Solving the backward problem in Riesz-Feller fractional diffusion by a new nonlocal regularization method

摘要

In this paper, we consider the backward problem introduced in [48] for Riesz-Feller fractional diffusion. To begin with, some basic properties of solution of the corresponding forward problem, such as the L-P estimates, symmetry property and asymptotic estimates, are established by Fourier analysis technique. And then, under various a priori bound assumptions, we give the L-2 conditional stability estimates for the solution of backward problem and also its symmetry property. Moreover, in order to overcome the ill-posedness of the backward problem, we propose a new nonlocal regularization method (NLRM) to solve it. That is, the following nonlocal variational functional is introducedJ(phi) = 1/2 parallel to(u(phi(x); x, T) - f (delta)(X)parallel to(2) + beta/2 parallel to[P-alpha 1 * phi](x)parallel to(2),where beta is an element of (0, 1) is a regularization parameter, "*" denotes the convolution operation and P-alpha 1 (x) is called a convolution kernel with parameter alpha(1), which will be selected properly later. The minimizer of above variational problem is defined as the regularization solution, and the L-2 estimates, symmetry property of regularization solution are given. These results actually show the well-posedness of nonlocal variational problem. Our idea is essentially that using this well-posed problem to approximate the backward (ill-posed) problem. Thus, under an a posteriori parameter choice rule, we deduce various convergence rate estimates under different a-priori bound assumptions for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective and adaptive for different a-priori information. (C) 2018 IMACS. Published by Elsevier B.V. All rights reserved.