Application of the Variational Iteration Method for System of Nonlinear Volterra’s Integro-Differential Equations

Saeid Abbasbandy,Elyas Shivanian

doi:10.3390/mca14020147

Abstract

In this paper, the variational iteration method is proposed to solve system of nonlinear Volterra's integro-differential equations. Four numerical examples are illustrated by this method. The results reveal that this method is very effective and highly promising in comparison with other numerical methods.

Highlights

The variational iteration method [1, 2], which is a modified general Lagrange multiplier method [3] has been shown to solve effectively, and accurately, a large class of nonlinear problems with approximations which converge quickly to accurate solutions
It was successfully applied to autonomous ordinary differential equation [4], to nonlinear partial differential equations with variable coefficients [5], to SchrödingerKDV, generalized KDV and shallow water equations [6], to Burgers' and coupled Burgers' equations [7], to the linear Helmoltz partial differential equation [8] and recently to nonlinear fractional differential equations with Caputo differential derivative [9], and other fields [10,11,12, 28,29,30,31]
There are various numerical and analytical methods to solve such problems, for example, the homotopy perturbation method [17], the Adomian decomposition method [18], but each method limits to a special class of integro-differential equations

Summary

1.INTRODUCTION

The variational iteration method [1, 2], which is a modified general Lagrange multiplier method [3] has been shown to solve effectively, and accurately, a large class of nonlinear problems with approximations which converge quickly to accurate solutions. He used the variational iteration method for solving some integro-differential equations [19]. This Chinese mathematician chooses [19] initial approximate solution in the form of exact solution with unknown constants. M. Ghasemi et al solved the nonlinear Volterra's integro-differential equations [26] by using homotopy perturbation method. In [21], the variational iteration method was applied to solve the system of linear integro-differential equations. Biazar et al solved systems of integro-differential equations by He's homotopy perturbation method [22]. The purpose of this paper is to extend the analysis of the variational iteration method to solve the system of general nonlinear Volterra's integro-differential equations which is as follows: u1(m.

BASIC IDEAS OF VARIATIONAL ITERATION METHOD

APPLICATIONS

CONCLUSIONS