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

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.