Sobolev‐type regularization method for the backward diffusion equation with fractional Laplacian and time‐dependent coefficient

Author:

Khieu Tran Thi12,Khanh Tra Quoc34ORCID

Affiliation:

1. Faculty of Mathematics and Computer Science University of Science Ho Chi Minh City Vietnam

2. Vietnam National University Ho Chi Minh City Vietnam

3. Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence Van Lang University Ho Chi Minh City Vietnam

4. Faculty of Fundamental Sciences Van Lang University Ho Chi Minh City Vietnam

Abstract

This work is concerned with an ill‐posed problem of reconstructing the historical distribution of a backward diffusion equation with fractional Laplacian and time‐dependent coefficient in multidimensional space. The investigated problem is regularized by a Sobolev‐type equation method. Unlike previous works, to prove the convergence of the regularized solution to the exact one, we only require a very weak and natural a priori condition that the solution belongs to the standard Lebesgue space . This is done by suitably employing the Lebesgue‐dominated convergence theorem. If we go further to impose a stronger a priori condition, one may know how fast the convergence is. Finally, some MATLAB‐based numerical examples are provided to confirm the efficiency of the proposed method.

Publisher

Wiley

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