Finite Difference Method Of A Class Of Fractional Differential Equations On The Non-uniform Grid

Finite difference method on the non-uniform grid have been well developed to approximate the classical integrations as well as derivatives. While it is not easy to generalize them into the fractional cases, since the operators are non-local. In this pa-per we find that fractional integrations/derivatives can be valued adaptively in some degree. Specifically, it is natural that the existing fractional difference methods, which can solve the fractional integrations/derivatives of a smooth function, own "short mem-ory" principle as the fractional calculus. Basing on this, a general algorithm is given to cut down the calculations of these methods. For non-smooth cases, by subdividing the integration interval in the neighborhood of the non-smooth point, it is turned out the integrations/derivatives can be computed effectively too. This method is applied into space fractional diffusion equations. Detailed numerical experiments are demonstrated to show the effectiveness of these algorithms.