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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 Grünwald difference operator, we establish schemes with temporal and spatial accuracy order equal to two and four respectively. [ABSTRACT FROM AUTHOR]