Numerical Solutions of a Quadratic Integral Equations by Using Variational Iteration and Homotopy Perturbation Methods

Hind Al-Badrani,Wafa Shammakh,F A Hendi

doi:10.5539/jmr.v9n2p134

Abstract

In this paper, the approximate solutions for quadratic integral equations (QIEs) are given by the variational iteration method(VIM) and homotopy perturbation method (HPM). These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with some examples that show simplicity and effectiveness.

Highlights

In this paper, the approximate solutions for quadratic integral equations (QIEs) are given by the variational iteration method (VIM) and homotopy perturbation method (HPM)
There are few papers which have dealt with the numerical solutions of QIEs such as Elsayed (El-Sayed et al, 2010) used the classical method of successive approximations Picard and Adomian decomposition method for solving QIEs, Avazzadeh (Avazzadeh, 2012) used the radial basis functions to obtain the approximate solutions of QIEs of Urysohn’s type. (He, 1999a; He, 1999b; He, 2000; He, 2003) was the first one who proposed the VIM and HPM to find the solutions of linear and nonlinear problems
This method presents significant enhancements over existing numerical and analytic technique like the perturbation, Adomian, Galerkin, finite differences methods, etc. These methods have dealt with ordinary, partial differential equations, the integro-differential equations (IDEs) and integral equations, in a direct way without needing to any specific restriction which may give the closed form of exact solution if there is an exact solution

Summary

Variational Iteration Method

Consider the following differential equation where L and N are linear and nonlinear operators respectively, and g(x) is the inhomogeneous source term. Where λ is a general Lagrange multiplier, noting that in this method λ may be a constant or a function, which can be identified perfectly by the variational theory and the subscript n denotes the nth-order approximation, un is considered as a restricted value that means it behaves as a constant, i.e. δun = 0. It was found in (Abdou & Soliman, 2005; Abulwafa et al, 2006; He & Wu, 2007).

Homotopy Perturbation Method

Findings

Conclusion