×

zbMATH — the first resource for mathematics

A time-space collocation spectral approximation for a class of time fractional differential equations. (English) Zbl 1267.35243
Summary: A numerical scheme is presented for a class of time fractional differential equations with Dirichlet’s and Neumann’s boundary conditions. The model solution is discretized in time and space with a spectral expansion of Lagrange interpolation polynomial. Numerical results demonstrate the spectral accuracy and efficiency of the collocation spectral method. The technique not only is easy to implement but also can be easily applied to multidimensional problems.

MSC:
35R11 Fractional partial differential equations
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
Software:
FODE; ma2dfc
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[2] W. Wyss, “The fractional diffusion equation,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2782-2785, 1986. · Zbl 0632.35031 · doi:10.1063/1.527251
[3] W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134-144, 1989. · Zbl 0692.45004 · doi:10.1063/1.528578
[4] F. Huang and F. Liu, “The time fractional diffusion equation and the advection-dispersion equation,” The Australian & New Zealand Industrial and Applied Mathematics Journal, vol. 46, no. 3, pp. 317-330, 2005. · Zbl 1072.35218 · doi:10.1017/S1446181100008282
[5] F. Huang and B. Guo, “General solutions to a class of time fractional partial differential equations,” Applied Mathematics and Mechanics, vol. 31, no. 7, pp. 815-826, 2010. · Zbl 1204.35170 · doi:10.1007/s10483-010-1316-9
[6] Y. Luchko and R. Goren^\circ o, “The initial value problem for some fractional differen-tial equations with the Caputo derivatives,” Preprint Series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat Berlin, 1998.
[7] N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517-529, 2002. · Zbl 1029.34003 · doi:10.1016/S0096-3003(01)00167-9
[8] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. · Zbl 0818.26003
[9] J. Chen, F. Liu, and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1364-1377, 2008. · Zbl 1138.35373 · doi:10.1016/j.jmaa.2007.06.023
[10] M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 118, pp. 283-299, 2004. · Zbl 1126.76346
[11] T. A. M. Langlands and B. I. Henry, “The accuracy and stability of an implicit solution method for the fractional diffusion equation,” Journal of Computational Physics, vol. 205, no. 2, pp. 719-736, 2005. · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[12] Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533-1552, 2007. · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001
[13] C.-M. Chen, F. Liu, and K. Burrage, “Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 754-769, 2008. · Zbl 1144.65057 · doi:10.1016/j.amc.2007.09.020
[14] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. Vinagre Jara, “Matrix approach to discrete fractional calculus. II. Partial fractional differential equations,” Journal of Computational Physics, vol. 228, no. 8, pp. 3137-3153, 2009. · Zbl 1160.65308 · doi:10.1016/j.jcp.2009.01.014
[15] G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications, vol. 48, no. 7-8, pp. 1017-1033, 2004. · Zbl 1069.65094 · doi:10.1016/j.camwa.2004.10.003
[16] J. P. Roop, “Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 243-268, 2006. · Zbl 1092.65122 · doi:10.1016/j.cam.2005.06.005
[17] W. H. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204-226, 2008/09. · Zbl 1416.65344 · doi:10.1137/080714130
[18] E. Hanert, “A comparison of three Eulerian numerical methods for fractional-order transport models,” Environmental Fluid Mechanics, vol. 10, no. 1, pp. 7-20, 2010. · doi:10.1007/s10652-009-9145-4
[19] E. Hanert, “On the numerical solution of space-time fractional diffusion models,” Computers and Fluids, vol. 46, no. 1, pp. 33-39, 2011. · Zbl 1305.65212 · doi:10.1016/j.compfluid.2010.08.010
[20] X. Li and C. Xu, “A space-time spectral method for the time fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 2108-2131, 2009. · Zbl 1193.35243 · doi:10.1137/080718942
[21] Y. Chen and T. Tang, “Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel,” Mathematics of Computation, vol. 79, no. 269, pp. 147-167, 2010. · Zbl 1207.65157 · doi:10.1090/S0025-5718-09-02269-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.