Inverse source problem with a posteriori interior measurements for space-time fractional diffusion equations

TL;DR

This paper addresses an inverse source problem for space-time fractional diffusion equations, establishing uniqueness via fractional derivative properties and proposing a Tikhonov regularization-based numerical method with numerical validation.

Contribution

It introduces a novel approach combining theoretical uniqueness results with an effective numerical reconstruction method for fractional diffusion inverse problems.

Findings

Uniqueness proven using fractional derivative memory effects.

Numerical method successfully reconstructs sources from noisy data.

Algorithm demonstrates high accuracy and efficiency in examples.

Abstract

This paper investigates an inverse source problem for space-time fractional diffusion equations from a posteriori interior measurements. The uniqueness result is established by the memory effect of fractional derivatives and the unique continuation property. For the numerical reconstruction, the inverse problem is reformulated as an optimization problem with the Tikhonov regularization. We use the Levenberg-Marquardt method to identity the unknown source from noisy measurements. Finally, we give some numerical examples to illustrate the efficiency and accuracy of the proposed algorithm.