Numerical Calculation of Fractional Derivatives for the Sinc Functions via Legendre Polynomials

Document Type : Original Scientific Paper


1 ‎Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎‎University of Kashan, ‎Kashan‎, ‎Iran

2 Department of Mathematics, Faculty of Sciences, Azarbaijan Shahid Madani University, Tabriz, Iran


‎This paper provides the fractional derivatives of‎ ‎the Caputo type for the sinc functions‎. ‎It allows to use efficient‎ ‎numerical method for solving fractional differential equations‎. ‎At‎ ‎first‎, ‎some properties of the sinc functions and Legendre‎ ‎polynomials required for our subsequent development are given‎. ‎Then‎ ‎we use the Legendre polynomials to approximate the fractional‎ ‎derivatives of sinc functions‎. ‎Some numerical examples are‎ ‎introduced to demonstrate the reliability and effectiveness of the‎ ‎introduced method‎.


[1] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.
[2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[3] M. A. Darani and N. Nasiri, A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations, Comput. Methods Differ. Equ. 1 (2) (2013) 96 − 107.
[4] M. A. Darani and A. Saadatmandi, The operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications, Comput. Methods Differ. Equ. 5 (1) (2017) 67 − 87.
[5] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. 59 (3) (2010) 1326 − 1336.
[6] A. Saadatmandi, Bernstein operational matrix of fractional derivatives and its applications, Appl. Math. Model. 38 (4) (2014) 1365 − 1372.
[7] A. Saadatmandi and M. Dehghan, A Legendre collocation method for fractional integro-differential equations, J. Vib. Control 17 (13) (2011) 2050 − 2058.
[8] A. Saadatmandi, M. Dehghan and M. R. Azizi, The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients, Commun. Nonlinear Sci. Numer. Simul. 17 (11) (2012) 4125 −4136.
[9] M. Abbaszadeh and A. Mohebbi, Fourth-order numerical solution of a fractional PDE with the nonlinear source term in the electroanalytical chemistry, Iranian J. Math. Chem. 3 (2012) 195 − 220.
[10] A. Mohebbi, M. Abbaszade and M. Dehghan, Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions(RBF) meshless method, Eng. Anal. Bound. Elem. 38 (2014) 72 − 82.
[11] S. Mashayekhi and M. Razzaghi, Numerical solution of the fractional Bagley Torvik equation by using hybrid functions approximation, Math. Methods Appl. Sci. 39 (3) (2016) 353 − 365.
[12] M. R. Azizi and A. Khani, Sinc operational matrix method for solving the Bagley-Torvik equation, Comput. Methods Differ. Equ. 5 (2017) 56 − 66.
[13] A. Saadatmandi, A. Asadi and A. Eftekhari, Collocation method using quintic B-spline and sinc functions for solving a model of squeezing flow between two infinite plates, Int. J. Comput. Math. 93 (2016) 1921 − 1936.
[14] S. Yeganeh, Y. Ordokhani and A. Saadatmandi, A sinc-collocation method for second-order boundary value problems of nonlinear integro-differential equation, J. Inform. Comput. Sci. 7 (2012) 151 − 160.
[15] E. Babolian, A. Eftekhari and A. Saadatmandi, A sinc-Galerkin technique for the numerical solution of a class of singular boundary value problems, Comput. Appl. Math. 34 (1) (2015) 45 − 63.
[16] K. Parand, M. Dehghan and A. Pirkhedri, The Sinc-collocation method for solving the Thomas-Fermi equation, J. Comput. Appl. Math. 237 (2013) 244−252.
[17] M. Nabati and M. Jalalvand, Solution of Troesch’s problem through double exponential sinc- Galerkin method, Comput. Methods Differ. Equ. 5 (2) (2017) 141 − 157.
[18] S. Alkan, K. Yildirim and A. Secer, An eflcient algorithm for solving fractional differential equations with boundary conditions, Open Phys. 14 (2016) 6−14.
[19] A. Secer, S. Alkan, M. A. Akinlar and M. Bayram, Sinc-Galerkin method for approximate solutions of fractional order boundary value problems, Bound. Value Probl. 2013 (2013) 281, 14 pp.
[20] A. Pirkhedri and H. H. S. Javadi, Solving the time-fractional diffusion equation via Sinc-Haar collocation method, Appl. Math. Comput. 257 (2015) 317 − 326.
[21] E. Hesameddini and E. Asadollahifard, A new reliable algorithm based on the sinc function for the time fractional diffusion equation, Numer. Algorithms 72 (2016) 893 − 913.
[22] N. H. Sweilam, A. M. Nagy and A. A. El-Sayed, Solving time-fractional order telegraph equation via sinc-Legendre collocation method, Mediterr. J. Math. 13 (6) (2016) 5119 − 5133.
[23] J. Lund and K. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992.
[24] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993.
[25] T. Okayama, T. Matsuo and M. Sugihara, Approximate formulae for fractional derivatives by means of Sinc methods, J. Concr. Appl. Math. 8 (3) (2010) 470 − 488.
[26] G. Baumann and F. Stenger, Fractional calculus and sinc methods, Fract. Calc. Appl. Anal. 143 (4) (2011) 568 − 622.
[27] F. B. Hildebrand, Introduction to Numerical Analysis, Dover Publications, New York, 1956.