (119887 minus 119909)119899minus119894 0 le 119894 le 119899 (5)

If the function 119891 of two real variables over the intervalsquare are

119878 [119886 119887] times [119888 119889] (6)

then the two-variable Bernstein polynomial of (119899119898)degree equivalent to the function 119891 is determined in thefollowing formula [20]

(119861119899119898119891) (119909 119905) =119899sum119894=0

119898sum119896=0

119891( 119894119899 119896119898)119875119894119899 (119909) 119875119896119898 (119905) (7)

where

119875119895119904 (119911) = (119904119895) 119911119895 (1 minus 119911)119904minus119895 (8)

We use (7) to convert the transcendental functions(trigonometric exponential and Logarithm functions) thatappear in homogenous and nonhomogenous PDEs such asthe sine-Gordon equation into the series approximation bymeans of Bernstein polynomial

The following nonlinear partial differential equation con-tains the transcendental functions term as follows

L (y) + N (y) + T (y) = g (x) (9)

such that 119879(119910) is the transcendental functionsUsing (2) we have the following

119910119899+1 (119909 119905) = 119910119899 (119909 119905) + int119905

0120582 (120591) [119871 (119910119899 (119909 119905))

+ 119873 (119910119899 (119909 119905)) + 119879 (119910119899 (119909 119905)) minus 119892 (119909)] 119889120591(10)

Therefore we approach the transcendent functions119879(119910119899(119909 119905)using Bernsteinrsquos approximation because it can only beintegrated by numerical methods

119879 (119910119899 (119909 119905)) = 119861119894119895119910119899 (119909 119905) (11)

International Journal of Mathematics and Mathematical Sciences 3

4 Analysis of the Convergence

The variational iteration method converts the partial differ-ential equations to a recurrence sequence of functions Thelimit of that sequence leads to the solution of the PDE

Now let

119865(119910 119879 (119910119899) 120597119910120597119905 120597119910120597119909 12059721199101205971199092

12059721199101205971199052

1205972119910120597119909120597119905) = 119892 (119909) (12)

be the second- order partial differential equation and119879(119910119899(119909 119905) be the transcendent functions Using VIM withBernstein polynomial in (12) we have

119910119899+1 (119909 119905) = 119910119899 (119909 119905) + int119905

0120582[119865(119910119899 119861119899119898 (119910119899) 120597119910119899120597120591

120597119910119899120597119909 12059721199101198991205971199092

12059721199101198991205971205912 1205972119910119899120597119909120597120591) minus 119892 (119909)] 119889120591

(13)

such that 119910119899 is Hersquos monographs ie 120575(119910119899) = 0 [21] To findthe value 120582 (13) becomes as follows

120575119910119899+1 (119909 119905) = 120575119910119899 (119909 119905) + 120575int119905

0120582[119865(119910119899 119861119899119898 (119910119899)

120597119910119899120597120591 120597119910119899120597119909

12059721199101198991205971199092 12059721199101198991205971205912

1205972119910119899120597119909120597120591) minus 119892 (119909)] 119889120591(14)

In fact the solution of (14) is considered as the fixed pointwith the initial condition 1199100(119909 119905) [21]Theorem 1 (see [22]) Let a sequence 119902119899 satisfy 0 lt 119902119899 lt 1 and119902119899 997888rarr 1 as 119899 997888rarr infin en for any function 119891 isin 119862[0 1]119861119899 (119891 119902119899 119909) 997888rarr997888rarr 119891 (119909) [119909 isin [0 1] 119899 997888rarr infin] (15)

In other words 119861119899(119891 119902119899 119909) is Bernstein polynomials anduniformly convergent to 119891(119909)Theorem2 (see [21] (Banachrsquos fixed point theorem)) Assumethat119883 is a Banach space and

119860 119883 997888rarr 119883 (16)

is a nonlinear mapping and suppose that1003817100381710038171003817119860 [119910] minus 119860 [119910]1003817100381710038171003817 le 120583 1003817100381710038171003817119910 minus 1199101003817100381710038171003817 119910 119910 isin 119883 (17)

for some constant 120583 lt 1 en A has a unique fixed pointFurthermore the sequence

119910119899+1 = 119860 [119910119899] (18)

with an arbitrary choice of 1199100 isin 119883 converges to the fixed pointof A and

1003817100381710038171003817119910119896 minus 1199101198971003817100381710038171003817 le 10038171003817100381710038171199101 minus 11991001003817100381710038171003817119896minus2sum119895=119897minus1

120583119895 (19)

and according to eorem 2 for the nonlinear mapping

119860 [119910]= 119910 (119909 119905)+ int1199050120582[119865(119910 120597119910120597120591

120597119910120597119909 12059721199101205971199092

12059721199101205971205912

1205972119910120597119909120597120591)]119889120591

(20)

a sufficient condition for convergence of the variational itera-tion method is strictly contraction of A

