HYBRID OF RATIONALIZED HAAR FUNCTIONS METHOD FOR SOLVING DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
الموضوعات : فصلنامه ریاضی
1 - Department of Applied Mathematics, Faculty of Mathematical Sciences, Alzahra
University, Tehran, Iran.
2 - Department of Applied Mathematics, Faculty of Mathematical Sciences, Alzahra
University, Tehran, Iran.
الکلمات المفتاحية: Hybrid, Differential equation, Riemann-Liouville integral, Caputo fractional derivative, Rationalized Haar, Fractional operational matrix,
ملخص المقالة :
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.
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.
