Study on multi-order fractional differential equations via operational matrix of hybrid basis functions

Document Type: Research Paper

Authors

1 School of‎ ‎Mathematics‎, ‎Iran University of Science & Technology‎, ‎Narmak‎, ‎Tehran 16846 13114‎, ‎Iran.

2 Department 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.

Keywords

Main Subjects


S. Abbasbandy, An approximation solution of a‎ ‎nonlinear equation with Riemann-Liouville's fractional‎ derivatives by He's variational iteration methodJ. 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 scalesBull. Iranian Math. Soc.  38  (2012), no. 1, 241--252.

D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods , World Scientific, Singapore, 2012.

C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods on Fluid Dynamics , Springer-Verlag, Berlin, 1988.

J. Deng and L. Ma, Existence and uniqueness of‎ ‎solutions of initial value problems for nonlinear fractional‎ ‎differential equationsAppl. Math. Lett. 23  (2010), no. 6, 676--680.

K. Diethelm, The Analysis of Fractional‎ ‎Differential Equations , Springer-Verlag, Berlin, 2010.

K. Diethelm and N.J. Ford, Numerical solution of‎ ‎the Bagley-Torvik equationBIT  42  (2002), no. 3, 490--507.

M. Di Paola, G. Failla and A. Pirrotta, Stationary‎ ‎and non-stationary stochastic response of linear fractional‎ ‎viscoelastic systemsProbabilist. Eng. Mech. 28  (2012) 85--90.

A.A. Elbeleze, A. Kilicman and B.M. Taib, Homotopy perturbation method for fractional Black-Scholes‎ ‎European option pricing equations using Sumudu transformMath. Probl. Eng.   2013  (2013), Article ID 524852, 7 pages.

A. El-Mesiry, A. El-Sayed and H. El-Saka, Numerical‎ ‎methods for multi-term fractional (arbitrary) orders  differential‎ ‎equationsAppl. Math. Comput.  160  (2005), no. 3, 683--699.

J.H. He, Homotopy perturbation method for‎ ‎bifurcation of nonlinear problemsInt. J. Nonlinear Sci. Numer. Simul.  (2005), no. 2, 207--208.

J.H. He, Some applications of nonlinear‎ ‎fractional differential equations and their approximationsBullSci. 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 orderAbstr. Appl. Anal.  2013  (2013), Article ID 916456, 15 pages.

A. Kilicman and Z.A. Al Zhour, Kronecker operational‎ ‎matrices for fractional calculus and some  applicationsAppl. Math. Comput.  187  (2007), no. 1, 250--265.

V.V. Kulish and J.L. Lage, Application of fractional‎ ‎calculus to fluid mechanicsJ. Fluid. Eng.  124  (2002) 803--806.

Y. Li, Solving a nonlinear fractional‎ ‎differential equation using Chebyshev waveletsCommun. 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 equationsAppl. 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 functionsPolitehn. 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 functionsMath. 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 equationsMediterr. J. Math.  13  (2016), no. 3, 1377--1390.

Z. Odibat and S. Momani, Numerical methods for‎ ‎nonlinear partial differential equations of fractional  orderAppl. Math. Model.  32  (2008) 28--39.

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.


Volume 43, Issue 2
March and April 2017
Pages 307-318
  • Receive Date: 28 February 2014
  • Revise Date: 24 September 2015
  • Accept Date: 14 November 2015