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 approximate solution of higher order boundary value problems via variational iteration method Zobia Hamid, Jamshad Ahmad* Department of Mathematics, Faculty of Science, University of Gujrat, Pakistan. *Email: jamshadahmadm@gmail.com Article history: Received 09 October, 2017; Accepted 31 October, 2017 DOI: http://dx.doi.org/10.3126/bibechana.v15i0.18347 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ Abstract In this paper, application of variational iteration method has been successfully extended to obtain approximate solutions of some higher order boundary value problems. We emphasize the power of the method by testing three different mathematical models of distinct orders. The results are obtained by using only little iteration. Keywords: Nonlinear BVP; Variational iteration method; Approximate solution. 1. Introduction The ordinary differential equations (ODE) with variable coefficients appear in many areas of applied sciences. Examples of these equations are Euler equation, Bessel equation and Legendre equation. Moreover, the nonlinear ordinary differential equations with variable coefficients, such as the Doffing equation, the Thomas-Fermi equation, and the Van der Pole equation, have been investigated in the literature. Linear and nonlinear ODEs with variable coefficients play a significant role in applied mathematics, physics, and engineering [1-5]. Researchers were aiming to establish reliable methods capable for solving a large class of linear or nonlinear differential and integral equations without the tangible restrictive assumptions or discretization of the variables. Recently, there has been great development of new powerful methods capable of handling linear and nonlinear equations that overcome most of the classical methods. The Adomian decomposition method, the variational iteration method, and the homotopy perturbation method are examples of the newly developed methods. The variational iteration method, now used by many researchers is capable for handling a large class of linear or nonlinear differential equations. The flexibility and adaptation provided by the method have made it readily applicable to cases where the solution is unknown in advance as is often the case in the applied sciences and engineering. The VIM provides efficient algorithm for analytic approximate solutions and numeric simulations for real-world applications in sciences [6–18]. Unlike the Adomian decomposition method, where computational algorithms are normally used to deal with the nonlinear terms, the VIM does not require the use of restrictive as-assumptions for the nonlinear terms which would complicate the analytic calculations. The VIM approaches linear and nonlinear problems directly in a like manner. The aim of this work is reconfirm the potential and applicability of the proposed method on higher order boundary value problems. Zobia Hamid, Jamshad Ahmad / BIBECHANA 15 (2018) 37-42: RCOST p.37 http://nepjol.info/index.php/BIBECHANA http://dx.doi.org/10.3126/bibechana.v15i0.18347 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ 2. The Analysis of Variational Iteration Method Consider the general differential equation Lu+Nu = g(x) (1) where L and N are linear and nonlinear operators respectively, and g(x) is the source inhomogeneous term. The variational iteration method admits the use of a correction functional for equation (1) in the form ,dt))t(g)t(uN)t(Lu()t()x(u)x(u nn x 0 n1n    (2) where  is a general Lagrange’s multiplier, which can be identified optimally via the variational theory, and 𝑢𝑛̅̅̅̅ as a restricted variation which means 𝛿𝑢𝑛̅̅̅̅ =0. The Lagrange multiplier  is crucial and critical in the