Commun Nonlinear Sci Numer Simulat

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Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: wwwelseviercom/locate/cnsns A note on the use of

abstract Article history: eceived 3 August 7 eceived in revised form January Accepted 7 April Available online xxxx PACS: 6 x 6Lj This paper extends an earlier work [Hosseini MM, Nasabzadeh H

1 Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: wwwelseviercom/locate/cnsns A note on the use of Adomian decomposition method for high-order system of nonlinear differential equations MM Hosseini *, M Jafari Department of Mathematics, Yazd University, PO Box , Yazd, Iran article info abstract Article history: eceived 3 August 7 eceived in revised form January Accepted 7 April Available online xxxx PACS: 6 x 6Lj This paper extends an earlier work [Hosseini MM, Nasabzadeh H Modified Adomian decomposition method for specific second order ordinary differential equations Appl Math Comput 7;6:7 3] to high order system of differential equations Solution of these problems is considered by proposed modification of Adomian decomposition method Furthermore, with providing some examples, the aforementioned cases are dealt with numerically Ó Published by Elsevier BV Keywords: Adomian decomposition method Singular initial value problems Nonlinear ordinary differential equations Introduction It is well known that the Adomian decomposition method ADM) its modifications [3 5,7,, 5] are efficient methods to solve linear nonlinear ODEs, DAEs, PDEs, SDEs, integral equations integro-integral equations The ADM has been applied to a wide class of problems in physics, biology chemical reaction The method provides the solution in a rapid convergent series with computable terms It is the purpose of this paper to introduce a new reliable modification of ADM For this reason, a new differential operator is defined which can be used for high-order system of differential equations In this manner, some examples are illustrated to show the advantages of using the proposed method to solve the initial value problems Modified ADM to solve high-order system of differential equations Consider the initial value problem in the n-order differential equation in the form: y ðnþ þ PðxÞy ðn Þ þ Ny ¼ gðxþ; yðþ ¼a ; y ðþ ¼a ; ; y ðn Þ ðþ ¼a n ; ðþ where N is a nonlinear differential operator of order less than n, PðxÞ gðxþ are given functions a ; a ; ; a n are given constants * Corresponding author address: hosse_m@yazduniacir MM Hosseini) 7-574/$ - see front matter Ó Published by Elsevier BV doi:6/jcnsns44, Commun Nonlinear Sci Numer Simulat ), doi:6/jcnsns44

2 MM Hosseini, M Jafari / Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx Here, we propose the new differential operator, as below:! L ¼ e PðxÞdx d dx e PðxÞdx d n : ðþ d n x So, the problem ) can be written as Ly ¼ gðxþ Ny: The inverse operator L is therefore considered a n-fold integral operator, as below: Z x Z x Z x Z x L ðþ ¼ e PðxÞdx e PðxÞdx ðþdx dx: By operating L on 3), we have ð3þ ð4þ such that yðxþ ¼UðxÞþL gðxþ L ðnyþ; LUðxÞ ¼: We write Ny ¼ P þ n¼ A n y ¼ P þ n¼ y n where the components of A n are the so-called Adomian polynomials, for each i, A i depends on y ; y ; ; y i only Now by considering 5), we have X n¼ y n ¼ UðxÞþL X gðxþ L n¼ A n : Through using Adomian decomposition method, the components y n ðxþ can be determined as y ¼ UðxÞþL gðxþ; y nþ ¼ L A n ; n P : If the series converges in a suitable way, then it can be seen y ¼ lim M!þ W MðxÞ; where W M ¼ P M n¼ y n Now an expression for the A i is required Specific algorithms were seen in [9,] to formulate Adomian polynomials The theoretical treatment of the convergence of ADM has been considered in [,,6,] The mentioned method can be used for solving system of differential equation in the following form see Example 3): y ðnþ þ pðxþy ðn Þ þ F ðx; y ; ; y ðn Þ ; y ; ; y ðn Þ ; ; y n ; ; y ðn Þ n Þ¼g ðxþ; >< y ðnþ þ pðxþy ðn Þ þ F ðx; y ; ; y ðn Þ ; y ; ; y ðn Þ ; ; y n ; ; y ðn Þ n Þ¼g ðxþ; y ðnþ n þ pðxþy ðn Þ n þ F n ðx; y ; ; y ðn Þ ; y ; ; y ðn Þ ; ; y n ; ; y ðn Þ n Þ¼g n ðxþ: To perform the ADM, in general, for an arbitrary natural number, m, gðxþ, pðxþ, e pðxþdx e pðxþdx are expressed in Taylor series, at x ¼ 3 Test problems In this section, nonsingular singular 3-order ODEs singular system of ODEs are considered these problems are solved by stard modified ADM which is presented in Section The algorithms are performed by Maple Example Consider the nonlinear initial value problem ð5þ ð6þ ð7þ y þ e x y þ 4x y þ x y 3 ¼ gðxþ; yðþ ¼y ðþ ¼y ðþ ¼; ðþ where gðxþ is compatible to exact solution yðxþ ¼x 3 e x : Here, we use Taylor series of gðxþ with order 9, gðxþ 6 þ 3x þ 4x þ 45x 3 þ 7 4 x4 þ 64 5 x5 þ 59 3 x6 þ x7 þ 344 x : ð9þ, Commun Nonlinear Sci Numer Simulat ), doi:6/jcnsns44

