A numerical solution for solving system of Fredholm integral equations by using homotopy perturbation method
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:whereIn 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)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 bywhere and are the exact solutions. Example 1 For first example consider the following system of Fredholm integral equations [27]:
Conclusions
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 [27] reveals that the present method is very effective and convenient.
References (27)
A coupling method of a homotopy technique and a perturbation technique for non-linear problems
Int. J. Non-Linear Mech.
(2000)The homotopy perturbation method for non-linear oscillators with discontinuities
Appl. Math. Comput.
(2004)Application of homotopy perturbation method to nonlinear wave equations
Chaos Solitons Fractals
(2005)Asymptotology by homotopy perturbation method
Appl. Math. Comput.
(2004)Homotopy perturbation method for solving boundary problems
Phys. Lett. A
(2006)Limit cycle and bifurcation of nonlinear problems
Chaos Solitons Fractals
(2005)Iterated He’s homotopy perturbation method for quadratic Riccati differential equation
Appl. Math. Comput.
(2006)- et al.
Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder
Phys. Lett. A
(2006) Exp-function method for nonlinear wave equations
Chaos Solitons Fractals
(2006)- et al.
Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation
Chaos Solitons Fractals
(2005)
A numerical simulation and explicit solutions of the generalized Burger–Fisher equation
Appl. Math. Comput.
Numerical solution of second kind Fredholm integral equations system by using a Taylor-series expansion method
Appl. Math. Comput.
Homotopy perturbation method and axisymmetric flow over a stretching sheet
Int. J. Nonlinear Sci. Numer. Simul.
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