1School of Mathematics, Iran University of Science & Technology, Narmak, Tehran 16846 13114, Iran.
2Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran.
Abstract
In this paper we apply hybrid functions of general block-pulse functions and Legendre polynomials for solving linear and nonlinear multi-order fractional differential equations (FDEs). Our approach is based on incorporating operational matrices of FDEs with hybrid functions that reduces the FDEs problems to the solution of algebraic systems. Error estimate that verifies a convergence of the approximate solutions is considered. The numerical results obtained by this scheme have been compared with the exact solution to show the efficiency of the method.
S. Abbasbandy, An approximation solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iteration method, J. Comput. Appl. Math.207 (2007), no. 1, 53--58.
A. Ahmadkhanlu and M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iranian Math. Soc. 38 (2012), no. 1, 241--252.
J.H. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol.15 (1999) 86--90.
A. Kazemi Nasab, A. Kilicman, Z. Pashazadeh Atabakan and S. Abbasbandy, Chebyshev wavelet finite difference method: A new approach for solving initial and boundary value problems of fractional order, Abstr. Appl. Anal. 2013 (2013), Article ID 916456, 15 pages.
A. Kilicman and Z.A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput. 187 (2007), no. 1, 250--265.
V.V. Kulish and J.L. Lage, Application of fractional calculus to fluid mechanics, J. Fluid. Eng. 124 (2002) 803--806.
Y. Li, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simulat.15 (2010), no. 9, 2284--2292.
Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput. 216 (2010), no. 8, 2276--2285.
K. Maleknejad and E. Hashemizadeh, A numerical approach for Hammerstein integral equation of mixed type using operational matrices of hybrid functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 73 (2011), no. 3, 95--104.
K. Maleknejad, M. Khodabin and M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block-pulse functions, Math. Comput. Model. Dyn. Syst. 55 (2012), no. 3-4, 791--800.
K. Maleknejad, K. Nouri and L. Torkzadeh, Operational matrix of fractional integration based on the shifted second kind Chebyshev polynomials for solving fractional differential equations, Mediterr. J. Math.13 (2016), no. 3, 1377--1390.
I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications , Academic Press, New York, 1999.
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