Spectral approximations to the fractional integral and derivative

Abstract

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods. Editorial Note: The first and third authors of this paper have recently received prestigeous awards at the 5th International Symposium "Fractional Differentiation and Applications' 2012" in China, May 14-17, 2012. Professor Changpin Li, together with his PhD student Yutian Ma, received the "Riemann-Liouville Award" for Best FDA Paper (theory), FDA #077: Fractional Dynamical System and Its Linearization Theorem. Professor Fawang Liu got the "Mittag-Leffler Award" which is given once per two years for a cumulative contribution to the field of Fractional Derivative and Applications, see details at http://em.hhu.edu.cn/fda12/Awards.html. © 2012 Diogenes Co., Sofia.