An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems

Mohammad Maleki,Saeid Abbasbandy,Ishak Hashim,Majid Tavassoli Kajani

doi:10.1155/2012/381708

Abstract

An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP) which involve Caputo-type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE). By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre-Gauss (ShLG) collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley-Torvik equation the proposed method is shown to be efficient and accurate.

Highlights

Due to the development of the theory of fractional calculus and its applications, such as in the fields of physics, Bode’s analysis of feedback amplifiers, aerodynamics and polymer rheology, and so forth, many works on the basic theory of fractional calculus and fractional order differential equations have been established 1–3 .In general, the analytical solutions for most of the fractional differential equations are not readily attainable, and the need for finding efficient computational algorithms for obtaining numerical solutions arises
The multi-term FBVP is first converted into a singular Volterra integrodifferential equation SVIDE
In this work a new adaptive pseudospectral method based on ShLG collocation points has been proposed for solving the multi-term FBVPs

Summary

Introduction

Due to the development of the theory of fractional calculus and its applications, such as in the fields of physics, Bode’s analysis of feedback amplifiers, aerodynamics and polymer rheology, and so forth, many works on the basic theory of fractional calculus and fractional order differential equations have been established 1–3. Abstract and Applied Analysis the solutions of initial value and boundary value problems for linear and nonlinear fractional differential equations. These methods include finite difference approximation method 4 , collocation method 5, 6 , the Adomian decomposition method 7, 8 , variational iteration method 9–12 , operational matrix methods 13–16 , and homotopy methods 17, 18. We intend to introduce an efficient adaptive pseudospectral method for multiterm fractional boundary value problems FBVP of the form. . ., ξl lie in 0, L , and Dαq denotes the Caputo-fractional derivative of order αq, defined as follows 23 : Dαq y x In this method, the multi-term FBVP is first converted into a singular Volterra integrodifferential equation SVIDE. The appendix is given which consists of the derivation of BagleyTorvik equation

Review of Legendre and Shifted Legendre Polynomials

Function Approximation

Convergence Rate For N 1 we introduce the piecewise polynomials space

Problem Replacement and the Solution Technique

Numerical Examples

Conclusion