Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.77 (Online Publication: Dec., 2015) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal Analytical solution of system of differential equations by variational iteration method Jamshad Ahmed1*, Faizan Hussain2, 3 1Department of Mathematics, Faculty of Sciences, University of Gujrat, Pakistan 2Department of Mathematics, University of Gujrat, (Sialkot Campus) Pakistan 3Department of Mathematics, NCBA&E, Gujrat (Sub-Campus) Pakistan *Email: jamshadahmadm@gmail.com Article history: Received 11 March, 2015; Accepted 17 September, 2015 DOI: http://dx.doi.org/10.3126/bibechana.v13i0.13394 Abstract In this paper, Varitational Iteration Method using He’s Polynomials is used to construct the exact as well as approximate solutions of differential equations. From the obtained numerical results, it has been observed that this proposed technique is very efficient and reliable for the solution of the linear and non- linear system of differential equations. Numerical results and graphical representation reflect the accuracy and effectiveness of the proposed modification. ©RCOST: All rights reserved. Keywords: Variational interation method; He’s Polynomials; System of linear and non-linear differential equations. 1. Introduction In recent years considerable interest in system of ordinary differential equations has been stimulated due to their numerous applications in the fields of physics and engineering. A huge number of research and investigations have been invested in these directions. In this paper we consider a very efficient and power full technique He’s variational iteration method for finding approximate solutions of systems of differential equations. This technique was first time introduce by the Chinese mathematician He [1]. The variational interaion method is used to investigate autonomous ordinary differential systems [2], Helmholtz equation [3], Burger’s and coupled Burger’s equation [4], Coupled Schrodinger-KdV equations and shallow water equations [4]. Also the application of the present technique to linear fractional partial differential equations arising from fluid mechanics is presented in [5]. The variational iteration method is investigated in [6-10] to solve parabolic integro-differential equations. This method is Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.78 (Online Publication: Dec., 2015) applied to find the solution of various classes of variational problem [11]. Some recent research works in this field are [12-20]. 2. Analysis of VIM for System of Differential Equations In case of m equations, we rewrite equations in the form ,,2,1,)()()( 21 mixfyyyNyL imiii   (1) where iL is a linear with respect to iy and iN is the nonlinear part of ith equation. In this case the correct functional are obtained as                    ,~,,~,~, 0 ,,2,1,1,   x nmnniiiinini dfyyyNyLxxyxy   (2) and the optimal values of ,,,2,1, mii  are obtained by taking the variation from both sides of the functional and finding stationary condition using .,,2,1,01, miy ni  Our goal in the paper is the use of the following system of sequences instead of the system which results from the Variational iteration method:                    ,,,,, 0 ,,2,1,1,   x nmnniiiinini dfyyyNyLxxyxy   (3) For .,,2 mi  In fact the updated values     ,,,, 1,12,21,1  ninn yyy  are used for finding  .1, niy This technique accelerates the convergence of the system of sequences. Therefore, using just few terms of the sequences, an accurate solution can be obtained for a large domain of the problem. 