Corollary 3 Using eorems 1 and 2 we conclude that

119860 [119910] = 119910 (119909 119905)+ int1199050120582[119865(119910 119861119899119898 (119910) 120597119910120597120591

120597119910120597119909 12059721199101205971199092

12059721199101205971205912

1205972119910120597119909120597120591)

minus 119892 (119909)] 119889120591(21)

such that 119861119899119898(119910) is Bernstein approximation then the suf-ficient condition for convergence of the variational iterationmethod is strictly contraction of A Furthermore the sequence(13) converges to the fixed point of A which is also the solutionof the partial differential equation (9)

5 Numerical Examples

In this section the following three examples of nonlinearPDEs are solved using the VIM with Bernstein polynomialsThe results are generated using Maple 13 The accuracy of theresults is determined when compared to the exact solutions

Example 1 Let us take the sine-Gordon equation [17]

119910119905119905 minus 119910119909119909 + sin (119910) = 0 119905 gt 0119910 (119909 0) = 0119910119905 (119909 0) = 8

119890119909 + 119890minus119909 (22)

the exact solution of (9) is as follows

119910 (119909 119905) = 4 arctan( 2119905119890119909 + 119890minus119909 ) (23)

Applying the Bernstein polynomials approximation forsin(y) in (7) with n=m=2 results in the following

1199100 = 8119905119890119909 + 119890minus119909

sin (1199100) = 1818594854119905 + 02808027720119909minus 017437779301199051199092 minus 25753973491199052+ 044350969581199052119909+ 0729022702611990521199092

(24)


Applying the VIM on (22) we get

119910119899+1 (119909 119905) = 119910119899 + int119905

0(119904 minus 119905) (119910119904119904 minus 119910119909119909 + sin (1199100)) 119889119904

119899 ge 1(25)

where the Lagrange multiplier is as follows

120582 (s) = s minus t (26)

Consider the first iterate numerical solution

1199100 = 8119905119890119909 + 119890minus119909

1199101 (119909 119905)= minus 1(1198901000000063119909 + 119890minus1000000063119909)3 (1 times 10

minus11) 16times 1012119904 + 8000001 times 10111199053 + 369591413times 10911990541199091198903000000189119909 minus 6438493370times 101011990541198901000000063119909 minus 643849337times 10101199054119890minus1000000063119909 + 909297427times 101011990531198901000000063119909 + 909297427times 10101199053119890minus1000000063119909 + sdot sdot sdot


minus20119905)sdot 666666745 times 10201199052 minus 76666686 times 10201199054 minus 1times 101011990541199091198903000000189119909 minus 16 times 10211198902000000126119909minus 145314825 times 101811990541198901000000063119909 minus 145314825times 10181199054119890minus1000000063119909 minus 833610388times 101911990531198901000000063119909 minus 833610388times 10191199053119890minus1000000063119909

(27)

The absolute error of VIM-Bernstein and the exact solutionare presented in Tables 1-2 and Figures 1-2

Tables 1-2 present the absolute error and mean squareerror of VIM with modified Bernstein polynomial whenn=m=2 and 119905 = 01 in Table 1 and n=m=1 and 119909 = 01 inTable 2 The maximum errors generated using the modifiedBernstein polynomial are of 10minus4Example 2 Consider the PDE [23]

119910119905 = minus119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909 + 119890minus119910119877 (119909 119905) 119910 (119909 0) = 119892 (119909) (28)

Table 1 The absolute and mean square errors using y2 for Sine-Gordon equation when n=m=2 and 119905 = 01X VIM-Berns Exact |VIMB-Exact|01 0397057362 0396702532 3548303times10minus402 0391225182 0390882208 3429731times10minus403 0381815965 0381490269 3256956times10minus404 0369256745 0368953065 3036803times10minus405 0354079776 0353801996 2777801times10minus406 0336872278 0336623332 2489467times10minus407 0318228605 0318010464 21814014times10minus408 0298710758 0298524478 1862801times10minus409 0278820327 0278666139 1541880times10minus410 0258982287 0258859720 1225671times10minus4MSE 7020528502times10minus8

Table 2 The absolute and mean square errors using y3 for Sine-Gordon equation when n=m=1 and 119909 = 01t VIM-Berns Exact |VIMB-Exact|001 003980022977 003979951651 7132600times10minus7002 007959685852 007959115424 5704280times10minus6003 01193862874 01193670439 1924350times10minus5004 01591649208 01591193346 4558630times10minus5005 01989291693 01988402037 8896560times10minus5006 02386754506 02385218656 1535849times10minus4007 02784001917 02781565807 2436109times10minus4008 03180998309 03177366650 3631659times10minus4009 03577708202 03572544980 5163222times10minus401 03974096254 03967025322 7070931times10minus4MSE 8825264703times10minus7

