Homotopy Analysis Method for Solving the Backward Problem for the Time-Fractional Diffusion Equation

Abstract

This paper deals with the backward problem of a nonhomogeneous time-fractional diffusion equation, that is, the problem of determining the past distribution of the substance from present measurements. By the separation of variables method, exact solutions of the forward and backward problems are obtained in terms of eigenfunctions and Mittag-Leffler functions. Contrary to the
forward problem, i.e., determining the present solution from given initial data, the backward problem, i.e., the problem of recovering the initial condition from noisy measurements of the final data, is proved to be ill-posed and highly unstable with respect to perturbations in the final data, and thus, some regularization technique is required. The novelty of the current work stems from utilizing the homotopy analysis method as a tool to obtain a regularization scheme to tackle the instability of the backward problem. Stability and convergence results of the proposed method are proved, and optimal convergence rates of the regularized solution are given under both a priori and a posteriori parameter choice rules. The resulted algorithm is very efficient and computationally inexpensive. Numerical examples are presented to illustrate the validity and accuracy of the proposed homotopy method.

Key words and phrases. Inverse problems, fractional diffusion, backward problem, homotopy analysis.

2010 Mathematics Subject Classification. 35R11, 65F22, 65J22, 47A52