3. Analysis of VIM using He’s Polynomials Variational Iteration Method using He’s polynomials [21] is a modified form of VIM. This modification is obtained by coupling the correction functional of VIM with He's polynomials and is given by                   midfyNpyLpxpxyxyp inini n n nii n n x ii n ni n ,,2,1,~, ,, 0 , 00 0, 0 1,                 By comparing the like powers of p, give solution of various order. 4. Numerical Applications Example 4.1 Consider the system of first order differential equations, ,cos31 xyy  (4) ,32 xeyy  (5) Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.79 (Online Publication: Dec., 2015) ,213 yyy  (6) subjected to the initial conditions   ,101 y   ,002 y   .203 y According to VIM, correction functional for Eq. (4), (5) & (6) can be writ ten as,                                                   ,~~,)()( ,~~,)()( ,~cos~, ,2,1 ,3 0 3,31,3 0 ,3 ,2 2,21,2 0 ,3 ,1 1,11,1            dyy d yd xxyxy dey d yd xxyxy dy d yd xxyxy nn n x nn x n n nn x n n nn (7) The Lagrange multiplier   3,2,1,, ixi  can be identified via variational theory; i.e. the multiplier should be chosen in such a way that the correction functional equation is stationary i.e.       .0&0,0 1,31,21,1   xyxyxy nnn  From Eq. (7)   ,1,1  x   ,1,2  x and   ,1,3  x Thus Eq. (7) becomes           , 1)()( 1)()( cos)1( ,2,1 ,3 0 ,31,3 0 ,3 ,2 ,21,2 0 ,3 ,1 ,11,1                                               dyy d yd xyxy dey d yd xyxy dy d yd xyxy nn n x nn x n n nn x n n nn (8) for ,0n Eq. (8) gives,              ,2 ,1 ,sin1 0,3 0,2 0,1 xy exy xxy x for ,1n Eq. (8) gives, Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.80 (Online Publication: Dec., 2015)                                              ,1)()( ,1)()( ,cos)1( 0,20,1 0,3 0 0,31,3 0 0,3 0,2 0,21,2 0 0,3 0,1 0,11,1         dyy d yd xyxy dey d yd xyxy dy d yd xyxy x x x (9)               , ,sin 2,3 2,2 2,1 x x exCosxy xxy exy Therefore, we get the result as,              , ,sin 3 2 1 x x exCosxy xxy exy This is the same result obtained by ADM in [22]. Example 4.2 Consider the non-linear system of differential equations, ,2 2 2 1 y xd yd  (10) ,1 2 ye xd yd x (11) ,32 3 yy xd yd  (12) subjected to the initial conditions   ,101 y   ,102 y   .003 y The exact solution is given, ,2 1 xey  ,2 xey  and .3 xexy  According to VIM, the correction functional for the Eq. (10), (11) & (12) can be writ ten as Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.81 (Online Publication: Dec., 2015)                                                    ,~~,)()( ,~,)()( ,~2, ,3,2 ,3 0 3,31,3 0 ,1 ,2 2,21,2 0 2 ,2 ,1 1,11,1            dyy d yd xxyxy dye d yd xxyxy dy d yd xxyxy nn n x nn x n n nn x n n nn (13) The Lagrange multiplier   3,2,1,, ixi  can be identified via Variational Theory; i.e. the multiplier should be chosen in such a way that the correction functional equation is stationary i.e.       0&0,0 1,31,21,1   xyxyxy nnn  So from Eq. (13), we get   ,1,1  x   ,1,2  x and   ,1,3  x Thus Eq. (13) becomes                                                      ,1)()( ,1)()( ,21 ,3,2 ,3 0 ,31,3 0 ,1 ,2 ,21,2 0 2 ,2 ,1 ,11,1         dyy d yd xyxy dye d yd xyxy dy d yd xyxy nn n x nn x n n nn x n n nn (14) According to VIMHP, Eq. (14) can be writ ten as                                                                         ,1)()( ,1)()( ,21 0 ,3 0 ,2 0 ,3 0 0,3 0 ,3 0 0 ,1 0 ,2 0,2 0 ,2 0 0 2 ,2 0 ,1 0,1 0 ,1         dypyp d yd ppxyxyp dype d yd ppxyxyp dyp d yd ppxyxyp n n n n n n n nn x n n n x n n n n nn n n n x n n n n nn n n n                                                                     ,1)( ,11)( ,211 0 ,3 0 ,2 0 ,3 00 ,3 0 0 ,1 0 ,2 0 ,2 0 0 2 ,2 0 ,1 0 ,1         dypyp d yd ppxyp dype d yd ppxyp dyp d yd ppxyp n n n n n n n nn x n n n x n n n n nn n n n x n n n n nn n n n (15) Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.82 (Online Publication: Dec., 2015) Now, comparing the co-efficient of like powers of p,         , 0 1 1 : 0,3 0,2 0,1 0         xy xy xy p               , )1( 1 21 : 0 0,30,2 0,3 1,3 0 0,1 0,2 1,2 0 2 0,2 0,1 1,1 1                                            dyy d yd xy dye d yd xy dy d yd xy p x x x       ,1 2 : 1,3 1,2 1,1          xxy exy xxy x         , 2 1 222 444 : 2 2,3 2 2,2 2,1 2               x xexy exexy exxy p x xx x         , 62 222 422 83121015 : 32 2 3,3 2 3,2 2 3,1 3               xx xeexexxy exexy exeexxy p xxx xx xxx Therefore, approximations to the solutions with the five terms are as follows:                               3888.106666.40083333.0111111.05.72312 ,5555.111312111111.0528784 888.1346666.5811111.3156431264 2532 3 32 2 322 1 xxxeexexxy exexexxy xeexxexxy xxx xxx xxx This is same as obtain by ADM in [22]. Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.83 (Online Publication: Dec., 2015) Table1. Numerical values of these solution. ix  ixy1  ixye 1  ixy2  ixye 2  ixy3  ixye 3 0 1.00008 0 1 0 0 0 0.1 1.22132 1.6535E-5 1.10516 2.9323E-6 0.110517 0 0.2 1.49186 5.3375E-5 1.22139 1.1211E-5 0.244275 0 0.3 1.82161 5.9740E-4 1.34974 1.1407E-4 0.404906 5.1165E-5 0.4 2.22249 3.1315E-3 1.49125 5.7298E-4 0.594560 2.7328E-4 0.5 2.70702 1.1341E-2 1.64676 1.9591E-3 0.823372 9.8853E-4 0.6 3.28813 3.2068E-2 1.81686 5.2497E-3 1.090460 2.8076E-3 0.7 3.97860 7.6660E-2 2.00184 1.1909E-2 1.402860 6.7609E-3 0.8 4.79050 6.6253E-1 2.20161 2.3929E-2 1.766030 1.1440E-2 0.9 5.73528 3.1445E-1 2.41576 4.3841E-2 2.185680 2.7962E-2 Graphical representation: Fig. 1: Comparison of exact and approximate solution of ,1y Fig. 2: Comparison of exact and approximate solution of ,2y Exact ______ y1 _ _ _ _ _ 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 Exact ______ y2 _ _ _ _ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.84 (Online Publication: Dec., 2015) Fig. 3: Comparison of exact and approximate solution of y3. Example 4.3 Consider a non-linear ordinary differential equation , 1 3 3 xd yd y xxd yd  (16) subject to the boundary conditions   ,00 y   ,10 y   .20 y Considering    ,1 xyxy     ,2 xyxy  and    ,3 xyxy  we convert Eq. (16) in the system of non- linear of three differential equation of order one, i.e.     ,21 xyxy  (17)    ,32 xyxy  (18)      ,1 313 xyxy x xy  (19) According to VIM, the correction functional for the Eq. (17), (18) & (19) can be writ ten as,                                                   ,~~1 ,)()( ,~,)()( ,~, ,3,1 ,3 0 ,31,3 0 ,3 ,2 ,21,2 0 ,2 ,1 ,11,1           dyy d yd xxyxy dy d yd xxyxy dy d yd xxyxy nn n x nn x n n nn x n n nn (20) Exact: ______ y3 : - - - - - 0.0 0.2 0.4 0.6 0.8 0.0 0.5 1.0 1.5 2.0 Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.85 (Online Publication: Dec., 2015) The Lagrange multiplier   3,2,1,, ixi  can be identified via variational theory. i.e. the multiplier should be chosen in such a way that the correction functional equation is stationary i.e.       0&0,0 1,31,21,1   xyxyxy nnn  So from Eq. (20), we get   ,1,1  x   ,1,2  x and   ,1,3  x Thus Eq. (20) becomes                                                   ,~~1 1)()( ,~1)()( ,~1 ,3,1 ,3 0 ,31,3 0 ,3 ,2 ,21,2 0 ,2 ,1 ,11,1        dyy d yd xyxy dy d yd xyxy dy d yd xyxy nn n x nn x n n nn x n n nn (21) Therefore, we get,              ,2 ,1 ,0 0,3 0,2 0,1 xy xy xy Let r r yyyyy ,12,11,10,1   is a notation for an approximation to the solution with p+1 term. Therefore, some computed approximations are as follows , 3 1 2 3        x xxy , 122 1 32 4        xx xxy , 6062 1 432 5        xxx xxy . , 360180 7 62 1 5432 6         xxxx xxy The closed form solution is   .xexxy  (22) This is the exact solution. Jamshad Ahmed and Faizan Hussain/ BIBECHANA 13 (2016) 77-86: RCOST p.86 (Online Publication: Dec., 2015) 4. 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