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

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.