Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

Emran Tohidi,M M Ezadkhah,S Shateyi

doi:10.1155/2014/162896

Abstract

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

Highlights

In real world, for modeling and analysing a huge size of problems we need fractional calculus
Several numerical examples are given to illustrate the accuracy and effectiveness of the proposed method, and all of them are performed on a computer using some programs written in MAPLE 13
The method can be extended to the system of nonlinear fractional integrodifferential equations, but some modifications are required

Summary

Introduction

For modeling and analysing a huge size of problems we need fractional calculus. For this purpose several techniques were proposed to solve the fractional order differential equations (or integro-differential equations). The Bernoulli (matrix and collocation) methods have been used to find the approximate solutions of differential and integro-differential equations [16,17,18]. According to the discussions in [18], Bernoulli polynomials have some certain properties that encourage us to use them for solving any applied mathematics problem These subjects motivate us to present a new numerical scheme for solving FIDEs. In this paper, by using the Bernoulli polynomials as the test functions and collocating the following FIDE (subject to sufficient initial or boundary conditions) at the Legendre. Gauss collocation points and approximating the existing integrals by the Gauss quadrature rule, we find the numerical solution of the following FIDE: x.

Preliminaries

Basic Idea

Numerical Examples

Conclusions