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A quadrature tau method for fractional differential equations with variable coefficients. (English) Zbl 1269.65068
Summary: We develop a direct solution technique for solving multi-order fractional differential equations (FDEs) with variable coefficients using a quadrature shifted Legendre tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the tau method for FDEs with variable coefficients is treated using the shifted Legendre-Gauss-Lobatto quadrature. Numerical results are given to confirm the reliability of the proposed method for some FDEs with variable coefficients.

65L05 Numerical methods for initial value problems
34A08 Fractional ordinary differential equations and fractional differential inclusions
34A30 Linear ordinary differential equations and systems, general
Full Text: DOI
[1] Garrappa, R.; Popolizio, M., On the use of matrix functions for fractional partial differential equations, Math. comput. simulation, 81, 1045-1056, (2011) · Zbl 1210.65162
[2] Podlubny, I., ()
[3] Ray, S.S.; Bera, R.K., Solution of an extraordinary differential equation by Adomian decomposition method, J. appl. math., 4, 331-338, (2004) · Zbl 1080.65069
[4] Li, C.; Taoa, C., On the fractional Adams method, Comput. math. appl., 58, 1573-1588, (2009) · Zbl 1189.65142
[5] Abdulaziz, O.; Hashim, I.; Momani, S., Application of homotopy-perturbation method to fractional ivps, J. comput. appl. math., 216, 574-584, (2008) · Zbl 1142.65104
[6] Yanga, S.; Xiao, A.; Su, H., Convergence of the variational iteration method for solving multi-order fractional differential equations, Comput. math. appl., 60, 2871-2879, (2010) · Zbl 1207.65109
[7] Odibat, Z.; Momani, S.; Xu, H., A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. math. model., 34, 593-600, (2010) · Zbl 1185.65139
[8] Bhrawy, A.H.; Abd-Elhameed, W.M., New algorithm for the numerical solutions of nonlinear third-order differential equations using jacobi – gauss collocation method, Math. probl. eng., 2011, (2011) · Zbl 1217.65155
[9] Coutsias, E.A.; Hagstrom, T.; Torres, D., An efficient spectral method for ordinary differential equations with rational function, Math. comp., 65, 611-635, (1996) · Zbl 0846.65037
[10] Doha, E.H.; Bhrawy, A.H.; Hafez, R.M., A Jacobi-Jacobi dual-petrov – galerkin method for third- and fifth-order differential equations, Math. comput. model., 53, 1820-1832, (2011) · Zbl 1219.65077
[11] Doha, E.H.; Bhrawy, A.H.; Saker, R.M., Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations, Appl. math. lett., 24, 559-565, (2011) · Zbl 1236.65091
[12] Funaro, D., ()
[13] Esmaeili, S.; Shamsi, M., A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. nonlinear sci. numer. simul., 16, 3646-3654, (2011) · Zbl 1226.65062
[14] Ghoreishi, F.; Yazdani, S., An extension of the spectral tau method for numerical solution of multi-order fractional differential equations with convergence analysis, Comput. math. appl., 61, 30-43, (2011) · Zbl 1207.65108
[15] Pedas, A.; Tamme, E., On the convergence of spline collocation methods for solving fractional differential equations, J. comput. appl. math., 235, 3502-3514, (2011) · Zbl 1217.65154
[16] Vanani, S.K.; Aminataei, A., Tau approximate solution of fractional partial differential equations, Comput. math. appl., (2011) · Zbl 1228.65205
[17] Doha, E.H.; Bhrawy, A.H.; Ezz-Eldien, S.S., Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. math. model., (2011) · Zbl 1228.65126
[18] Doha, E.H.; Bhrawy, A.H.; Hafez, R.M., A Jacobi dual-petrov – galerkin method for solving some odd-order ordinary differential equations, Abstr. appl. anal., 2011, (2011) · Zbl 1216.65086
[19] Deng, J.; Ma, L., Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. math. lett., 23, 676-680, (2010) · Zbl 1201.34008
[20] EL-Sayed, A.M.A.; EL-Mesiry, A.E.M.; EL-Saka, H.A.A., Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. appl. math., 23, 33-54, (2004) · Zbl 1213.34025
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