Regularization Methods For The Two Inverse Problems Connected With Fractional Differentiation

In this paper, we investigate two classical ill-posed problems related to the fractional differentiation. The first is the problem of fractional numerical differentiation,the second is a non-characteristic Cauchy problem for a fractional diffusion equation. We propose Fourier truncation method for solving the first problem. Since the classical Tikhonov regularization over-smooths, we present a modified version of Tikhonov regularization technique for the second problem. For these inverse problems, stability estimates are obtained and error bounds with optimal order are proved.