004000000000000000000000000000000000000000000000000000000000000000000000 4

03

02

01

0

02

04

06

081

y

x t

006

008

010

0

Figure 1Numerical solution for (22) usingVIMBwhenn=m=2 andt=01

where k is constant and R(x t) is a given function Thusapplying the Bernstein approximation when n=m=8

1199100 = 119871119899 (119909) 119909 = 01198901199100 = 000000444811990581199098 + 000064506811990581199092


004000 0000000 0000 00000 00000000 00 000000 00000 00 00 00000000 00000 0000000000000000000000000000 00 0000000444444444444444444444

0020

0204

06

081

x

00003

00002

00001

Error

0

006008

010

t

Figure 2 Absolute error for (22) using VIMB when n=m=2 andt=01

minus 000004780811990581199097 minus 0000234661199058119909minus 00010031411990581199093 + 0000965711990581199094minus 000058958411990581199095 + 0000223202411990581199096+ 00040125611990571199093 + 00009388641199057119909minus 000258027211990571199092 minus 0003862811990571199094 + sdot sdot sdot

119890minus1199100 = 3046076038 + 0000001144211990581199098+ 0000597149611990581199092 minus 0000015325611990581199097minus 000026115361199058119909 minus 0000763419211990581199093+ 000059795411990581199094 minus 0000294511211990581199095+ 0000089345611990581199096 + 0003053676811990571199093+ 000104461441199057119909 minus 0002388598411990571199092minus 000239181611990571199094 + sdot sdot sdot

(29)

and the variational iteration method on (28) with 120582 = minus1k = 1 R(x t) = (1 + x + t) and 119905 isin [0 1] 119909 isin [1 2]

119910119899+1 (119909 119905)= 119910119899 (119909 119905)minus int1199050(minus119890119910119910119909 + (119910119909)2 + 119910119909119909 + 119890minus119910 (1 + 119909 + 119905)) 119889119904

1199100 = 119871119899 (119909)

(30)

1199101= 1119909 (1 times 10minus16) minus 410577777810101199059 + 488162times 101011990510119909 + 893456 times 1010119905101199097 minus 153256

Table 3The absolute andmean square errors using y1 for (28) whenn=m=8 and 119905 = 001X VIM-Berns Exact |VIMB-Exact|11 0104613190 0104360015 2531755times10minus412 0190945182 0190620359 3248227times10minus413 0270371291 0270027137 3441544times10minus414 0343920261 0343589704 3305573times10minus415 0412406467 0412109650 2968162times10minus416 0476485538 0476234179 2513593times10minus417 0536693084 0536493370 1997142times10minus418 0593472312 0593326845 1454676times10minus419 0647194150 0647103242 9090796times10minus520 0698172179 0698134722 3745730times10minus5MSE 6193132times10minus8

times 1010119905101199098 + 11442 times 109119905101199099 minus 2945112times 1011119905101199096 minus 2611536 times 10111199051011990925971496times 1011119905101199093 minus 7634192 times 1011119905101199094 + 597954times 1011119905101199095 + sdot sdot sdot

(31)

the exact solution for (28) is as follows

y (x t) = Ln (x + t) (32)

Tables 3-4 present the absolute error and mean squareerror of VIM with modified Bernstein polynomial whenn=m=8 and 119905 = 001 in Table 3 and n=m=2 and 119909 = 2in Table 4 The absolute errors generated using the modifiedBernstein polynomial are of 10minus4Example 3 The inhomogeneous nonlinear equation (28)with initial condition [23]

119910119905 = 119865 (119909 119905) minus 119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909+ 119890minus119910119877 (119909 119905)

119910 (119909 0) = 1(33)

where 119865(119909 119905) = minus2119905 + 1199092(1 minus 41199052) minus 119890minus(1+1199051199092) + 21199051199091198901+1199051199092 and119877(119909 119905) = 1 is a given function k=1

Using the Bernstein approximation when n=m=2 we getthe following

1198650 (119909 119905) = 051199092 + 051199092 + 5333341592119909119905+ 19195351101199051199092 minus 018990639001199052119909+ 394768604611990521199092 minus 2 times 10minus101199052minus 03678794412 minus 2119905


Table 4The absolute andmean square errors using y3 for (28) whenn=m=2 and 119909 = 2t VIM-Berns Exact |VIMB-Exact|001 06981473299 06981347221 1260780times10minus5002 07031437324 07030975114 4622100times10minus5003 07081301493 07080357931 9435620times10minus5004 07131001195 07129498079 1503116times10minus4005 07180469602 07178397932 2071670times10minus4006 07229637696 07227059828 2577866times10minus4007 07278434274 07275486073 2948201times10minus4008 07326785978 07323678937 3107041times10minus4009 07374617300 07371640660 2976640times10minus401 07421850610 07419373447 2477163times10minus4MSE 2330795times10minus8

