On a time–space fractional backward diffusion problem with inexact orders

Abstract

In this paper, we focus on the backward diffusion problem with the Caputo fractional derivative operator in time and a general spatial nonlocal operator. For T>0 and s∈[0,T), we consider the problem (Ps) of recovering the distribution u(x,s) from a measure of the final data u(x,T) for the following non-homogeneous time–space fractional diffusion equation Dtαu(x,t)+KβLγu(x,t)=f(x,t)inRn×(0,T)subject to the final condition u(x,T)=uT(x)inRn. The derivative orders and the nonlocal operator are perturbed with noises. Firstly, for 0<s<T, we prove the well-posedness of Problem (Ps) by studying the unique existence and continuity with respect to the derivative orders, the source term as well as the final value of the solution. Secondly, for s=0, we verify the ill-posedness of Problem (P0) and use the method of modified iterated Lavrentiev to construct a regularization solution from inexact data and inexact derivative orders. We apply a modified form of the discrepancy principle to choose regularization parameter and establish new optimal convergence estimates between the exact solution and its regularized approximation.