Variational Iteration Method for Volterra Functional Integrodifferential Equations with Vanishing Linear Delays

Ali Konuralp,H Hilmi Sorkun

doi:10.1155/2014/678989

Abstract

Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay functionθ(t)vanishes inside the integral limits such thatθ(t)=qtfor0<q<1,t≥0. Either the approximate solutions that are converging to the exact solutions or the exact solutions of three test problems are obtained by using this presented process. The numerical solutions and the absolute errors are shown in figures and tables.

Highlights

Nowadays, understanding from the work of Brunner [1], we are faced with some important problems including the numerical analysis of Volterra functional equations with vanishing delays
The pantograph differential equations are employed for their numerical solutions by using various methods such as Taylor matrix method [4], variational iteration method [5, 6], differential transform method [7], and methods in other papers [8,9,10,11,12]
In [6], the variational iteration method is applied to some examples in order to obtain the numerical or exact solutions of the multipantograph equation where the coefficient a1(t) of u(t) is a constant and the Lagrange multiplier is given for (1) without terms (2) and (3), that is, only for the extended multipantograph equation

Summary

Introduction

Nowadays, understanding from the work of Brunner [1], we are faced with some important problems including the numerical analysis of Volterra functional equations with vanishing delays. In [6], the variational iteration method is applied to some examples in order to obtain the numerical or exact solutions of the multipantograph equation where the coefficient a1(t) of u(t) is a constant and the Lagrange multiplier is given for (1) without terms (2) and (3), that is, only for the extended multipantograph equation. In addition to [28], in this paper, the procedure of the variational iteration method is presented for the Volterra functional integrodifferential equations with vanishing delays (1), where the Volterra integral terms are as in (2) and the delayed Volterra integral terms are as in (3); this extended scheme is applied to three test problems for showing the applicability of the procedure and the convergent numerical solutions to the exact solutions. The numerical data for different parameter values are given by tables and figures

The Application Process of Variational Iteration Method to VFIDE

Numerical Examples

Conclusion