Solution of the System of Ordinary Differential Equations by Combined Laplace Transform–adomian Decomposition Method

In this paper, combined Laplace transform–Adomian decomposition method is presented to solve differential equations systems. Theoretical considerations are being discussed. Some examples are presented to show the ability of the method for linear and non-linear systems of differential equations. The results obtained are in good agreement with the exact solution and Runge-Kutta method.


INTRODUCTION
A system of ordinary differential equations of the first order can be considered as: where each equation represents the first derivative of each unkown functions as a mapping depending on the independent variable x , and n unknown functions The main purpose of this paper is to extend the application of combined Laplace transform-Adomian decomposition method [1,2,3,4] to obtain an approximate solution of differential equations systems .The paper is organised as follows: In Section 2, how to use of combined laplace transform-adomian decomposition method is presented.In Section3, combined laplace transform-adomian decomposition method is demonstrated by applying it on three problems and conclusion is given at the last section.

THE USE OF COMBINED LAPLACE TRANSFORM-ADOMIAN DECOMPOSITION METHOD
We can present the system (1), by using the th i equation as: respectively.To solve the system of ordinary differential equations of the first order (1) by using the combined Laplace transform-Adomian decomposition method, we recall that the Laplace transform of the derivative of i y  are defined by Applying the Laplace transform to both sides of (2) gives , This can be reduced to The Adomian decomposition method and the Adomian polynomials can be used to handle (4) and to address the nonlinear term   where the so-called Adomian polynomials ik A can be evaluated for all forms of nonlinearity.In other words, assuming that the nonlinear functions is   Substituting ( 5) and ( 6) into (4) leads to , Matching both sides of ( 9) yields the following iterative algorithm.
Applying the inverse Laplace transform to the first part of (10) gives 0 i y , that will define A that will allows us to determine 2 i y ,and so on.This successively will lead to the complete determination of the components of ,0 ik yk  upon using the second part of (10).The series solution follows immediately after using equation (5).The combined Laplace transform-Adomian decomposition method to solve systems of differential equations of the first and second order are illustrated by studying the following examples.

NUMERICAL EXAMPLES
Three examples are presented in this part.The first and second examples are considered to illustrate the method for linear and non-linear ordinary differential equations systems of order one while in third example a differential equations system of order two is solved.Example 1.In this example we solve the following non-linear system of differential equations, with initial values   A are Adomian polynomials defined by The absolute error involved in the combined Laplace transform-Adomian decomposition method along with the exact solution for Example 1 We obtain the following procedure by using the combined Laplace transform-Adomian decomposition method.Approximations to the solutions with five terms are as follows: Table 1 shows the results of from the solution of Example 1 and illustrates the absolute errors between exact solution and combined Laplace transform-Adomian decomposition method, respectively.We achieved a good approximation with combined Laplace transform-Adomian decomposition method with only six iterations.

A and 2k
A are Adomian polynomials defined by We obtain the following procedure by using the combined Laplace transform-Adomian decomposition method.
Approximations to the solutions with five terms are as follows:   Table 2 shows the results of the solution of Example 2 and illustrates the absolute errors between solution obtained by using Runge-Kutta fourth order method and solution obtained by using combined Laplace transform-Adomian decomposition method.We achieved a good approximation by using presented method with only six iterations.
We obtain the following procedure by using the combined Laplace transform-Adomian decomposition method.
Table 3 The absolute error involved in the combined Laplace transform-Adomian decomposition method along with the exact solution for Example 3 Table 3 shows the results of the solution of Example 3 and illustrates the absolute errors between the exact solution and the solution of presented method.We achieved a good approximation with the presented method with only six iterations.

CONCLUSION
Combined Laplace transform-Adomian decomposition method has been applied to linear and non-linear systems of ordinary differential equations.Numerical examples have been presented to show that the approach is promising and the research is worth continue in this direction.All the calculations are performed easily.The calculated results are quite reliable.Since every ordinary differential equations of order n can be written as a linear algebraic equation by using Laplace Transform, this method is very useful and reliable for any order ordinary differential equation systems.Therefore, this method can be applied to many complicated linear and non-linear ODEs.
side of equation (5) will be represented by an infinite series of the Adomian polynomials ik A in the form ,

Table 2 .
The absolute error involved in the combined Laplace transform-Adomian decomposition method along with the result obtained by the Runge-Kutta fourth order method for Example 2