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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Solving Nonlinear Integral Equations by using Adomian Decomposition Method

Mohedul Hasan Md* and Abdul Matin Md

Department of Mathematics, Dhaka University, Dhaka, Bangladesh

*Corresponding Author:
Mohedul Hasan Md
Department of Mathematics
Dhaka University
Dhaka, Bangladesh
Tel: 880-2-8628758
E-mail: [email protected]; [email protected]

Received date: February 24, 2016; Accepted date: March 17, 2017; Published date: March 24, 2017

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

Copyright: © 2017 Hasan M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

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.

Keywords

Nonlinear Fredholm integral equation; Adomian decomposition method; Adomian polynomials; Approximate solutions; OriginPro 8 and MATHEMATICA v9 softwares

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].

Consider the general form non-linear Fredholm integral equation of the second kind

equation

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 R2 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

equation   (1)

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

equation   (2)

The comparisons of like powers of p give solutions of various orders and the best approximation is

equation

The nonlinear term G(y(t)) can be expressed in Adomian polynomials [3-5] as

equation (3)

equation

where Hk’s are the so called Adomian polynomials which can be calculated by using the formula

equation   (4)

Using (2), (3) and (4) into (1), we have

equation   (5)

Equating the term with identical power of p in equation (5),

equation

equation

and so on.

and in general form we have

equation   (6)

Using the recursive scheme (6), the n-term approximation series solution can be obtained as follows:

equation   (7)

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

equation   (8)

where the exact solution of the equation is y(x)=x. In the following, we will compute Adomian polynomials for the nonlinear terms y2(t) that arises in nonlinear integral equation.

For k=0, equation (4) becomes

equation

equation

The Adomian polynomials for G(y)=y2 are given by

equation

By using the MATHEMATICA software, the next few terms, we have

equation

and so on.

Applying the technique as stated above in equation (6), we have

equation

equation

equation

In a similar manner, we stop the iteration at the tenth step. Therefore we can write

equation

equation

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).

Nodes (x) Exact solutions Approximate solutions Absolute error
0 0 0 0
0.10 0.100000 0.0999947 0.0000053
0.20 0.200000 0.1999890 0.0000110
0.30 0.300000 0.2999840 0.0000160
0.40 0.400000 0.3999790 0.0000210
0.50 0.500000 0.4999740 0.0000260
0.60 0.600000 0.5999680 0.0000320
0.70 0.700000 0.6999630 0.0000370
0.80 0.800000 0.7999580 0.0000420
0.90 0.900000 0.8999520 0.0000480
1.0 1.000000 0.9999470 0.0000530

Table 1: Numerical and exact solutions to the integral equation (8).

applied-computational-mathematics-numerical-integral-equation

Figure 1: Numerical and exact solutions to the integral equation (8).

Example 2

Consider the following nonlinear Fredholm integral equation of the second kind

equation   (9)

Applying above procedure, we have

equation

equation

In a similar manner, we stop the iteration at the tenth step. Therefore we can write

equation

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).

Nodes (x) Exact solutions Approximate solutions Absolute error
0 3 3 0
0.10 3.100000 3.10349 0.00349
0.20 3.200000 3.20699 0.00699
0.30 3.300000 3.31048 0.01048
0.40 3.400000 3.41397 0.01397
0.50 3.500000 3.51747 0.01747
0.60 3.600000 3.62096 0.02096
0.70 3.700000 3.72445 0.02445
0.80 3.800000 3.82794 0.02794
0.90 3.900000 3.93144 0.03144
1.0 4.000000 4.03493 0.03493

Table 2: Numerical and exact solutions to the integral equation (9).

applied-computational-mathematics-numerical-integral-equation

Figure 2: Numerical and exact solutions to the integral equation (9).

Example 3

Consider the following nonlinear Fredholm integral equation

equation   (10)

The exact solution of the equation (10) is equation. In the following, we will calculate Adomian polynomials for the nonlinear terms y3(t) that arises in nonlinear integral equation.

For k=0, equation (4) becomes

equation

equation

equation

The Adomian polynomials for G(y)=y3 are given by

equation

equation

By using the MATHEMATICA v9 software, the next few terms, we have

equation

and so on.

Applying the procedure as stated above in equation (6), we have

equation

In a similar manner, we stop the iteration at the ninth step. Therefore we can write

equation

equation

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).

Nodes Exact values Approximate values Absolute error
0.00 0.07542668890493687 0.07542668888687896 1.8057902×10-11
0.05 0.23093252624133365 0.23093252622349808 1.7835566×10-11
0.10 0.38075203836055493 0.3807520383433809 1.7174039×10-11
0.15 0.5211961716517719 0.5211961716356822 1.6089685×10-11
0.20 0.6488067254459994 0.6488067254313902 1.4609202×10-11
0.25 0.7604415043936764 0.7604415043809076 1.2768786×10-11
0.30 0.8533516897425216 0.8533516897319074 1.0614176×10-11
0.35 0.9252495243780194 0.9252495243698213 8.198108×10-12
0.40 0.9743646449962113 0.9743646449906311 5.580202×10-12
0.45 0.9994876743237375 0.9994876743209126 2.824962×10-12
0.50 1 1 0
0.55 0.9758890068665379 0.9758890068693629 2.824962×10-12
0.60 0.9277483875940958 0.927748387599676 5.580202×10-12
0.65 0.8567635239987162 0.8567635240069144 8.198108×10-12
0.70 0.7646822990073733 0.7646822990179875 1.0614176×10-11
0.75 0.6537720579794186 0.6537720579921874 1.2768786×10-11
0.80 0.526763779138947 0.5267637791535562 1.4609202×10-11
0.85 0.38678482782732176 0.38678482784341145 1.6089685×10-11
0.90 0.23728195038933994 0.237281950406514 1.7174067×10-11
0.95 0.08193640383912822 0.08193640385696378 1.7835566×10-11
1.00 -0.07542668890493687 -0.07542668888687896 1.8057902×10-11

Table 3: Numerical and exact solutions to the integral equation (10).

applied-computational-mathematics-numerical-integral-equation

Figure 3: Numerical and exact solutions to the integral equation (10).

Conclusion

This paper presents a technique to find the result of a nonlinear Fredholm integral equation by Adomian decomposition method (ADM). The estimated solutions obtained by the ADM are compared with exact solutions. It can be concluded that the ADM is effective and accuracy of the numerical results demonstrations that the proposed method is well suited for the solution of such kind problems.

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