Fractional Regularization Methods For Inverse Problems Of Time-fractional Diffusion Equation

The problem of recovering a space-dependent source for the time-fractional diffusion equation and the backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain is an important ill-posed problem in the inverse problem.Due to the approximate solution given by the classical regularization method is excessively smooth,for example,the jump of the exact solution,the classical method can not construct the feature of the solution,we considers that the fractional regularization method not only contains the classic regularization method,but also overcomes the excessive smoothness of the approximate solution.Although the fractional regularization method has been studied,until recently the theoretical results of most of these works are limited to the a-priori regularization parameter choice rule for compact operator equations Ax = y.We use fractional Tikhonov regularization method to identify a space-dependent source for the time-fractional diffusion equation and use fractional Landweber regularization method to solve the backward time-fractional diffusion problem.We discuss the selection of regularization parameters for the proposed methods and give the corresponding error estimates.Furthermore,numerical results show that the fractional regularization methods provided is effective and feasible.