On a space fractional backward diffusion problem and its approximation of local solution

Abstract

This article deals with a backward diffusion problem for an inhomogeneous backward diffusion equation with fractional Laplacian in R : u t ( x , t ) + − Δ α u ( x , t ) = f ( x , t ) , ( x , t ) ∈ R × [ 0 , T ] , u ( x , T ) = g ( x ) , x ∈ R , lim x → ± ∞ u ( x , t ) = 0 . This problem is an ill-posed problem due to the instability in solution. The goal of this paper is not only to provide a simple but effective regularization scheme to obtain the Hölder convergence rate, but also to give an approximation of solution of the equation with fractional diffusion to the one of the equation with Laplacian in both L 2 ( R ) and L p ( R ) setting. This result holds, in particular, when f ( x , t ) is spatially compactly supported, in which the difficulties due to the fractional Laplacian have been successfully overcome thanks to an additional condition on Fourier transform of f . We further study the convergence of solution of inhomogeneous problem to that of the homogeneous problem. Finally, numerical simulations, with finite difference schemes, based on Discrete Fourier Transform (DFT) algorithm are also presented to illustrate the theoretical results.