3 MM Hosseini, M Jafari / Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx 3 Stard Adomian decomposition method: we put LðÞ ¼ d3 dx 3 ðþ; so Z x Z x L ðþ ¼ Z x ðþdxdxdx: In an operator form, Eq ) becomes Ly ¼ e x y 4x y x y 3 þ gðxþ: ðþ By applying L to both sides of ) we have y ¼ yðþþxy ðþþ x y ðþ L ðe x y Þ 4L ðx y Þ L ðx y 3 ÞþL ðgðxþþ: Proceeding as before we obtained the recursive relationship y ¼ L ðgðxþþ; y nþ ¼ L ðf ðxþy n Þ 4L ðx y n Þ L ðx A n Þ; n P ; ðþ where f ðxþ is obtained by using Taylor series of e x at x ¼ A n s are Adomian polynomials of nonlinear term y 3, as below: f ðxþ þ x þ x þþx! A ¼ y 3 ; >< A ¼ 3y y ; A ¼ 3y y þ 3y y ; So, by substituting ) 3) into ), we have y ¼ x 3 þ 5 4 x4 þ 4 5 x5 þ; >< y þ y ¼ x 3 þ x 4 þ 9 x5 þ x6 þ; y þ y þþy 7 ¼ x 3 þ x 4 þ x5 þ 6 x6 þþ 54 x þ x þ Note that, the Taylor series of exact solution 9) with order 9 is as below: yðxþ ¼x 3 þ x 4 þ x5 þ 6 x6 þþ 54 x þ 43 x þ ð5þ It is easy to see that the stard ADM is slowly convergent to exact solution of this problem 3 Modified Adomian decomposition method ðþ ð3þ ð4þ Here, we use such that Z L ðþ ¼ Z x Z x Z pðxþdx ¼ e pðxþdx ¼ e ex e PðxÞdx e x dx ¼ e x ; Z x e PðxÞdx ðþdxdxdx; ð6þ e pðxþdx ¼ e ex :, Commun Nonlinear Sci Numer Simulat ), doi:6/jcnsns44

4 4 MM Hosseini, M Jafari / Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx Now by substituting the Taylor series of e ex e ex with order 9, into 6), we obtain In addition p e ex þ x þ x þ 5 6 x3 þ 5 x4 þ 3 3 x5 þ 3 7 x6 þ x7 þ 3 4 x ; q e ex x þ 6 x3 þ 4 x4 6 x5 x6 56 x7 þ 5 43 x : L ðþ ¼ Z x Z x ðqþ Z x according to 7) we have y ¼ L gðxþ; y nþ ¼ L A n ; n P : ðpþdxdxdx Thus, y ¼ x 3 þ x 4 þ x5 þ 6 x6 þ 3 4 x7 ; y þ y ¼ x 3 þ x 4 þ x5 þ 6 x6 þþ 54 x x þ ð7þ The comparison between 4) 7) shows that the rate of convergence of modified Adomian method is faster than stard Adomian method for this problem Example Consider the nonlinear singular initial value problem y þ x y þ 4xy þ lnðyþ ¼gðxÞ; yðþ ¼; y ðþ ¼; y ðþ ¼ ; ðþ where gðxþ is compatible to exact solution yðxþ ¼ þ x : ð9þ Here, we use Taylor series of gðxþ with order 9, gðxþ 4 x þ 4x 9x x 3 þ 33 x4 þ 44x x6 9x 7 þ 9 4 x : 3 Stard Adomian decomposition method According to example ) it can be easily seen that the Z x Z x Z x y ¼ L ðgðxþþ ¼ 4 x þ 4x 9x x 3 þ 33 x4 þ 44x x6 9x 7 þ 9 4 x dxdxdx is undefined So, the stard ADM can not solve the above problem 33 Modified Adomian decomposition method According to example ) by considering 6), we have A ¼ y lnðy Þ; >< A ¼ y lnðy Þ; A ¼ y lnðy Þþ y ; y Through applying modified Adomian decomposition method to problem ), we obtain: >< y ¼ x þ x 4 9 x5 x 6 þ; y þ y ¼ x þ x 4 x 6 þ x þ; y þ y þ y ¼ x þ x 4 x 6 þ x x 74 x þ; which y þ y þ y is quite close to Taylor expansion of exact solution 9), Commun Nonlinear Sci Numer Simulat ), doi:6/jcnsns44