1199101 (119909 119905) = 1 minus 0063302129991199053119909 + 131589534911990531199092minus 6666666666 times 10minus111199053+ 26666707961199052119909 + 0959767555011990521199092minus 11199052 minus 232 times 10minus8119905 + 051199051199092 + 05119909119905

(34)

1198901199101 = 2718282 + 7241711382119909119905+ 87373913841199051199092 + 96504550351199052119909+ 330833462711990521199092 minus 051571820761199052minus 1202563816119905

119890minus1199101 = 1 times 10minus101199092 + 0367879418 minus 1 times 10minus101199092+ 03678794180 minus 1 times 10minus10119909minus 08709878762119909119905+ +012414624211199051199092minus 091728420351199052119909+ 0666923305811990521199092 + 042314637031199052+ 02089742357119905


1199102 (119909 119905) = 1 + 050000000011199051199092 + 0499999999119909119905minus 25811067351199053119909 minus 199859924011990531199092+ 08720358581199052119909 minus 13515999461199053



minus 00475992401199057119909 + 0989474611511990571199092+ 26268957021199056119909 + 0941352393911990561199092minus 145114071611990561199093 minus 16757225011199055119909minus 182061139511990551199092 minus 131608286511990551199093minus 53033696701199054119909 minus 521071454611990541199092minus 869012980611990541199093 minus 291246379411990531199093+ 000057245138041199057minus 0061709657301199056 + 16693913671199055+ 22338088621199054 + 102184067611990521199092minus 10750833821199052 minus 232 times 10minus8119905

(35)

The exact solution is 119910(119909 119905) = 1 + 1199051199092Tables 5-6 present the absolute error and mean square

error of VIM with modified Bernstein polynomial whenn=m=2 and 119905 = 001 in Table 5 andwhen n=m=3 and 119905 = 001


x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t

Figure 3 Numerical solution for (28) using VIMB when n=m=8and 119905 = 001

00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010

Figure 4 Absolute error for (28) using VIMB when n=m=8 and119905 = 001

in Table 6 The absolute errors generated using the modifiedBernstein polynomial are of 10minus3

These examples are solved by the VIMwith the Bernsteinpolynomials on the bound region 119886 le 119909 le 119887 and 119888 le 119905 le 119889The obtained results are listed in Tables 1ndash6 and Figures 1ndash6at 119905 = 01 and 119905 = 001 The results that were obtainedshowed that VIMwith the Bernstein approximation solutionsconverges to the exact solution with less iterations It isobvious that the approximate solutions foundwhen using ourproposed method are significantly accurate when increasingthe number of iterations within the smallest value of timeWe notice that the solution becomes faster by transformingthe transcendent functions which are difficult to integrate insome cases into a series of polynomials where the solutionbecomes simpler and faster

6 Conclusions

VIM is more reliable with Bernstein polynomial approxima-tion when compared to dependent variable transcendentalfunctions in differential equations Comparing the aboveillustrated numerical results as shown in tables and figures

x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

00008

00006

00004

00002


we conclude that VIMB provides reliably accurate results thatare characterized by stability for all 119909 at different values of119905 The proposed method can be also applied to other differ-ential equations that may include trigonometric functions ofdependent variables All the computations were carried outwith the aid of the Maple 13 software The VIM with theBernstein polynomials used in this paper was applied directlywithout resorting to linearization or any kind of confiningassumptions and was successful in determining approximateanalytic solutions of nonlinear PDEs Comparing numericalresults of VIM with the Bernstein polynomial and the exactsolutions proves power ofVIMas amathematical tool to solvenonlinear PDEs and with results that converge rapidly to theexact solution

Data Availability

The data used to support the findings of this study areavailable from the corresponding authors upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper


Acknowledgments

The research is supported by College of Computer Sciencesand Mathematics University of Mosul Republic of Iraqunder Project no 21558430

References

[1] M M Aziz and S F AL-Azzawi ldquoHybrid chaos synchro-nization between two different hyperchaotic systems via twoapproachesrdquoOptik - International Journal for Light and ElectronOptics vol 138 pp 328ndash340 2017

[2] A M Siddiqui and M Hameed ldquoAdomian decompositionmethod applied to study nonlinear equations arising in non-Newtonian flowsrdquoAppliedMathematical Sciences vol 6 no 97-100 pp 4889ndash4909 2012

[3] A F Qasim and E S AL-Rawi ldquoAdomian decompositionmethod with modified Bernstein polynomials for solving ordi-nary and partial differential equationsrdquo in Journal of AppliedMathematics vol 2018 p 9 2018

