Identification of the initial value for a space-time fractional diffusion equation

TL;DR

This paper addresses the inverse problem of determining the initial condition in a space-time fractional diffusion equation, demonstrating its ill-posedness and proposing a regularized conjugate gradient method for stable solutions.

Contribution

It proves the uniqueness and ill-posedness of the inverse problem and introduces a novel regularization approach with numerical validation.

Findings

The inverse problem is uniquely solvable from final data.

The problem is ill-posed, lacking continuous dependence on data.

The proposed method achieves high accuracy with noisy data.

Abstract

In this paper, we study an inverse problem for identifying the initial value in a space-time fractional diffusion equation from the final time data. We show the identifiability of this inverse problem by proving the existence of its unique solution with respect to the final observed data. It is proved that the inverse problem is an ill-posed problem. Namely, we prove that the solution to the inverse problem does not depend continuously on the measured data. The inverse problem is formulated as a regularized optimization one minimizing a least-squares type cost functional. Then the conjugate gradient method combined with Morozov's discrepancy is proposed for finding a stable approximate solution to the regularized variational problem. Numerical examples with noise-free and noisy data illustrate the applicability and high accuracy of the proposed method to some extent.