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Abstract and Applied Analysis
Volume 2012, Article ID 381708, 19 pages
Research Article

An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems

1School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi Selangor, Malaysia
2Department of Mathematics, Islamic Azad University, Khorasgan Branch, Isfahan, Iran
3Department of Mathematics, Imam Khomeini International University, Ghazvin 34149-16818, Iran

Received 3 June 2012; Revised 17 August 2012; Accepted 30 August 2012

Academic Editor: Dumitru Bǎleanu

Copyright © 2012 Mohammad Maleki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.