[4] L Cveticanin ldquoHomotopyndashperturbation method for pure non-linear differential equationrdquo Chaos Solitons amp Fractals no 6pp 4787ndash4800 2012

[5] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied Mathematicsand Computation vol 208 no 1 pp 434ndash439 2009

[6] Z Odibat and S Momani ldquoModified homotopy perturbationmethod application to quadratic Riccati differential equationof fractional orderrdquo Chaos Solitons amp Fractals vol 36 no 1 pp167ndash174 2008

[7] Y Do and B Jang ldquoNonlinear Klein-Gordon and schrodingerequations by the projected differential transform methodrdquoAbstract and Applied Analysis vol 2012 Article ID 150527 2012

[8] T M Elzaki ldquoSolution of nonlinear differential equationsusing mixture of Elzaki transform and differential transformmethodrdquo International Mathematical Forum Journal foreoryand Applications vol 7 no 13-16 pp 631ndash638 2012

[9] H Taghvafard and G H Erjaee ldquoTwo-dimensional differentialtransform method for solving linear and non-linear Goursatproblemrdquo World Academy of Science Engineering and Technol-ogy vol 39 2009

[10] J H He G C Wu and F Austin ldquoThe variational iterationmethod which should be followedrdquoNonlinear Science Letters AMathematics Physics andMechanics vol 1 no 1 pp 1ndash30 2010

[11] B Batiha ldquoA numeric-analytic method for approximatingquadratic Riccati differential equationrdquo International Journal ofApplied Mathematical Research vol 1 no 1 pp 8ndash16 2012

[12] J H He ldquoA variational iteration approach to nonlinear prob-lems and its applicationsrdquoMechanical Application vol 20 no 1pp 30-31 1998

[13] E H Doha A H Bhrawy and M A Saker ldquoIntegrals ofBernstein polynomials an application for the solution of higheven-order differential equationsrdquoAppliedMathematics Lettersvol 24 no 4 pp 559ndash565 2011

[14] Y Khan F Naeem and Z Smarda ldquoA novel iterative schemeand its application to differential equationsrdquoeScientificWorldJournal vol 2014 Article ID 605376 4 pages 2014

[15] M O Olayiwola F O Akinpelu and A W Gbolagade ldquoMod-ified variational iteration method for the solution of nonlinearpartial differential equationsrdquo International Journal of Scientificamp Engineering Research vol 2 no 10 2011

[16] M Abolhasani S Abbasbady and T Allahviranloo ldquoA newvariational iterationmethod for a class of fractional convection-diffusion equations in large domainsrdquo Mathematics vol 5 no2 article no 26 2017

[17] U Saeed ldquoModified variational iteration method for sine-gordon equationrdquo TEM Journal vol 5 no 3 pp 305ndash312 2016

[18] J A Aniqa ldquoThe variational iteration method on K-S equationusing hersquos polynomialsrdquo Journal of Science and Arts vol 3 no40 pp 443ndash448 2017

[19] A S J AL-Saif and T A K Hattim ldquoVariational iterationmethod for solving some models of nonlinear partial differen-tial equationsrdquo International Journal of Pure and Applied Scienceand Technology vol 4 no 1 pp 30ndash40 2011

[20] F L Martinez ldquoSome properties of two-dimensional Bernsteinpolynomialsrdquo Journal of Approximation eory vol 59 no 3pp 300ndash306 1989

[21] M Tatari and M Dehghan ldquoOn the convergence of Hersquosvariational iteration methodrdquo Journal of Computational andApplied Mathematics vol 207 no 1 pp 121ndash128 2007

[22] A Ilrsquoinskii and S Ostrovska ldquoConvergence of generalizedBernstein polynomialsrdquo Journal of Approximation eory vol116 no 1 pp 100ndash112 2002

[23] D Kaya ldquoThe use of Adomian decomposition method for solv-ing a specific nonlinear partial differential equationsrdquo Bulletinof the Belgian Mathematical Society - Simon Stevin vol 9 no 3pp 343ndash349 2002

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nonlinear terms The solutions obtained using our modi-fied method are comparable to VIM In our approach weapplied the Bernstein polynomial approximations to non-linear functions in the correction functions before iteratingthe numerical solution of PDEs The main objective is toachieve reliable accurate PDEs solutions if using VIM leadsto unstable solutions The feasibility of our approach isexpounded through a number of examples

The rest of the paper is divided into five sections Section 2introduces brief ideas of VIM Section 3 presents one- andtwo-dimensional Bernstein polynomials In Section 4 ananalysis of the convergence is presented Section 5 elucidatesthree examples that expound the feasibility of using VIMwith the Bernstein polynomials Section 6 is dedicated for theconclusions

2 VIM Basics

The concept of VIM is based on the general Lagrangersquosmultiplier method The key advantage of this method isreaching a mathematical problem solution by linearizationassumption to be used as an initial approximation thatpromptly converges to an exact solution [19]

