This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation. They are considered under mass/energy specification and Stefan conditions. The main issue with this problem is that the solution is unstable and sensitive to small contamination of noise in the input data. The Crank-Nicolson finite difference method (FDM) is utilized to solve the direct problem, whilst the inverse problem is viewed as a nonlinear optimization problem. The latter problem is solved numerically using the routine optimization toolbox lsqnonlin from MATLAB. Consequently, the Tikhonov regularization method is used in order to gain stable solutions. The results were compared with their exact solution and tested via

     The problem of reconstruction of a timewise dependent coefficient and free boundary at once in a nonlocal diffusion equation under Stefan and heat Flux as nonlocal overdetermination conditions have been considered. A Crank–Nicolson finite difference method (FDM) combined with the trapezoidal rule quadrature is used for the direct problem. While the inverse problem is reformulated as a nonlinear regularized least-square optimization problem with simple bound and solved efficiently by MATLAB subroutine lsqnonlin from the optimization toolbox. Since the problem under investigation is generally ill-posed, a small error in the input data leads to a huge error in the output, then Tikhonov’s regularization technique is app