Numerical Fractional Differentiation: Stability Estimate and Regularization

It is well known that the problem of fractional differentiation n is an ill-posed problem. So far there exists many approximation methods for solving this problem. In this paper we prove a stability estimate for a problem of fractional differentiation. Based on the obtained stability estimate, we present a Tikhonov regularization method and obtain the error estimate. According to the optimality theory of regularization, the error estimates are order optimal. Numerical experiment shows that the regularization works well.