Inverse problem for nonlinear backward space-fractional diffusion equation

Abstract

Abstract In this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) with nonlinear source in a strip is investigated. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order γ ∈ ( 0 , 2 ] {\gamma\in(0,2]} . We show that such a problem is severely ill-posed and further propose a new modified 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. Our method improves some results of a previous paper, including the earlier paper [28] and some other papers. A general case of nonlinear terms for this problem is also considered.