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

Transcription

1 Applied Mathematical Sciences, Vol. 6, 212, no. 3, Modified Adomian Decomposition Method for Solving Particular Third-Order Ordinary Differential Equations P. Pue-on 1 and N. Viriyapong 2 Department of Mathematics, Faculty of Science Mahasarakham University, Mahasarakham, 4415, Thailand 1 prapart.p@msu.ac.th, 2 nhongsit@yahoo.co.uk Abstract This manuscript presents a modification Adomian decomposition method for solving third-order ordinary differential equations. The method can be applied to singular and nonsingular problems. Some examples are illustrated efficiency of the method. Mathematics Subject Classification: 34A34 Keywords: third order ordinary differential equation, modification Adomian decomposition method 1 Introduction Many mathematical modelling which explain natural phenomena are usually formulated in term of nonlinear differential equations, both ordinary and partial. However, most of the method developed in mathematics are used in solving linear differential equation. In recent years a semi-analytical method named Adomian decomposition method proposed by G. Adomian( ) have been attracting the attention of many mathematicians, physicist and engineers. The method has the adventage of converging to the exact solution and can easily handle a wide class of both linear and nonlinear differential and integral equations. There are many literatures developed concerning Adomian decomposition method [1, 4] and the related modification to investigate various sciencetific model [2, 3, 7, 9] The theoretical treatment of the convergence of Adomian decomposition method has been considered in [5, 8] The purpose of this research is to present a new reliable modification of Adomian decomposition method for solving third order ordinary differential

2 1464 P. Pue-on and N. Viriyapong equation. For applying the method to singular and nonsingular problem, a new differential operator will be established. Moreover, some examples are illustrative for demonstrating the advantage of the new method. 2 Modified Adomian decomposition method Let us consider the initial value problem in the third order ordinary differential equation in the form y = p(x)y + f(x, y)+g(x), y() = A, y () = B, y () = C, (1) where f(x, y) is real function, p(x) and g(x) are given functions and A, B and C are arbitrary constants. We propose the new differential operator, as below So, the problem (1) can be written as L = e p(x)dx d ( e p(x)dx d2 ). (2) dx dx 2 Ly = f(x, y)+g(x). (3) Therefore, the inverse operator L 1 is consider a three-fold integral operator, as below x x ( ( L 1 x (.) = e p(x)dx e p(x)dx (.)dx )) dxdx. (4) By applying L 1 on (3), one have y(x) = Φ(x)+L 1 g(x)+l 1 f(x, y) (5) such that LΦ(x) =. The Adomian decomposition method is presented the solution y(x) and nonlinear function f(x, y) by infinite series and y(x) = y n (x) (6) n= f(x, y) = A n (7) n=

3 Modified ADM for solving particular third-order ODEs 1465 where the components y n (x) of the solution y(x) can be determined recurrently. Specific algorithms were seen in [1, 2] to formulate Adomian polynomials. The following algorithm: A = F (u ), A 1 = F (u )u 1, A 2 = F (u )u 2 + 1F (u 2 )u 2 1, A 3 = F (u )u 3 + F (u )u 1 u F (u 3! )u 3 1, (8). can be used to construct Adomian polynomial, when F (u) is a nonlinear function. By substituting (6) and (7) into (5), y n (x) = Φ(x)+L 1 g(x)+l 1 A n. (9) n= n= Through using Adomian decomposition method, the components y n (x) can be determined as y =Φ(x)+L 1 g(x), y 1 = L 1 A, y 2 = L 1 A 1, y 3 = L 1 A 2,. (1) From (8) and (1), we can determine the components y n (x), and hence the series solution of y(x) in (6) can be immediately obtained. For numerical purposes, the n term approximant n Ψ n = y k (x), (11) k= can be used to approximate the exact solution. 3 Illustrative examples In this section, two initial value problems are considered to show the efficiency of the propose method, which can be applied to singular and nonsingular problem. Example 1. Consider linear singular initial value problem in third order ordinary differential equation y + cos x sin x y = sin x cos x, y() = 1, y () = 2, y (12) () =.

4 1466 P. Pue-on and N. Viriyapong Standard Adomian Decomposition Method: Here, we put so In an operator form, Eq. (12) becomes L( ) = d3 dx 3 ( ), x x x ( )dxdxdx. Ly = cos x sin x y + sin x cos x. (13) By applying L 1 on both sides of (13), one obtains y = y() + y ()x + y ()x 2 + L 1( cos x sin x y ) + L 1 (sin x cos x). Proceeding as before we obtained the recursive relationship y = y() + y ()x + L 1 (sin x cos x), y n+1 = L 1( cos x sin x n) y, n. Hence, we have y =1 2x x x x x1 +..., y + y 1 =1 2x x x x x1 +..., y + y 1 + y 2 =1 2x x x x x1 +...,. Note that the Taylor series of the exact solution y(x) =1 2x + x sin2 x with order 1 is as below, y(x) = 1 2x x x x x1 + O(x 11 ). Modified Adomian Decomposition Method. According to (2), one puts So, L( ) = x x In an operator form, Eq. (12) becomes 1 d ( d 2 sin x sin x dx dx 2 ( )). 1 x sin x [ sin x(.)dx]dxdx. Ly = sin x cos x. (14)

5 Modified ADM for solving particular third-order ODEs 1467 Now by applying L 1 on both side of (14), one gets L 1 Ly = x x 1 [ x sin x(sin x cos x)dx ] dxdx. sin x and this implies y(x) = y() + y ()x + x sin2 x, = 1 2x + x sin2 x. Therefore, the exact solution is easily obtained by the proposed method. Example 2. Consider nonlinear initial value problem in third order ordinary differential equation y xy + x 2 y 2 = x sin x cos x + x 2 sin 2 x y() =, y () = 1, y () =, (15) with exact solution y = sin x. Standard Adomian Decomposition Method: we put so In an operator form, Eq.(15) becomes L( ) = d3 dx 3 ( ), x x x ( )dxdxdx. Ly = xy x 2 y 2 + x sin x cos x + x 2 sin 2 x (16) By applying L 1 on both sides of (16), one obtains y = y() + y ()x + y ()x 2 + L 1 (xy ) L 1 (x 2 y 2 )+L 1 (x sin x cos x + x 2 sin 2 x). Proceeding as before we obtained the recursive relationship y = y() + y ()x + L 1 (x sin x cos x + x 2 sin 2 x), y n+1 = L 1 (xy n ) L 1 (A n ), n when A n s are Adomian polynomials of nonlinear term x 2 y 2, as below [6] A = x 2 y 2 A 1 = x 2 (2y y 1 ) A 2 = x 2 (2y y 2 + y1 2) A 3 = x 2 (2y y 3 +2y 1 y 2 ).

6 1468 P. Pue-on and N. Viriyapong Hence, we have y = x 1 6 x x x x x , y + y 1 = x 1 6 x x x x x , y + y 1 + y 2 = x 1 6 x x x x x ,. Note that the Taylor series of the exact solution y(x) = sin x with order 1 is as below sin x = x 1 6 x x x x9 + O(x 11 ). Modified Adomian Decomposition Method. According to (2), we put So, and we have L( ) = e x2 2 d ( e x2 d 2 2. dx dx 2 ( )), x x [ e x2 x 2 e x2 2 (.)dx ] dxdx, y = y() + y ()x + L 1 g(x), y n+1 = L 1 (A n ), n. By using Taylor series of g(x), e x2 2 and e x2 2 with order 1 and Adomian polynomials mentioned above, one obtains, y = x 1 6 x x x x y + y 1 = x 1 6 x x x x x x y + y 1 + y 2 = x 1 6 x x Conclusion 12 x x x x x13 Modified Adomian decomposition method is a powerful device to solve many functional equations. Here we use the method for solving particular third-order ordinary differential equations. In above examples, it is demonstrated that the method has the ability of applying to singular and nonsingular problem. For linear singular problem, as in Example 1, and for nonlinear problem, as in Example 2, the results show that the rate of convergence of Modified Adomian is faster than standard Adomian method. ACKNOWLEDGEMENTS. This work has been supported by Mahasarakham University. The authors are deeply indebted to Chawalit Boonpok and Chokchai Viriyapong for their suggestions and comments.

7 Modified ADM for solving particular third-order ODEs 1469 References [1] A.M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, 1997 [2] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl Math Comput, 111(1) (2), [3] A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl Math Comput, 12, (1999), 77. [4] G. Adomian, A review of the decomposition method and some recent results for nonlinear equation, Math Comput Model, 13(7) (1992) 17. [5] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition method,comput Math Appl, 29(1995), [6] M.M. Hosseini, Adomian decomposition method with Chebyshev polynomials, Appl Math Comput, 175(26), [7] M.M. Hosseini, H Nasabzadeh, Modified Adomian decomposition method for specific second order ordinary differential equations, Appl. Math. Comput, 186 (27), [8] M.M. Hosseini, H. Nasabzadeh, On the convergence of Adomian decomposition method, Appl Math Comput, 182(26), [9] Y.Q. Hasan, L.M. Zhu, Solving singular boundary valued problems of higher-order ordinary differential equations by modified Adomian decomposition method, Commun Nonlinear Sci Numer Simulat, 14 (29), Received: August, 211