A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method

doi:10.1016/j.amc.2006.12.070

Applied Mathematics and Computation

Volume 189, Issue 2, 15 June 2007, Pages 1921-1928

https://doi.org/10.1016/j.amc.2006.12.070 Get rights and content

Abstract

Homotopy perturbation method is applied to the numerical solution for solving system of Fredholm integral equations. Comparison of the result obtained by the present method with that obtained by Taylor-series expansion method [K. Maleknejad, N. Aghazade, M. Rabbani, Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput. 175 (2006) 1229–1234] reveals that the present method is very effective and convenient.

Introduction

A new perturbation method called homotopy perturbation method (HPM) [1] was proposed by He in 1997 and systematical description in 2000 which is, in fact, a coupling of the traditional perturbation method and homotopy in topology. This new method was further developed and improved by He and applied to nonlinear oscillators with discontinuities [2], nonlinear wave equations [3], asymptotology [4], boundary value problem [5], Limit cycle and bifurcation of nonlinear problems [6] and many other subjects. After that, many researchers applied the method to various nonlinear problems (see [7], [8], [9], [10], [11]). Recently various powerful mathematical methods such as variational iteration method [12], [13], [14], [15], [16], [17], [18], [19], [20], Exp-function method [21], [22], F-expansion method [23], Adomian decomposition method [24] and others [25], [26] have been proposed to obtain exact and approximate analytic solutions for linear and nonlinear problems.

This paper, applies the homotopy perturbation method [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] to the discussed problem.

Section snippets

Homotopy perturbation method

Consider the following system of Fredholm integral equations: $F (x) = G (x) + \int_{0}^{1} K (x, t) F (t) d t, 0 ⩽ x ⩽ 1,$ where $\begin{matrix} K (x, t) = [k_{ij} (x, t)] = [(x - t)^{q_{ij}}], \\ F (x) = [f_{1} (x), f_{2} (x), \dots, f_{n} (x)], \\ G (x) = [g_{1} (x), g_{2} (x), \dots, g_{n} (x)] . \end{matrix}$ In Eq. (1) the functions K and G are given, and F the solution to be determined [27]. We assume that (1) has the unique solution. Consider the ith equation of (1) $f_{i} (x) = g_{i} (x) + \int_{0}^{1} \sum_{j = 0}^{n} k_{ij} (x, t) f_{j} (x) d t, i = 1, 2, \dots n .$ By the homotopy technique [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], we construct a homotopy for

Implementation of the method

In this section, in order to verify numerically whether the proposed methodology leads to higher accuracy, we evaluate the numerical solution of the problem (1). To show the efficiency of the present method for our problem in comparison with the exact solution we report absolute error which defined by $‖ f_{i}^{N} (x) ‖ = | f_{i} (x) - f_{i}^{N} (x) |, i = 1, 2, \dots, n,$ where $f_{i}^{N} (x) = \sum_{m = 0}^{N} f_{im}$ and $f_{i} (x), i = 1, 2, \dots, n$ are the exact solutions.

Example 1

For first example consider the following system of Fredholm integral equations [27]: $\{\begin{matrix} f_{1} (x) = g_{1} (x) + \int_{0} \end{matrix}$

Conclusions

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Homotopy perturbation method and axisymmetric flow over a stretching sheet

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A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method

Abstract

Introduction

Section snippets

Homotopy perturbation method

Implementation of the method

Conclusions

Int. J. Non-Linear Mech.

Appl. Math. Comput.

Chaos Solitons Fractals

Appl. Math. Comput.

Phys. Lett. A

Chaos Solitons Fractals

Appl. Math. Comput.

Phys. Lett. A

Chaos Solitons Fractals

Chaos Solitons Fractals

Appl. Math. Comput.

Appl. Math. Comput.

Homotopy perturbation method and axisymmetric flow over a stretching sheet

Int. J. Nonlinear Sci. Numer. Simul.