^{1}

^{*}

^{2}

The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear integro-differential equations that will facilitate the calculations. In this modification, compared to the standard Adomian decomposition method, the size of calculations was reduced. This modification also avoids computing Adomian polynomials. Numerical results are given to show the efficiency and performance of this method.

In 1999, Wazwaz [

In spite of the fact that the “Modified Decomposition Method” of wazwaz has shown to be computationally efficient in some applications, the criterion of separating the function

We consider the Integro-differentil equation of the form

with initial condotions

Where

will be determined,

Let

initial conditions, we obtain

For nonlinear equations, the nonlinear operator

The standard Adomian method defines the solution

where the components

The main ideas of the proposed “Two-Step Adomian Decomposition Method” are:

(1) Applying the inverse operator

where the function

where _{o} is defined as:

where

(2) We set

Compared to the common “Adomian Decomposition Method” and the “Modified Decomposition Method”, it is clear that the “Two-Step Decomposition Method” may produce the solution by using only one iteration. It is worthy to note that the Procedure of verification in the first step can be larg effective in many cases. This can be note through the following examples. Further, the “Two-Step Decomposition Method” avoids the difficulties arising in the modified method. Also the number of the terms in

Example 1

Consider nonlinear Volterraintegro-differential equation [

With the exact solution is

The modified decomposition method: Using the modified recursive relation (10), and by selecting

In view of (12), the exact solution is given by

It is to be noted that if we select

putational work required compared to the standard Adomian method.

The (TSADM), using the scheme (7) gives

By selecting

However, we use the standard Adomian method to find:

In view of (14), the modified method also requires a huge size of computational work to obtain few terms of the series. Moreover, the same as the standard Adomian decomposition method, the modified method requires the use of the Adomian polynomials for nonlinear models. However, using the two-step Adomian decomposition method, there is no need to use the Adomian polynomials.

Example 2

Consider nonlinear Fredholmintegro-differential equation

With the exact solution is

Applying

The modified decomposition method: Using the modified recursive relation (15), and by selecting

In view of (17), the exact solution is given by

It is to be noted that if we select

work required compared to the standard Adomian method.

The (TSADM), using the scheme (7) gives

By selecting

However, we use the standard Adomian method to find:

In view of (19), the modified method also requires a huge size of computational work to obtain few terms of the series. Moreover, the same as the standard Adomian decomposition method, the modified method requires the use of the Adomian polynomials for nonlinear models. However, using the two-step Adomian decomposition method, there is no need to use the Adomian polynomials.

Example 3

Consider the system of nonlinear Volterraintegro differential equation [

With the exact solution are

Applying

The modified decomposition method: Using the modified recursive relation (20), and by selecting

In view of (22), the exact solution is given by

It is to be noted that if we select

The (TSADM), using the scheme (7) gives

By selecting

and by verifying that

However, we use the standard Adomian method to find:

In view of (26), the modified method also requires a huge size of computational work to obtain few terms of the series. Moreover, the same as the standard Adomian decomposition method, the modified method requires the use of the Adomian polynomials for nonlinear models. However, using the two-step Adomian decomposition method, there is no need to use the Adomian polynomials.

Example 4

Consider the system of nonlinear Fredholmintegro-differential equation [

With exact solution

The modified decomposition method: Using the modified recursive relation (27), and by selecting

In view of (29), the exact solution is given by

It is to be noted that if we select

size of computational work required compared to the standard Adomian method.

The (TSADM), using the scheme (7) gives

By selecting

and by verifying that

However, we use the standard Adomian method to find:

In view of (33), the modified method also requires a huge size of computational work to obtain few terms of the series. Moreover, the same as the standard Adomian decomposition method, the modified method requires the use of the Adomian polynomials for nonlinear models. However, using the two-step Adomian decomposition method, there is no need to use the Adomian polynomials.

In this paper, we have applied two-step Adomian Decomposition Method (TSAM) to obtain the solutions of nonlinear integro-differential equations. Some examples have been discussed as illustrations. In this work, we show that TSADM is convenient to solve integro-differential equations and reduce the size of calculations compared to the standard Adomian decomposition method and modified decomposition method. This modification also avoids computing Adomian polynomials. The TSADM produce the solution by using only two iterations, if compared with the common Adomian method and the modified method. Moreover, the TSADM overcomes the difficulties arising in the modified decomposition method.

Al-Mazmumy, M. and Almuhalbedi, S.O. (2016) Solution of Nonlinear Integro Differential Equations by Two-Step Adomian Decomposition Method (TSAM). International Journal of Modern Nonlinear Theory and Application, 5, 248- 255. http://dx.doi.org/10.4236/ijmnta.2016.54022