The Simplified Tikhonov Regularization Method for Solving a Riesz–Feller Space-Fractional Backward Diffusion Problem

Abstract

In this paper, we consider a backward diffusion problem for a space-fractional diffusion equation. Such a problem is obtained from the classical diffusion equation in which the second-order space derivative is replaced with a Riesz–Feller derivative of order \(\alpha \in (0,2]\). The temperature is sought from a measured temperature history at a fixed time \(t=T\). This problem is ill-posed, i.e., the solution (if it exists) does not depend on the data. The simplified Tikhonov regularization method is proposed to solve this problem. Under the a priori bound assumptions for the exact solution, the convergence estimate is presented. Moreover, a posteriori parameter choice rule is proposed and the convergence estimate is also obtained. All estimates are Holder type. Numerical examples are presented to illustrate the validity and effectiveness of this method.