**Authors**

Department of Applied Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran.

**Abstract**

Abstract. In this paper, we implement numerical solution of diﬀerential equations of frac- tional order based on hybrid functions consisting of block-pulse function and rationalized Haar functions. For this purpose, the properties of hybrid of rationalized Haar functions are presented. In addition, the operational matrix of the fractional integration is obtained and is utilized to convert computation of fractional diﬀerential equations into some algebraic equa- tions. We evaluate application of present method by solving some numerical examples.

**Keywords**

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations, Johnwiley and Sons, New York, 1993.

I. Podlubny, Fractional Diﬀerential Equations: An Introduction to Fractional Derivatives, Fractional Diﬀerential Equations, to Methods of their Solution and Some of Their Application, Mathematics in Science and Engineering, Academic Press, New York, Volume 198, 1999.

M. U. Rehman and R. A. Khan, The Legendre wavelet method for solving fractional diﬀerential equations, Commun Nonlinear Sci Numer Simmulat, 16 (2011) 4163-4173.

H. Jafari and S.A. Youse, M. A. Firroozjaee, S. Momani, C. M. Khalique, Application of Legendre wavelet for solving fractional diﬀerential equations, Computers and Mathematics with Applications, 62 (2011) 1033-1045.

Y. Li and W. Zhao, Haar Wavelet operational matrix of fractional order integration and its application in solving the fractional order diﬀerential equations, Applied Mathmatics and Computation, 216 (2010) 2276-2285.

Y. Li, Solving a nonlinear fractional diﬀerential equations using Chebyshev wavelets, Communication in Nonlinear Science and Numerical Simulation, 15 (2010) 2284-2292.

S. A. El-Wakil, A. Elhanblay and M. A. Abdou, Adomian decomposition method for solving fractional nonlinear diﬀerential equations, Applied Mathematical and Computation, 182 (2006) 313-324.

S. Momani and Z. Odibat, Numerical comparison of methods for solving linear diﬀerential equations of fractional order, Chos Solitons and Fractals, 31 (2007) 1248-1255.

A. Arikoglu and I. Ozkol, Solution of fractional diﬀerential equations by using diﬀerential transform method, Chos Solitons and Fractals, 34 (2007) 1473-1481.

A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional -order diﬀerential equations, Computers and Mathematics with Applications, 59 (2010) 1326-1336.

M. U. Rehman and R. A. Khan, A numerical method for solving boundary value problems for frac- tional diﬀerential equations, Applied Mathematical Modelling, 36 (2011) 894-907.

M. M. Khader, T. S. El danaf and A. S. Hendy, Eﬃcient spectral collocation method for solving multi-term fractional diﬀerential equations based on the generalized Lagurre polynomials, Journal of Fractional Calculus and Application, 3 (12) (2012) 1-14.

J. D. Munkhammar, Riemann-Liouville Fractional Derivatives and The Taylor-Riemmann Series, U. U. D. M. Project Report, 2004.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Diﬀerential Equations, North-Holland Mathematic Studies, Elsvier, 204 (2006).

B. Arabzadeh, M. Razzaghi and Y. Ordukhani, Numerical solution of linear time-varying diﬀeren- tial equations using hybrid of block-pulse and rationalized Haar functions, Journal of Vibration and Control, 12 (2006) 1081-1092.

Y. Ordokhani, Solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via rational- ized Haar functions, Applied Mathmatics and Computation, 187 (2006) 436-443.

G. M. Philips and P.J. Taylor, Theory and Application of Numerical Analysis, Academic Press, New York, 1973.

A. Kilicman and Z. A. A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Applied Mathematics and Computation, 187 (2007) 250-265.

Volume 6, 2 (SPRING)

Winter 2016

Pages 149-158