Numerical identification of the time-dependent coefficient in the heat equation with fractional Laplacian

TL;DR

This paper develops a stable numerical method for identifying a time-dependent source coefficient in a fractional Laplacian heat equation, ensuring accuracy and robustness even with noisy data.

Contribution

It introduces a novel finite-difference scheme with rigorous stability analysis and an efficient algorithm for inverse coefficient identification in fractional PDEs.

Findings

Proved uniqueness and stability of the inverse problem.

Developed a stable, convergent Crank-Nicolson scheme.

Validated robustness with numerical experiments under noise.

Abstract

We address the inverse problem of identifying a time-dependent source coefficient in a one-dimensional heat equation with a fractional Laplacian subject to Dirichlet boundary conditions and an integral nonlocal data. An a priori estimate is established to ensure the uniqueness and stability of the solution. A fully implicit Crank-Nicolson (CN) finite-difference scheme is proposed and rigorously analysed for stability and convergence. An efficient noise-stable computation algorithm is developed and verified through numerical experiments, demonstrating accuracy and robustness under noisy data.