A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

Abstract

In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE) in x ∈ [0, L]. This problem is ubiquitous in science and engineering applications. The illposedness of the Cauchy problem is explained by its solution in frequency domain. Furthermore, the problem is formulated into a minimization problem with a modified Tikhonov regularization method. The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem. The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions. Finally, numerical examples illustrate the effectiveness of the proposed method.