A REGULARIZATION METHOD FOR BACKWARD PROBLEMS OF SINGULARLY PERTURBED PARABOLIC AND FRACTIONAL DIFFUSION EQUATIONS

In this paper, backward problems of singularly perturbed parabolic and fractional diffusion equations are studied from the additional temperature data at fixed time $ t=T$. We analyze the ill-posedness of these two inverse problems, and apply the quasi-reversibility regularization method to solve these problems. Then we obtain the convergence rates of logarithmic and Hölder types for the backward problems. Finally, several one- and two-dimensional numerical examples are given to verify the effectiveness and feasibility of the proposed method.