Abstract: A new implicit difference method is proposed for a time fractional diffusion equation, in which space derivatives are discretized by the central difference method. For a time fractional derivative, the Caputo fractional derivative is transformed into the Riemman-Liouville fractional derivative, and further forming it in the form of the Hadamard finite part integral. The finite part integral is then approximated by piecewise quadratic polynomials. A new 3-α order approximation scheme to the Riemman-Liouville fractional derivative can be derived as result of the approximation. Consequently, an implicit difference scheme for fractional diffusion equations, which is unconditionally stable and convergent, can be obtained. Our numerical experiments verify effectiveness of the implicit difference scheme.

Key words: fractional derivatives, Hadamard finite-part integral, piecewise quadratic interpolation polynomials, stability