Numerical Solution of Coupled System of Nonlinear Partial Differential Equations Using Laplace-Adomian Decomposition Method

Aim of the paper is to investigate applications of Laplace Adomian Decomposition Method (LADM) on nonlinear physical problems. Some coupled system of non-linear partial differential equations (NLPDEs) are considered and solved numerically using LADM. The results obtained by LADM are compared with those obtained by standard and modified Adomian Decomposition Methods. The behavior of the numerical solution is shown through graphs. It is observed that LADM is an effective method with high accuracy with less number of components. Mathematics Subject Classification: 45J05, 45B05, 45D05 and 45A05


INTRODUCTION
The partial differential equations (PDEs) have so many essential applications of science and engineering such as wave propagation, shallow water waves, fluid mechanics, thermodynamic, chemistry and micro electro mechanic system, etc. It is difficult to handle nonlinear part of these systems, although most of the scientists applied numerical methods to find the solutions of these systems that based on linearization, perturbed and discretizations. Debnath [1] applied the characteristics method and Logan [2] used the Rieman invariants method to handle systems of PDEs. Wazwaz [3] used the Adomian decomposition method (ADM) to handle the systems of PDEs. Laplace Decomposition Method (LDM) is free of any small or large parameters and has advantages over other approximation techniques like perturbation, LDM requires no discretization and linearization, therefore, results obtained by LDM are more efficient and realistic. This method has been used to obtain approximate solutions of a class of nonlinear ordinary and PDEs [4][5][6][7]. In this paper, we compute numerical solutions to nonlinear systems of PDEs by using LADM. The numerical solutions become easier and higher accuracy than the standard Adomian Decomposition Method (ADM).

LADM for NONLINEAR COUPLED of PDEs
Consider the general nonlinear coupled of PDEs written in an operators form (see [8]) 1 1 1 1 Subject to the initial conditions  N symbolized the nonlinear differential operators and 1 2 ( , ), ( , ) f x t f x t are given functions. The method consists of first applying the Laplace transform to both sides of equations in system (1) and then by using initial conditions, we have in the Laplace decomposition method we assume the solution is in an infinite series, given as follows where ( , ) and ( , ) kk u x t v x t are to be recursively computed. Also the nonlinear terms N are decomposed as an infinite series of Adomian polynomials (see [9,10]), Substituting (4) and (6) into (3) and applying the linearity of the Laplace transform, we get the following recursively formula and     Applying the inverse Laplace transform, we can evaluate with the following initial conditions The exact solution of the system (9) is [11].
( , y, ) ; ( , , ) ; ( , , ) Solution: To solve the system of equations (9)-(10) by means of LADM, we construct a correctional functional which We can define the Adomian polynomials as follows x y t w We define an iterative scheme Applying the inverse Laplace transform, finally, According to (11)    x v x t x e   (see [11] and [12] ). Solution: To solve the system of equation (14) by means of LADM , we construct a correctional functional which reads We can define the Adomian polynomials as follows   We define an iterative scheme        (  3  3), According to (7), the 0 th components 0 ( , ) u x t and 0 ( , ) v x t written as follows: , where, 0 n  . So, we get the following components: Similarly, we can also find other components, and the approximate solution for calculating 5 th . Using (4), the series solutions are therefore given by  Example 3: we consider the nonlinear system [13] , t ; The exact solution given as ( , y, ) ; v(x, y, ) , 0 Solution: Taking the Laplace transform on both sides of Eqs. (10) then, by using the differentiation property of Laplace transform and initial conditions, as the same procedure in the above example when we using equations (9)-(10) for the system (10) and according to (3)  Similarly, we can also find other components, and the approximate solution for calculating more 20 th . Using (3), the series solutions are therefore given by   (18) (see [14] and [16]).
u(x,y,t) LADM v(x,y,t) ) LADM u(x,y,t) and v(x,y,t) exact   We note from the above results, the absolute error obtained by the proposed algorithm LADM as compared with the absolute error of the standard algorithm for Adomian's polynomials ADM and modified ADM given results more accurate.

Example 4:
The mathematical models on many phenomena in applied sciences lead to non-linear PDEs such as the homogeneous form of the system of two dimensional Burger's equations which is proposed as mathematical model of free turbulence [15,16]