Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.