On three explicit difference schemes for fractional diffusion and diffusion-wave equations

Three explicit difference schemes for solving fractional diffusion and fractional diffusion-wave equations are studied. We consider these equations in both the Riemann–Liouville and the Caputo forms. We find that the Gorenflo et al (2000 J. Comput. Appl. Math. 118 175) and the Yuste–Acedo (2005 SIAM J. Numer. Anal. 42 1862) methods when applied to fractional diffusion equations are equivalent when BDF1 coefficients are used to discretize the fractional derivative operators, but that this is not the case for fractional diffusion-wave equations. The accuracy and stability of the three methods are studied. Surprisingly, the third method, that of Ciesielski–Leszczynski (2003 Proc. 15th Conf. on Computer Methods in Mechanics), although closely related to the Gorenflo et al method, is the least accurate, especially for short times. The stability analysis is carried out by means of a procedure close to the standard von Neumann method. We find that the stability bounds of the three methods are the same.

[1]  Diego A. Murio,et al.  Implicit finite difference approximation for time fractional diffusion equations , 2008, Comput. Math. Appl..

[2]  S. Ray.  Exact solutions for time-fractional diffusion-wave equations by decomposition method , 2006 .

[3]  Fawang Liu,et al.  New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation , 2008, SIAM J. Numer. Anal..

[4]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[5]  I. Sokolov,et al.  Anomalous transport : foundations and applications , 2008 .

[6]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[7]  K. Morton,et al.  Numerical Solution of Partial Differential Equations , 1995 .

[8]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[9]  I. Podlubny.  Fractional differential equations , 1998 .

[10]  Hossein Jafari,et al.  SOLVING FRACTIONAL DIFFUSION AND WAVE EQUATIONS BY MODIFIED HOMOTOPY PERTURBATION METHOD , 2007 .

[11]  H. R. Hicks,et al.  Numerical methods for the solution of partial difierential equations of fractional order , 2003 .

[12]  Santos B. Yuste,et al.  Weighted average finite difference methods for fractional diffusion equations , 2004, J. Comput. Phys..

[13]  Rudolf Gorenflo,et al.  Convergence of the Grünwald-Letnikov scheme for time-fractional diffusion , 2007 .

[14]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[15]  Target problem with evanescent subdiffusive traps. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  C. Lubich.  Discretized fractional calculus , 1986 .

[17]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

[18]  Mariusz Ciesielski,et al.  Numerical treatment of an initial-boundary value problem for fractional partial differential equations , 2006, Signal Process..

[19]  L. Acedo,et al.  Some exact results for the trapping of subdiffusive particles in one dimension , 2004 .

[20]  Zhi‐zhong Sun,et al.  A fully discrete difference scheme for a diffusion-wave system , 2006 .

[21]  Fawang Liu,et al.  A Fourier method for the fractional diffusion equation describing sub-diffusion , 2007, J. Comput. Phys..

[22]  R. Gorenflo,et al.  Time Fractional Diffusion: A Discrete Random Walk Approach , 2002 .

[23]  Santos B. Yuste,et al.  On an explicit finite difference method for fractional diffusion equations , 2003, ArXiv.

[24]  Weihua Deng,et al.  Remarks on fractional derivatives , 2007, Appl. Math. Comput..

[25]  Shaher Momani,et al.  Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation , 2007 .

[26]  Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  I. Turner,et al.  A fractional-order implicit difference approximation for the space-time fractional diffusion equation , 2006 .

[28]  Katja Lindenberg,et al.  Subdiffusive target problem: survival probability. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  O. Agrawal.  Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain , 2002 .