Two regularization methods for solving a Riesz–Feller space-fractional backward diffusion problem

Abstract

In this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) in a strip is investigated. 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 beta is an element of (0, 2]. We show that such a problem is severely ill-posed and further propose a new regularization method and apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical methods are effective.