Solving Nonlinear Integral Equations by using Adomian Decomposition Method

In this paper, we propose a numerical method to solve the nonlinear integral equation of the second kind. We intend to approximate the solution of this equation by Adomian decomposition method using He’s polynomials. Several examples are given at the end of this paper with the exact solution is known. Also the error is estimated. Citation: Hasan M, Matin A (2017) Solving Nonlinear Integral Equations by using Adomian Decomposition Method. J Appl Computat Math 6: 346. doi: 10.4172/2168-9679.1000346


Introduction
Several scientific and engineering applications are usually described by integral equations. Integral equations arise in the potential theory more than any other field. Integral equations arise also in diffraction problems, conformal mapping, water waves, scattering in quantum mechanics, and population growth model. The electrostatic, electromagnetic scattering problems and propagation of acoustical and elastically waves are scientific fields where integral equations appear [1]. The Fredholm integral equation is of widespread use in many realms of engineering and applied mathematics [2]. where y(x) is the unknown solution, a and b are real constants. The kernel K(x,t) and f(x) are known smooth functions on R 2 and R, respectively. The parameter λ is a real (or complex) known as the eigenvalue when λ is a real parameter, and G is a nonlinear function of y.

Adomian Decomposition Method
Consider the following non-linear Fredholm integral equation of the second kind of the form We assume G(y(t)) is a nonlinear function of y(x). That means that the nonlinear Fredholm integral equation (1) contains the nonlinear function presented by G(y(t)). Assume that the solution of equation (1) can be written in the form ( ) The nonlinear term G(y(t)) can be expressed in Adomian polynomials [3][4][5] Using (2), (3) and (4) into (1), we have Equating the term with identical power of p in equation (5), Using the recursive scheme (6), the n-term approximation series solution can be obtained as follows:

Numerical Implementations
In this section, we will apply the Adomian decomposition method to compute a numerical solution for non-linear integral equation of the Fredholm type. Then we will compare between the results which we obtain by the numerical solution technique and the results of the exact solution. To illustrate this, we consider the following example:

Example 1
Consider the following nonlinear Fredholm integral equation of the second kind ( ) where the exact solution of the equation is y(x)=x. In the following, we will compute Adomian polynomials for the nonlinear terms y 2 (t) that arises in nonlinear integral equation.
For k=0, equation (4) becomes The Adomian polynomials for G(y)=y 2 are given by By using the MATHEMATICA software, the next few terms, we have  Applying the technique as stated above in equation (6), we have In a similar manner, we stop the iteration at the tenth step. Therefore we can write ( ) The table under shows the approximate solutions obtained by applying the Adomian Decomposition method giving to the value of x, which is in the interval [0-1] (Table 1 and Figure 1).

Example 2
Consider the following nonlinear Fredholm integral equation of the second kind The exact solution of the equation is 3+x. The table below shows the approximate solutions obtained by applying the Adomian decomposition method according to the value of x, which is confined between zero and one. We compared these results with the results which were obtained by the exact solution (Table 2 and Figure 2).

Example 3
Consider the following nonlinear Fredholm integral equation . In the following, we will calculate Adomian polynomials for the nonlinear terms y 3 (t) that arises in nonlinear integral equation.
For k=0, equation (4) becomes The Adomian polynomials for G(y)=y 3 are given by ( ) The table under shows the approximate solutions obtained by applying the Adomian decomposition method according to the value of x, which is in the interval [0-1] (Table 3 and Figure 3).