Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomial

D Sh Mohammed

doi:10.1155/2014/431965

Abstract

We investigate the numerical solution of linear fractional integro-differential equations by least squares method with aid of shifted Chebyshev polynomial. Some numerical examples are presented to illustrate the theoretical results.

Highlights

Many problems can be modeled by fractional Integrodifferential equations from various sciences and engineering applications
The authors in [12, 13] give an application of nonlinear fractional differential equations and their approximations and existence and uniqueness theorem for fractional differential equations with integral boundary conditions
In this paper least squares method with aid of shifted Chebyshev polynomial is applied to solving fractional Integro-differential equations

Summary

Introduction

Many problems can be modeled by fractional Integrodifferential equations from various sciences and engineering applications. The authors in [1, 2] applied collocation method for solving the following: nonlinear fractional Langevin equation involving two fractional orders in different intervals and fractional Fredholm Integro-differential equations. Chebyshev polynomials method is introduced in [3,4,5] for solving multiterm fractional orders differential equations and nonlinear Volterra and Fredholm Integro-differential equations of fractional order. The authors in [6] applied variational iteration method for solving fractional Integro-differential equations with the nonlocal boundary conditions. In this paper least squares method with aid of shifted Chebyshev polynomial is applied to solving fractional Integro-differential equations. We are concerned with the numerical solution of the following linear fractional Integro-differential equation: Dαφ (x) = f (x) + ∫ K (x, t) φ (t) dt, 0 ≤ x, t ≤ 1, (1). Where Dαφ(x) indicates the αth Caputo fractional derivative of φ(x); f(x), K(x, t) are given functions, x and t are real variables varying in the interval [0, 1], and φ(x) is the unknown function to be determined

Basic Definitions of Fractional Derivatives

Solution of Linear Fractional Integro-Differential Equation

Numerical Examples