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

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 ( P s ) 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 D t α u ( x , t ) + K β L γ u ( x , t ) = f ( x , t ) in R n × ( 0 , T ) subject to the final condition u ( x , T ) = u T ( x ) in R n . The derivative orders and the nonlocal operator are perturbed with noises. Firstly, for 0 s T , we prove the well-posedness of Problem ( P s ) 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 ( P 0 ) 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.