Stability estimate and the modified regularization method for a Cauchy problem of the fractional diffusion equation

In this paper we investigate a non-characteristic Cauchy problem for a fractional diffusion equation. Using the Fourier transformation technique, we give a conditional stability estimate on the solution. Since the problem is highly ill-posed in the Hadamard sense, a modified version of the Tikhonov regularization technique is devised for stable numerical reconstruction of the solution. An error bound with optimal order is proven. For illustration, several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method.