Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation

In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.


Introduction
The Laplace decomposition method (LDM) is one of the efficient analytical techniques to solve linear and nonlinear equations [1][2][3].LDM is free of any small or large parameters and has advantages over other approximation techniques like perturbation.Unlike other analytical techniques, 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 partial differential equations [1][2][3][4].See for example, the Duffing equation [4] and the Klein-Gordon equation [3].In this paper, the LDM is applied to, the Whitham-Broer-Kaup shallow water model [5] , , with exact solution are given in [5] as 2 t a n h ò , and the coupled nonlinear reaction diffusion equations [6] with exact solution are given in [6] as and thesystem of Hirota-Satsuma coupled KdV [7].
with exact solution are given in [7] as we discuss how to solve Numerical solution of nonlinear system of parial differential equations by using LDM.The results of the present technique have close agreement with approximate solutions obtained with the help of the Adomian decomposition method [8].

Laplace Decomposition Method
where , , , , the method consists of first applying the Laplace transformation to both sides of ( 7) using the formulas of the Laplace transform, we get in the Laplace decomposition method we assume the solution as an infinite series, given as follows , where the terms are to be recursively computed.
n Also the linear and nonlinear terms i and is decomposed as an infinite series of Adomian polynomials (see [8,9]).Applying the inverse Laplace transform, finally we get

The Pade Approximant
Here we will investigate the construction of the Pade approximates [10] for the functions studied.The main advantage of Pade approximation over the Taylor series approximation is that the Taylor series approximation can exhibit oscillati which may produce an approximation error bound.Moreover, Taylor series approximations can never blow-up in a fin region.To overcome these demerits we use the Pade approximations.The Pade approximation of a function is given by ratio of two polynomials.The coefficients of the polynomial in both the numerator and the denominator are determined using the coefficients in the Taylor series expansion of the function.The Pade approximation of a function, symbolized by [m/n], is a rational function defined by where we considered b 0 = 1, and the numerator and denominator have no common factors.In the LD-PA method we use the method of Pade approximation as an aftertreatment method to the solution obtained by the Laplace decomposition method.This after-treatment method improves the accuracy of the proposed method.

Application
In this section, we demonstrate the analysis of our numerical methods by applying methods to the system of partial differential Equations ( 1), ( 3) and ( 5).A comparison of all methods is also given in the form of graphs and tables, presented here.

The Laplace Decomposition Method
Exampe 1.The Whitham-Broer-Kaup model [5] To solve the system of Equation ( 1) by means of Laplace decomposition method, and for simplicity, we , we construct a correctional functional which reads we can define the Adomian polynomial as follows: we define an iterative scheme

Example 2. coupled nonlinear RDEs [6]
To solve the system of Equation ( 3) by means of Laplace decomposition method, and for simplicity, we take , we construct a correctional functional which reads we can define the Adomian polynomial as follows: we define an iterative scheme applying the inverse Laplace transform, finally we get similarly, we can also find other components, and the approximate solution for calculating 16 th terms as follows: and Figures 2(a) and (b) show the exact and numerical solution of system (3) with 16 th terms by (LDM).

Example 3. Hirota-Satsuma coupled KdV System [7]
To solve the system of Equation ( 5) by means of Laplace decomposition method, and for simplicity, we take   we can define the Adomian polynomial as follows: we define an iterative scheme applying the inverse Laplace transform, finally we get  similarly, we can also find other components, and the approximate solution for calculating 16 th terms as follows: and Figures 3(a)-(c) show the exact and numerical solution of system (5) with 16 th terms by (LDM).

Conclusion
The Laplace decomposition method is a powerful tool   which is capable of handling nonlinear system of partial differential equations.In this paper the (LDM) and Pade approximant has been successfully applied to find ap-proximate solutions for,the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and thesystem of Hirota-Satsuma coupled ,