  In this paper, compact finite difference schemes for the modified anomalous
fractional sub-diffusion equation and fractional diffusion-wave equation are
studied. Schemes proposed previously can at most achieve temporal accuracy of
order which depends on the order of fractional derivatives in the equations and
is usually less than two. Based on the idea of weighted and shifted Grunwald
difference operator, we establish schemes with temporal and spatial accuracy
order equal to two and four respectively.
