Numerical Solution of Inverse Coefficient Problem Governing by Heat Equation under Mixed Boundary Conditions and General Integral Extra Data

Abstract

     This study investigates the numerical identification of time-dependent coefficients, numerically in a one-dimensional parabolic partial differential equation subject to nonlocal initial, mixed boundary, and overdetermination conditions of general integral type. The Crank-Nicolson finite difference method (FDM) was employed to solve the direct (forward) problem, whilst the inverse problem (IP) was reshaped as a nonlinear regularized least-square optimization problem and solved iteratively using the lsqnonlin routine from MATLAB's optimization toolbox, given the ill-posed nature of the IP, where small perturbations in input data may cause large deviations in the solution. Therefore, Tikhonov regularization (TR) was applied to ensure stability and improve the reliability of the numerical results.