 Applied Mathematics, 2011, 2, 975-980 doi:10.4236/am.2011.28134 Published Online August 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Adomian Decomposition Method for Solving Goursat’s Problems Mariam A. Al-Mazmumy Mathematics Department, Science College, King Abdulaziz University, Jeddah, Saudi Arabia E-mail: mome0505@hotmail.com Received May 18, 201 1; revised June 22, 2011; accepted June 29, 2011 Abstract In this paper, Goursat’s problems for: linear and nonlinear hyperbolic equations of second-order, systems of nonlinear hyperbolic equations and fourth-order linear hyperbolic equations in which the attached conditions are given on the characteristics curves are transformed in such a manner that the Adomian decomposition method (ADM) can be applied. Some examples with closed-form solutions are studied in detail to further illustrate the proposed technique, and the results obtained indicate this approach is indeed practical and effi-cient. Keywords: Goursat’s Problem, Linear and Nonlinear Hyperbolic Equation of Second and Fourth-Orders, System of Linear Hyperbolic Equations of Second Order, Adomian Decomposition Method 1. Introduction The simple Goursat’s problem concerns a class of linear hyperbolic equations of second-order in two independent variables with given values on two characteristics curves . For example, for the linear hyperbolic equation =,tt xxuu uftx Goursat’s problem is posed as follows:   01=,,,=,tt xxuu uftxutxxfor xtutxxfor xt 0=0. (1) Several numerical methods such as Range-Kutta method, finite difference method and finite elements method have been used to approach the problem. The general difficult which arises to us is the p resence of the attached conditions on two characteristics curves and which complicates the application of numerical methods and Adomian decomposition me- thod [2-12]. =0xt=0xtThe clue of this one consists in transforming this type of problems into classical problems where the conditions can be converted into initial conditions. In this tech n ique, we use the variablesand Hence we show that the linear Goursat models will be approached more effectively and rapidly by using the Adomian decomposition method (ADM) to obtain the exact solu- tions to this type of prob lems. In a manner parallel to this problem, we study a class of Goursat’s problems for non- linear hyperbolic equations, systems of nonlinear hyper- bolic equations and fourth-order linear hyperbolic equa- tions. Our techniques are easily applicable and offer a very direct way to determine the solution s. =wxt=zxtThe present pa per extends som e result s of [ 13] . 2. The Goursat’s Problem for Linear Hyperbolic Equations of Second Order In this section consider Goursat’s prob lem (1). Our first approach consists in converting problem (1) into a classical problem by introducing new variables =wxt and =zxt. The second one is that in order to find the solution for the given problem we consider this form as the most general form for ADM. Make the substitutions and =wxt=zxt. into the first Equation of (1), and applying the chain rule to obtain =uuuxwz =uuutwz 22 2222=2uu uwz 2uxwz 976 M. A. Al-MAZMUMY 22 2222=2uu uwztw z  2u Then the first Equation of (1) becomes 24= ,22uwzwzfuwz  (2) Also the given conditions of (1) can be converted into 0,0 =2wuw  (3) 10, =2zuz  (4) Then, Lemma 1. Goursat’s problem (1) is equivalent to (2)-(4). Now we shall use the Adomian’s decomposition method [2-12] for solving (2)- (4 ). Consider Equation (2) in an operator form as 1=44wzLuf u (5) where 2=wz uLu wz Operating with the inverse operator we have 100(.) =(.)dd,wzwzL wz 010000,= 0,0221,d422,dd4wzwzwzuwz uwzwz dfwzuwz wz    (6) where .  010,0 =00uFollowing the Adomian decomposition method the unknown solution is assumed to be given by a series uof the form where the com-  =0,= ,nnuwzu wzponents are going to be determined recurrently. ,nuwzThus, we get the schem e 00 100100=0221,d422=dd,14wzwznnwzuuwzwz,0dfwzuuwzn    (7) Now, by summing the first 1n terms of =0,= ,nnuwzu wz we obtain the approxi- nthmation to the solution  as =0=,nniiSun0i (8) or 0=1=nniSu u (9) By substitution of the recursive scheme (7) into this sum, we conclude that the ADM for Goursat’s problem (1) can be converted to an equivalent problem, which we state as follows Theorem 1. The ADM for Goursat’s problem (1) is equivalent to the following problem: Find the sequence Sn such that and satisfies 01=nnSuu u 00 10001000,0221,d422=dd,4wzwznnwzuuwzwz d1fwzSu Swzn    (10) Also, the result concerning the convergence analysis of the ADM for problem (1) can be stated as follows Theorem 2. Let be a sequence defined by (10). If nS0