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 $L^2\left({ℝ}^d\right)$. 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.