The following nonlinear differential equation elucidatesthe basic concept of the VIM as follows

L (y) +N (y) = g (x) (1)

where L and N are the linear and nonlinear operatorsrespectively while g(x) is an inhomogeneous term VIM isused to formalize the following correction function

119910119899+1 (119909 119905)= 119910119899 (119909 119905)+ int1199050120582 (120591) [119871 (119910119899 (119909 119905)) + 119873 (119910119899 (119909 119905)) minus 119892 (119909)] 119889120591

(2)

where 120582(120591) is a general Lagrangian multiplier which can bedetermined through the variational theory and integration byparts yn represents a restricted variation (ie 120575 yn = 0) andthe subscript n symbolizes the nth-order approximation

The Lagrange multiplier 120582(120591) is optimally determinedthrough integration by parts The consecutive approxima-tions yn+1 n ge 0 of the solution yn(x t) are determinedusingthe Lagrange multiplier from the first step using any selectivefunction y0 Thus the solution will be as follows

119910 (119909 119905) = lim119899997888rarrinfin

119910119899 (3)

3 Bernstein Polynomials

Polynomials are considered easily defined calculated differ-entiated and integrated mathematical tools The Bernsteinbased polynomials are tested to approximate the functionsas they result in improved approximations to a function witha few terms This feature has led to their widespread use inappliedmathematics physics and computer based geometric

designs combined with other methods like Galerkin andcollocation to solve both differential and integral equations

The nth degree Bernstein Polynomials form a completebasis over [0 1] formalized as

119861119894119899 (119909) = (119899119894) 119909119894 (1 minus 119909)119899minus119894 0 le 119894 le 119899 (4)

in which ( ni ) = ni(n minus 1) represent the binomialcoefficients

Based on [13] we use the generalized Bernstein polyno-mials of nth degree which form a complete basis over [a b] asfollows

119861119894119899 (119909) = 1(119887 minus 119886)119899 (

119899119894) (119909 minus 119886)

119894 (119887 minus 119909)119899minus119894 0 le 119894 le 119899 (5)

If the function 119891 of two real variables over the intervalsquare are

119878 [119886 119887] times [119888 119889] (6)

then the two-variable Bernstein polynomial of (119899119898)degree equivalent to the function 119891 is determined in thefollowing formula [20]

(119861119899119898119891) (119909 119905) =119899sum119894=0

119898sum119896=0

119891( 119894119899 119896119898)119875119894119899 (119909) 119875119896119898 (119905) (7)

where

119875119895119904 (119911) = (119904119895) 119911119895 (1 minus 119911)119904minus119895 (8)

We use (7) to convert the transcendental functions(trigonometric exponential and Logarithm functions) thatappear in homogenous and nonhomogenous PDEs such asthe sine-Gordon equation into the series approximation bymeans of Bernstein polynomial

The following nonlinear partial differential equation con-tains the transcendental functions term as follows

L (y) + N (y) + T (y) = g (x) (9)

such that 119879(119910) is the transcendental functionsUsing (2) we have the following

119910119899+1 (119909 119905) = 119910119899 (119909 119905) + int119905

0120582 (120591) [119871 (119910119899 (119909 119905))

+ 119873 (119910119899 (119909 119905)) + 119879 (119910119899 (119909 119905)) minus 119892 (119909)] 119889120591(10)

Therefore we approach the transcendent functions119879(119910119899(119909 119905)using Bernsteinrsquos approximation because it can only beintegrated by numerical methods

119879 (119910119899 (119909 119905)) = 119861119894119895119910119899 (119909 119905) (11)




Now let

119865(119910 119879 (119910119899) 120597119910120597119905 120597119910120597119909 12059721199101205971199092

12059721199101205971199052

1205972119910120597119909120597119905) = 119892 (119909) (12)


119910119899+1 (119909 119905) = 119910119899 (119909 119905) + int119905

0120582[119865(119910119899 119861119899119898 (119910119899) 120597119910119899120597120591

120597119910119899120597119909 12059721199101198991205971199092

12059721199101198991205971205912 1205972119910119899120597119909120597120591) minus 119892 (119909)] 119889120591

(13)


120575119910119899+1 (119909 119905) = 120575119910119899 (119909 119905) + 120575int119905

0120582[119865(119910119899 119861119899119898 (119910119899)

120597119910119899120597120591 120597119910119899120597119909

12059721199101198991205971199092 12059721199101198991205971205912

1205972119910119899120597119909120597120591) minus 119892 (119909)] 119889120591(14)



119860 119883 997888rarr 119883 (16)



119910119899+1 = 119860 [119910119899] (18)



120583119895 (19)


119860 [119910]= 119910 (119909 119905)+ int1199050120582[119865(119910 120597119910120597120591