method, and it can be a constant or a function. Having  determined, an iteration formula should be used for the determination of the successive approximations 0;)(1  nxun of the solution u(x). The zeroth approximation u0 can be any selective function. However, using the initial values u(0); );0(u and )0(u  are preferably used for the selective zeroth approximation 0u as will be seen later. Consequently, the solution is given by )x(uLim)x(u n n   (3) 3. Numerical Applications Problem 3.1 Consider a fifth order non-linear BVP ( ) ( )( ) ( ),v x iiu x e u x (4) with boundary conditions (0) (0) (0) 1 , (1) (1) 1.u u u u u       (5) The correctional functional is given as ,dt)}t(ue)t(u{)x(u)x(u x 0 )ii( n t)v( nn1n     (6) where ‘λ’ is langrage multiplier, which is identified as 1( 1) ( ) . ( 1) m mt x m         (7) And the initial approximation is ,)(0 xexu  For n=0,the equation (6) gives, ,dt)}t(ue)t(u{)xt( 24 1 )x(u)x(u )ii( 0 t)5( 0 x 0 4 01   ,dt)1e()xt( 24 1 e t x 0 4x   120 x 24 x 6 x 2 x x1 5432  For n=1, ,dt)}t(ue)t(u{)xt( 24 1 )x(u)x(u )ii( 1 t)5( 1 x 0 4 12    ,x 2 21 e56xe21ex3e 6 x x34 2 x3 24 x5 57 2xxx2x 334   Zobia Hamid, Jamshad Ahmad / BIBECHANA 15 (2018) 37-42: RCOST p.38 The approximate solution is 14111310129 118 10 96 8 765 43 2 n x105587471470745608.1x103331866059043833.1x10867510876756987.2 x10441725052108385.2 2628800 x x1099667557319236.2 x0000248016.0 5040 x 720 x 120 x x0416667.0x166667.0 2 x x1)x(u      Fig 1: Comparison of exact and approximate solution. Problem 3.2 Consider a non-linear BVP ( ) 2( ) ( ),vi xu x e u x (8) 1 1 1(0) 1, (0) 1, (0) 1, (1) , (1) , (1) .u u u u e u e u e             (9) The correctional functional is given as ( ) ( ) 1 0 ( ) ( ) { ( ) ( )} , x v x ii n n n nu x u x u t e u t dt      (10) where ‘λ’ is langrage multiplier ,which is calculated by 1 5( 1) ( ) ( ) , ( 1) 5 m mt x t x m             (11) And the initial approximation is ,e)x(u x 0  (12) For n=0, the equation (10) becomes, dt))}t(u(e)t(u{)xt( 120 1 )x(u)x(u )ii( 0 t)vi( o x 0 5 01   , dt)}e(ee{)xt( 120 1 e ttt x 0 5x    ,          5 x 4 x 3 x 2 x x1 5432 , Closed Approximate 2 2 4 6 8 100 200 300 400 500 600 700 Zobia Hamid, Jamshad Ahmad / BIBECHANA 15 (2018) 37-42: RCOST p.39 dt))}t(u(e)t(u{)xt( 120 1 )x(u)x(u )ii( 1 t)vi( 1 x 0 5 12   , 209 20 x x 12 7 x6x 2 69 xe84 24 ex ex 6 7 ex14x127e210 5 432x x4 x3x2x        14111310 129118107 876 543 2 x107297251470745597.1x106059043836.1 x108681750876756987.2x10565052108667.2x10760537557319336.2 40320 x 5040 x 720 x x0083333.0x0416667.0x166667.0 2 x x1)x(u The closed solution of this problem is e-x . Fig.2: Comparison of closed and approximate solution. Problem 3.3 Consider a six order non-linear BVP )x(ueu )ii(x)vi(  (13) with boundary conditions 1)0(u)0(u)0(u )iv(  (14) e)1(u)1(u)1(u )iv(  (15) The correctional functional is given as ,dt)}t(ue)t(u{)x(u)x(u x 0 )ii( n t)vi( nn1n     (16) where ‘λ’ is langrage multiplier, which is identified as ; 5 )xt( )1m( )xt()1( 51mm         (17) And the initial approximation is ,)(0 xexu  (18) For n=0,the equation (17) becomes, ,dt)}e(ee{)xt( 120 1 e)x(u ttt x 0 5x 1   6 4 2 2 4 50 100 150 Closed Approximate Zobia Hamid, Jamshad Ahmad / BIBECHANA 15 (2018) 37-42: RCOST p.40            6 x 5 x 4 x 3 x 2 x x1 65432 ,dt)}t(ue)t(u{)xt( 120 1 )x(u)x(u 1 t)vi( 1 x 0 5 12     4 2 3 2 3 4 5 7 69 127 210 84 14 6 6 24 2 7 1 12 20 x x x x x x e x e xe x e x e x x x x               14111310129 11810796 87 6 5 4 3 2 x101470745597.1x106059043867.1x100876756988.2 x105052109648.2x107557319224.2x107557318996.2 x0000248016.0x000198413.0 720 x x00833333.0 24 x x166667.0 2 x x1)x(u      Fig.3: Comparison of exact and approximate solution. 4. 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