The backward problem of a stochastic space-fractional diffusion equation driven by fractional Brownian motion

Abstract

This paper is concerned with a backward problem of a stochastic space-fractional diffusion equation. The source term is driven by fractional Brownian motion. The well-posedness of the forward problem is studied at first. The backward problem is ill-posed, i.e., the solution of this problem does not depend continuously on the data. The instability is discussed in the sense of expectation and variance. A truncated regularization method is used to solve the backward problem. Under the a priori and the a posteriori regularization parameter choice rules, the error estimates between the regularization solution and the exact solution are obtained, respectively. Different numerical examples are presented to illustrate the validity and effectiveness of our method.