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.