120597119910120597119909 12059721199101205971199092

12059721199101205971205912

1205972119910120597119909120597120591)]119889120591

(20)



119860 [119910] = 119910 (119909 119905)+ int1199050120582[119865(119910 119861119899119898 (119910) 120597119910120597120591

120597119910120597119909 12059721199101205971199092

12059721199101205971205912

1205972119910120597119909120597120591)

minus 119892 (119909)] 119889120591(21)





119910119905119905 minus 119910119909119909 + sin (119910) = 0 119905 gt 0119910 (119909 0) = 0119910119905 (119909 0) = 8

119890119909 + 119890minus119909 (22)


119910 (119909 119905) = 4 arctan( 2119905119890119909 + 119890minus119909 ) (23)


1199100 = 8119905119890119909 + 119890minus119909

sin (1199100) = 1818594854119905 + 02808027720119909minus 017437779301199051199092 minus 25753973491199052+ 044350969581199052119909+ 0729022702611990521199092

(24)



119910119899+1 (119909 119905) = 119910119899 + int119905

0(119904 minus 119905) (119910119904119904 minus 119910119909119909 + sin (1199100)) 119889119904

119899 ge 1(25)


120582 (s) = s minus t (26)


1199100 = 8119905119890119909 + 119890minus119909





(27)



119910119905 = minus119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909 + 119890minus119910119877 (119909 119905) 119910 (119909 0) = 119892 (119909) (28)



004000000000000000000000000000000000000000000000000000000000000000000000 4

03

02

01

0

02

04

06

081

y

x t

006

008

010

0



1199100 = 119871119899 (119909) 119909 = 01198901199100 = 000000444811990581199098 + 000064506811990581199092


004000 0000000 0000 00000 00000000 00 000000 00000 00 00 00000000 00000 0000000000000000000000000000 00 0000000444444444444444444444

0020

0204

06

081

x

00003

00002

00001

Error

0

006008

010

t




(29)



1199100 = 119871119899 (119909)

(30)




(31)


y (x t) = Ln (x + t) (32)


119910119905 = 119865 (119909 119905) minus 119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909+ 119890minus119910119877 (119909 119905)

119910 (119909 0) = 1(33)







(34)

1198901199101 = 2718282 + 7241711382119909119905+ 87373913841199051199092 + 96504550351199052119909+ 330833462711990521199092 minus 051571820761199052minus 1202563816119905







(35)




x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t


00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010




6 Conclusions


x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

00008

00006

00004

00002



Data Availability





Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











Now let

119865(119910 119879 (119910119899) 120597119910120597119905 120597119910120597119909 12059721199101205971199092

12059721199101205971199052

1205972119910120597119909120597119905) = 119892 (119909) (12)


119910119899+1 (119909 119905) = 119910119899 (119909 119905) + int119905

0120582[119865(119910119899 119861119899119898 (119910119899) 120597119910119899120597120591

120597119910119899120597119909 12059721199101198991205971199092

12059721199101198991205971205912 1205972119910119899120597119909120597120591) minus 119892 (119909)] 119889120591

(13)


120575119910119899+1 (119909 119905) = 120575119910119899 (119909 119905) + 120575int119905

0120582[119865(119910119899 119861119899119898 (119910119899)

120597119910119899120597120591 120597119910119899120597119909

12059721199101198991205971199092 12059721199101198991205971205912

1205972119910119899120597119909120597120591) minus 119892 (119909)] 119889120591(14)



119860 119883 997888rarr 119883 (16)



119910119899+1 = 119860 [119910119899] (18)



120583119895 (19)


119860 [119910]= 119910 (119909 119905)+ int1199050120582[119865(119910 120597119910120597120591

120597119910120597119909 12059721199101205971199092

12059721199101205971205912

1205972119910120597119909120597120591)]119889120591

(20)



119860 [119910] = 119910 (119909 119905)+ int1199050120582[119865(119910 119861119899119898 (119910) 120597119910120597120591

120597119910120597119909 12059721199101205971199092

12059721199101205971205912

1205972119910120597119909120597120591)

minus 119892 (119909)] 119889120591(21)





119910119905119905 minus 119910119909119909 + sin (119910) = 0 119905 gt 0119910 (119909 0) = 0119910119905 (119909 0) = 8

119890119909 + 119890minus119909 (22)


119910 (119909 119905) = 4 arctan( 2119905119890119909 + 119890minus119909 ) (23)


1199100 = 8119905119890119909 + 119890minus119909

sin (1199100) = 1818594854119905 + 02808027720119909minus 017437779301199051199092 minus 25753973491199052+ 044350969581199052119909+ 0729022702611990521199092

(24)



119910119899+1 (119909 119905) = 119910119899 + int119905

0(119904 minus 119905) (119910119904119904 minus 119910119909119909 + sin (1199100)) 119889119904

