Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8901639 | Journal of Computational and Applied Mathematics | 2019 | 27 Pages |
Abstract
This article deals with a backward diffusion problem for an inhomogeneous backward diffusion equation with fractional Laplacian in R: ut(x,t)+âÎαu(x,t)=f(x,t),(x,t)âRÃ[0,T],u(x,T)=g(x),xâR,limxâ±âu(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 L2(R) and Lp(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.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Triet Le Minh, Tran Thi Khieu, Tra Quoc Khanh, Hoang-Hung Vo,