A backward problem for distributed order diffusion equation: uniqueness and numerical solution

Abstract

In this paper we consider the identification of the initial condition for a distributed order diffusion equation. We first prove the unique existence and the regularity properties of the strong solution on the bounded temporal-spacial domain based on the eigenfunction expansions. The ill-posedness of the backward problem is interpreted by the compactness of the observation operator. Next the Laplace transformation technique and analytic continuation method are adopted to prove the uniqueness of the backward problem. Then for stabilizing the ill-posed problem, the backward problem is formulated as a Tikhonov type optimization problems, and the conjugate gradient method is adopted to solve the optimization problem with the help of the variational adjoint technique. Finally four numerical examples are given to show the efficiency and stability of the proposed method.