119899 ge 1(25)


120582 (s) = s minus t (26)


1199100 = 8119905119890119909 + 119890minus119909





(27)



119910119905 = minus119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909 + 119890minus119910119877 (119909 119905) 119910 (119909 0) = 119892 (119909) (28)



004000000000000000000000000000000000000000000000000000000000000000000000 4

03

02

01

0

02

04

06

081

y

x t

006

008

010

0



1199100 = 119871119899 (119909) 119909 = 01198901199100 = 000000444811990581199098 + 000064506811990581199092


004000 0000000 0000 00000 00000000 00 000000 00000 00 00 00000000 00000 0000000000000000000000000000 00 0000000444444444444444444444

0020

0204

06

081

x

00003

00002

00001

Error

0

006008

010

t




(29)



1199100 = 119871119899 (119909)

(30)




(31)


y (x t) = Ln (x + t) (32)


119910119905 = 119865 (119909 119905) minus 119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909+ 119890minus119910119877 (119909 119905)

119910 (119909 0) = 1(33)







(34)

1198901199101 = 2718282 + 7241711382119909119905+ 87373913841199051199092 + 96504550351199052119909+ 330833462711990521199092 minus 051571820761199052minus 1202563816119905







(35)




x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t


00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010




6 Conclusions


x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

00008

00006

00004

00002



Data Availability





Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018










119910119899+1 (119909 119905) = 119910119899 + int119905

0(119904 minus 119905) (119910119904119904 minus 119910119909119909 + sin (1199100)) 119889119904

119899 ge 1(25)


120582 (s) = s minus t (26)


1199100 = 8119905119890119909 + 119890minus119909





(27)



119910119905 = minus119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909 + 119890minus119910119877 (119909 119905) 119910 (119909 0) = 119892 (119909) (28)



004000000000000000000000000000000000000000000000000000000000000000000000 4

03

02

01

0

02

04

06

081

y

x t

006

008

010

0



1199100 = 119871119899 (119909) 119909 = 01198901199100 = 000000444811990581199098 + 000064506811990581199092


004000 0000000 0000 00000 00000000 00 000000 00000 00 00 00000000 00000 0000000000000000000000000000 00 0000000444444444444444444444

0020

0204

06

081

x

00003

00002

00001

Error

0

006008

010

t




(29)



1199100 = 119871119899 (119909)

(30)




(31)


y (x t) = Ln (x + t) (32)


119910119905 = 119865 (119909 119905) minus 119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909+ 119890minus119910119877 (119909 119905)

119910 (119909 0) = 1(33)







(34)

1198901199101 = 2718282 + 7241711382119909119905+ 87373913841199051199092 + 96504550351199052119909+ 330833462711990521199092 minus 051571820761199052minus 1202563816119905







(35)




x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t


00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010




6 Conclusions


x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

00008

00006

00004

00002



Data Availability





Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









004000 0000000 0000 00000 00000000 00 000000 00000 00 00 00000000 00000 0000000000000000000000000000 00 0000000444444444444444444444

0020

0204

06

081

x

00003

00002

00001

Error

0

006008

010

t




(29)



1199100 = 119871119899 (119909)

(30)




(31)


y (x t) = Ln (x + t) (32)


119910119905 = 119865 (119909 119905) minus 119890119910119910119909 + 119896 (119910119909)2 + 119896119910119909119909+ 119890minus119910119877 (119909 119905)

119910 (119909 0) = 1(33)







(34)

1198901199101 = 2718282 + 7241711382119909119905+ 87373913841199051199092 + 96504550351199052119909+ 330833462711990521199092 minus 051571820761199052minus 1202563816119905







(35)




x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t


00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010




6 Conclusions


x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

00008

00006

00004

00002



Data Availability





Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018











(34)

1198901199101 = 2718282 + 7241711382119909119905+ 87373913841199051199092 + 96504550351199052119909+ 330833462711990521199092 minus 051571820761199052minus 1202563816119905







(35)




x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t


00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010




6 Conclusions


x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

00008

00006

00004

00002



Data Availability





Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









x

y

06

05

04

03

02

01

0

0002

0004

0006

0008

0010

218

16 14t


00040000000000000000000000000000 000000000 00000000000 000 00 00000000 044444444444

x

Error

218

16

1412

t

00003

00002

00001

00

00060008

0010




6 Conclusions


x

Y

06

08

04

020

0002

0004

0006

0008

00101

t

00

1008

1006

1004

1002


x

Error

0608

0402

0002

1t

00040006

00080010

x

Error

0666666666608

0402

0000000000000000000000000000000000000000000000002002000020002020000202000200000000000

1t

0004000666666666

0008001

0

00012

00010

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Data Availability





Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018









Acknowledgments


References































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








treating transcendental functions in partial differential...

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