Signal Processing 86 (2006) 2602–2610 An approximate method for numerical solution of fractional differential equations Pankaj Kumar, Om Prakash Agrawal à Mechanical Engineering, Southern Illinois University, Carbondale, IL 62901, USA Received 25 May 2005; received in revised form 15 October 2005; accepted 6 December 2005 Available online 2 March 2006 Abstract This paper presents a numerical solution scheme for a class of fractional differential equations (FDEs). In this approach, the FDEs are expressed in terms of Caputo type fractional derivative. Properties of the Caputo derivative allow one to reduce the FDE into a Volterra type integral equation. Once this is done, a number of numerical schemes developed for Volterra type integral equation can be applied to find numerical solution of FDEs. In this paper the total time is divided into a set of small intervals, and between two successive intervals the unknown functions are approximated using quadratic polynomials. These approximations are substituted into the transformed Volterra type equation to obtain a set of equations. Solution of these equations provides the solution of the FDE. The method is applied to solve two types of FDEs, linear and nonlinear. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is stable. r 2006 Published by Elsevier B.V. Keywords: Fractional differential equation; Caputo derivative; Fractional-order system 1. Introduction This paper deals with numerical solution of fractional differential equations (FDEs). Fractional derivatives (FDs) and fractional integrals (FIs) have received considerable interest in recent years. In many applications, FDs and FIs provide more accurate models of systems under consideration. For example, FDs have been used successfully to model frequency dependent damping behavior of many viscoelastic materials. Bagley and Torvik [1–3] provided a review of work done in this area prior to 1980, and showed that half-order FD models describe the frequency dependence of the damping materials very well. Other authors have demonstrated applications of FDs and FIs in the areas of electrochemical processes [4,5], dielectric polarization [6], colored noise [7], viscoelastic materials [8–11] and chaos [12]. Mainardi [13] and Rossikhin and Shitikova [14] presented survey of the application of FDs and FIs, in general to solid mechanics, and in particular to modeling of viscoelastic damping. Magin [15–17] presented a three part critical review of applications of frac- tional calculus in bioengineering. Applications of FDs and FIs in other fields and related mathema- tical tools and techniques could be found in [15–20]. ARTICLE IN PRESS www.elsevier.com/locate/sigpro 0165-1684/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.sigpro.2006.02.007 à Corresponding author. Tel.: +1 618 453 7090; fax: +1 618 453 7658. E-mail addresses: pkumar@engr.siu.edu (P. Kumar), om@engr.siu.edu (O.P. Agrawal).