Hostname: page-component-758b78586c-t6f8b Total loading time: 0 Render date: 2023-11-29T22:15:28.628Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

Published online by Cambridge University Press:  28 May 2015

Jincheng Ren*
College of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450000, China Department of Mathematics, Southeast University, Nanjing 210096, China
Zhi-zhong Sun*
Department of Mathematics, Southeast University, Nanjing 210096, China
Corresponding author. Email address:
Corresponding author. Email address:


Core share and HTML view are not possible as this article does not have html content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Some efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and L1 approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

Research Article
Copyright © Global-Science Press 2014


[1]Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.Google Scholar
[2]Meerschaert, M.M., Benson, D. and Baeumer, B., Operator Lévy motion and multiscaling anomalous diffusion, Phys. Rev. E 63, 11121117 (2001).Google Scholar
[3]Solomon, T.H., Weeks, E.R. and Swinney, H.L., Observations of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow, Phys. Rev. Lett. 71, 39753979 (1993).Google Scholar
[4]Yuste, S.B. and Lindenberg, K., Subdiffusion-limited A+A reactions, Phys. Rev. Lett. 87, 118301 (2001).Google Scholar
[5]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Phys. A 314, 749755 (2002).Google Scholar
[6]Mainardi, F., Raberto, M., Gorenflo, R. and Scalas, E., Fractional calculus and continuous-time finance II: the waiting-time distribution, Phys. A 287, 468481 (2000).Google Scholar
[7]Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M., Application of a fractional advection-dispersion equation, Water Resources Res. 36, 14031412 (2000).Google Scholar
[8]Vong, S. and Wang, Z.B., High order difference schemes for a time fractional differential equation with Neumann boundary conditions, East Asian J. Appl. Math. 4 (to appear).Google Scholar
[9]Bagley, R.L. and Torvik, P.J., On the appearance of the fractional derivative in the behaviour of real materials, J. Appl. Mech. 51, 294298 (1984).Google Scholar
[10]Luchko, Y., Initial-boundary-value problems for the generalized multi-term time fractional diffusion equation, J. Math. Anal. Appl. 374, 538548 (2011).Google Scholar
[11]Daftardar-Gejji, V. and Bhalekar, S., Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl. 345, 754765 (2008).Google Scholar
[12]Jiang, H., Liu, F., Turner, I. and Burrage, K., Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain, J. Math. Anal. Appl. 389, 11171127 (2012).Google Scholar
[13]Jiang, H., Liu, F., Meerschaert, M.M. and McGough, R., Fundamental solutions for the multi-term modified power law wave equations in a finite domain, Elec. J. Math. Anal. Appl. 1, 5566 (2013).Google Scholar
[14]Chen, C., Liu, F., Turner, I. and Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys. 227, 886897 (2007).Google Scholar
[15]Lynch, V.E., Carreras, B.A. and Del-Castill-Negrete, D., Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. 192, 406421 (2003).Google Scholar
[16]Yuste, S.B. and Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42, 18621874 (2005).Google Scholar
[17]Yuste, S.B., Weighted average finite difference metods for fractional diffusion equations, J. Comput. Phys. 216, 264274 (2006).Google Scholar
[18]Zhuang, P., Liu, F., Anh, V. and Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal. 46, 10791095 (2008).Google Scholar
[19]Zhang, Y.N., Sun, Z.Z. and Wu, H.W., Error estimates of Crank-Nicolson-type difference scheme for the subdiffusion equation, SIAM J. Numer. Anal. 49, 23022322 (2011).Google Scholar
[20]Zhao, X. and Sun, Z.Z., A box-type scheme for the fractonal sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys. 230, 60616074 (2011).Google Scholar
[21]Ren, J.C., Sun, Z.Z. and Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys. 232, 456467 (2013).Google Scholar
[22]Li, C.P., Zhao, Z.G. and Chen, Y.Q., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl. 62, 855875 (2011).Google Scholar
[23]Deng, W.H., Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys. 227, 15101522 (2007).Google Scholar
[24]Ervin, V.J., Heuer, N. and Roop, J.P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal. 45, 572591 (2007).Google Scholar
[25]Ford, N.J., Xiao, J.Y. and Yan, Y.B., Stability of a numerical method for a space-time fractional telegraph equation, Comput. Meth. Appl. Math. 12, 116 (2012).Google Scholar
[26]Liu, F., Yang, C. and Burrage, K., Numerical method and analytical technique of the modifed anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231, 160176 (2009).Google Scholar
[27]Liu, Q., Liu, F., Turner, I. and Anh, V., Finite element approximation for a modifed anomalous subdiffusion equation, Appl. Math. Model. 35, 41034116 (2011).Google Scholar
[28]Liu, F., Meerschaert, M.M., McGough, R., Zhuang, P. and Liu, Q., Numerical methods for solving the multi-term time fractional wave equations, Fractional Calculus & Applied Analysis, 16, 925 (2013).Google Scholar
[29]Yang, S.P., Xiao, A.G. and Su, H., Convergence of the variational iteration method for sovling multi-order fractional differential equations, Comput. Math. Appl. 60, 28712879 (2010).Google Scholar
[30]Lin, Y. and Xu, C.J., Finite difference/spectral approximations for the time fractional diffusion equation, J. Comput. Phys. 225, 15331552 (2007).Google Scholar
[31]Sun, Z.Z. and Wu, X.N., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56, 193209 (2006).Google Scholar
[32]Sun, Z.Z., Compact difference schemes for heat equation with Neumann boundary conditions, Numer. Methods Partial Differential Equations 25, 13201341 (2009).Google Scholar
[33]Ren, J.C. and Sun, Z.Z., Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions, J. Sci. Comput. 56, 381408 (2013).Google Scholar
[34]Sun, Z.Z., Numerical Methods of Partial Differential Equations (2nd edition), Science Press, Beijing, 2012.Google Scholar
[35]Conte, S.D. and de Boor, C., Elementary Numerical Analysis: An Algorithmic Approach (3rd edition), McGraw-Hill, New York, 1980.Google Scholar