5 MM Hosseini, M Jafari / Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx 5 Example 3 Consider the nonlinear system of differential equation, y þ tanðxþy þ z ¼ gðxþ; yðþ ¼; y ðþ ¼; z þ z þ y ¼ hðxþ; zðþ ¼; z ðþ ¼; ðþ where gðxþ hðxþ are compatible to exact solutions yðxþ ¼x sinðxþ zðxþ ¼x tanðxþ: ðþ Here, we use Taylor series of gðxþ, hðxþ tanðxþ with order 9 Stard Adomian decomposition method Here, we have y ¼ L ðgðxþþ ¼ x þ 4 x6 þ 3 4 x þ; z ¼ L ðhðxþþ ¼ x þ 3 x3 þ 3 x4 þ 3 x5 þ 6 x6 4 x7 þ 5 x þ ðþ y nþ ¼ L ðf ðxþy n Þ L ða n Þ; n P ; z nþ ¼ L ðz n Þ L ðb n Þ; n P ; ð3þ where A n B n are the Adomian polynomials of nonlinear terms y z Also, f ðxþ denoted the taylor series of tanðxþ with order 9 In this case, through considering ) 3), we have y ¼ x þ 4 x6 þ 3 4 x þ; >< y þ y ¼ x 6 x4 7 x6 63 x7 þ; y þ y þþy 6 ¼ x 6 x4 þ x6 54 x þ x 9 þ z ¼ x þ 3 x3 þ 3 x4 þ 3 x5 þ; >< z þ z ¼ x 33x x6 x7 þ; z þ z þþz 6 ¼ x þ 3 x4 þ 5 x6 þ 7 35 x þ 35 x 9 þ 567 So, the stard Adomian decomposition method converges to Taylor expansion of exact solution ) 34 Modified Adomian decomposition method By applying modified Adomian decomposition method to problem ), we obtain: y ¼ x 6 x4 þ 4 x6 þ 4 54 x þ y þ y ¼ x 6 x4 þ x6 54 x x þ z ¼ x þ 3 x4 þ 6 x6 þ z þ z ¼ x þ 3 x4 þ 5 x6 þ 7 35 x þ x þ; which is quite close to Taylor expansion of exact solution ) 4 Conclusion In this paper, modification of Adomian decomposition method was proposed The merit of this method is that it is more efficient than the stard Adomian decomposition method when high order system of differential equations are given The advantage of using the proposed algorithm of this paper is clearly demonstrated for Examples 3 The obtained results show that the rate of convergence of modified Adomian decomposition method is higher than stard Adomian decomposition method for these problems, Commun Nonlinear Sci Numer Simulat ), doi:6/jcnsns44

6 6 MM Hosseini, M Jafari / Commun Nonlinear Sci Numer Simulat xxx ) xxx xxx eferences [] Abbaoui K, Cherruault Y New ideas for proving convergence of decomposition methods Comput Math Appl 995;9:3 [] Cherruault Y, Adomian G, Abbaoui K, ach Further remarks on convergence of decomposition method Bio-Med Comput 995;3:9 93 [3] Hosseini MM Adomian decomposition method with Chebyshev polynomials Appl Math Comput 6;75:65 93 [4] Hosseini MM Adomian decomposition method for solution of differential-algebraic equations J Comput Appl Math 6;97:495 5 [5] Hosseini MM Adomian decomposition method for solution of nonlinear differential-algebraic equations Appl Math Comput 6;: [6] Hosseini MM, Nasabzadeh H On the convergence of Adomian decomposition method Appl Math Comput 6;: [7] Hosseini MM, Nasabzadeh H Modified Adomian decomposition method for specific second order ordinary differential equations Appl Math Comput 7;6:7 3 [] Lesnic D Convergence of Adomian decomposition method: periodic temperatures Comput Math Appl ;44:3 4 [9] Seng V, Abbaoui K, Cherruault Y Adomian s polynomials for nonlinear operators Math Comput Model 996;4:59 65 [] Wazwaz AM A reliable modification of Adomian decomposition method Appl Math Comput 999;:77 6 [] Wazwaz AM A new algorithm for calculating Adomian polynomials for nonlinear operators Appl Math Comput ;:53 69 [] Wazwaz AM A new method for solving singular initial value problems in the second-order ordinary differential equations Appl Math Comput ;:45 57 [3] Wazwaz AM The existence of noise terms for systems of inhomogeneous decomposition integral equations Appl Math Comput 3;46: 9 [4] Wazwaz AM Adomian decomposition for a reliable treatment of the Emden Fowler equation Appl Math Comput 5;6:543 6 [5] Wazwaz AM, El-Sayed SM A new modification of the Adomian decomposition method for linear nonlinear operators Appl Math Comput ;:393 45, Commun Nonlinear Sci Numer Simulat ), doi:6/jcnsns44

Modified Adomian Decomposition Method for Solving Particular Third-Order Ordinary Differential Equations

Applied Mathematical Sciences, Vol. 6, 212, no. 3, 1463-1469 Modified Adomian Decomposition Method for Solving Particular Third-Order Ordinary Differential Equations P. Pue-on 1 and N. Viriyapong 2 Department