Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method Xinxiu Li School of Information Science and Engineering, Southeast University, Nanjing 210096, China article info Article history: Received 20 September 2011 Received in revised form 6 January 2012 Accepted 8 February 2012 Available online 24 February 2012 Keywords: Caputo derivative Cubic B-spline function Wavelet collocation method Interpolating condition abstract Physical processes with memory and hereditary properties can be best described by frac- tional differential equations due to the memory effect of fractional derivatives. For that rea- son reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction In the past several decades, the stud1 11y of fractional calculus has turned to practical application from pure mathemat- ical theory. Compared to integer order differential equation, fractional differential equation has the advantage that it can bet- ter describe some natural physics processes and dynamic system processes [19,30] because the fractional order differential operators are non-local operators. In general, it is not easy to derive the analytical solutions to most of the fractional differ- ential equations. Therefore, it is vital to develop some reliable and efficient techniques to solve fractional differential equa- tions. And the numerical solution of fractional differential equations has attached considerable attention from many researchers. During the past decades, an increasing number of numerical schemes are being developed. These methods in- clude finite difference approximation methods [11,19,33,42,44], fractional linear multi-step methods [24–26], collocation method [1,2,8,17,32], the Adomian decomposition method [34,35,43], variational iteration method [9,41,43], and opera- tional matrix method [3,5,6,10,12,16,20–22,31,39,40]. Taking advantage of the fact that fractional differential equation can be reduced to Volterra type integral equation, Kumar and Agrawal presented polynomial approximated methods to a class of fractional differential equation [27–29]. According to quadrature formula approach, Diethelm proposed an implicit algorithm for the approximated solution to an important class of fractional differential equation and gave error analysis [14,15]. Agrawal and Atanackovic transformed fractional differential equation into a system of integer order differential equation [18,36]. Fast wavelet collocation method is proposed by Cai and Wang in [38]. The wavelet has properties of localization in both time and frequency domains and multi-resolution analysis, which makes wavelet collocation method have the following superior computational properties [7]. Firstly, it makes uniform approximation because there is no accumulative error. Secondly, it can effectively treat the singularities. Thirdly, it can flexibly handle various boundary conditions of partial 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2012.02.009 E-mail address: lixinxiu410@163.com Commun Nonlinear Sci Numer Simulat 17 (2012) 3934–3946 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns