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An extension of the spectral tau method for numerical solution of multi-order fractional differential equations with convergence analysis. (English) Zbl 1207.65108
Summary: The main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations (FDEs) based on a spectral Tau method. An extension of the operational approach of the Tau method with the orthogonal polynomial bases is proposed to convert FDEs to its matrix-vector multiplication representation. The fractional derivatives are described in the Caputo sense. The spectral rate of convergence for the proposed method is established in the \(\mathcal L^2\) norm. We tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence.

MSC:
65L99 Numerical methods for ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
45J05 Integro-ordinary differential equations
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[1] Alcoutlabi, M.; Martinez-Vega, J.J., Application of fractional calculus to viscoelastic behaviour modelling and to the physical ageing phenomenon in glassy amorphous polymers, Polymer, 39, 6269-6277, (1998)
[2] J.H. He, Nonlinear oscillation with fractional derivative and its applications, in: International Conference on Vibrating Engineering, vol. 98, Dalian, China, 1998, pp. 288-291.
[3] He, J.H., Some applications of nonlinear fractional differential equations and their approximations, Bull. sci. technol., 15, 2, 86-90, (1999)
[4] Momani, S.; Odibat, Z.; Erturk, V.S., Generalized differential transform method for solving a space and time-fractional diffusion-wave equation, Phys. lett. A, 370, 5-6, 379-387, (2007) · Zbl 1209.35066
[5] Abbasbandy, S., An approximation solution of a nonlinear equation with riemann – liouville’s fractional derivatives by he’s variational iteration method, J. comput. appl. math., 207, 53-58, (2007) · Zbl 1120.65133
[6] Sweilam, N.H.; Khader, M.M.; Al-Bar, R.F., Numerical studies for a multi-order fractional differential equation, Phys. lett. A, 371, 1-2, 26-33, (2007) · Zbl 1209.65116
[7] El-Sayed, A.M.A.; Gaber, M., The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. lett. A, 359, 3, 175-182, (2006) · Zbl 1236.35003
[8] Jafari, H.; Daftardar-Gejji, V., Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. comput. appl. math., 196, 2, 644-651, (2006) · Zbl 1099.65137
[9] Diethelm, K.; Walz, G., Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms, 16, 3-4, 231-253, (1997) · Zbl 0926.65070
[10] Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. lett. A, 365, 345-350, (2007) · Zbl 1203.65212
[11] Arikoglu, A.; Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos solitons fractals, 34, 5, 1473-1481, (2007) · Zbl 1152.34306
[12] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods fundamentals in single domains, (2006), Springer-Verlag Berlin · Zbl 1093.76002
[13] Ortiz, E.L.; Samara, H., An operational approach to the tau method for the numerical solution of nonlinear differential equations, Computing, 27, 1, 15-25, (1981) · Zbl 0449.65053
[14] Ortiz, E.L.; Samara, H., Numerical solution of differential eigenvalue problems with an operational approach to the tau method, Computing, 31, 2, 95-103, (1983) · Zbl 0508.65045
[15] Ortiz, E.L.; Pun, K.S., Numerical solution of nonlinear partial differential equations with the tau method, Proceedings of the international conference on computational and applied mathematics, Leuven, 1984, J. comput. appl. math., 12-13, 511-516, (1985) · Zbl 0579.65124
[16] El-Daou, M.K.; Khajah, H.G., Iterated solutions of linear operator equations with the tau method, Math. comp., 66, 217, 207-213, (1997) · Zbl 0855.47006
[17] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent. part II, Geophys. J. R. astron. soc., 13, 529-539, (1967)
[18] Oldham, K.B.; Spanier, J., ()
[19] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons, Inc. New York · Zbl 0789.26002
[20] Mason, J.C.; Handscomb, D.C., Chebyshev polynomials, (2003), Chapman-Hall, CRC Press · Zbl 1015.33001
[21] Kanwal, R.P., Linear integral equations, (1971), Birkhäuser Boston, Inc. Boston, MA · Zbl 0219.45001
[22] Gogatishvill, A.; Lang, J., The generalized Hardy operator with kernel and variable integral limits in Banach function spaces, J. inequal. appl., 4, 1, 1-16, (1999) · Zbl 0947.47020
[23] Chen, Y.; Tang, T., Convergence analysis of the Jacobi spectral collocation methods for Volterra integral equations with a weakly singular kernel, Math. comp., 79, 269, 147-167, (2010) · Zbl 1207.65157
[24] Naylor, A.W.; Sell, G.R., ()
[25] Guo, Ben-Yu; Shen, Jie; Wang, Li-Lian, Generalized Jacobi polynomials/functions and their applications, Appl. numer. math., 59, 5, 1011-1028, (2009) · Zbl 1171.33006
[26] Phillips, G.M., Interpolation and approximation by polynomials, (2003), Springer Verlag New York · Zbl 1023.41002
[27] Momani, S.; Odibat, Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos solitons fractals, 31, 5, 1248-1255, (2007) · Zbl 1137.65450
[28] El-Mesiry, A.E.M.; El-Sayed, A.M.A.; El-Saka, H.A.A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. math. comput., 160, 3, 683-699, (2005) · Zbl 1062.65073
[29] Sun Don, W.; Gottlieb, D., The chebyshev – legendre method: implementing Legendre methods on Chebyshev points, SIAM J. numer. anal., 31, 6, 1519-1534, (1994), (English summary) · Zbl 0815.65106
[30] Alpert, B.K.; Rokhlin, V., A fast algorithm for the evaluation of Legendre expansions, SIAM J. sci. stat. comput., 12, 1, 158-179, (1991) · Zbl 0726.65018
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