Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation
This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and heat source coefficients in the one-dimensional parabolic heat equation. This mathematical formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data lead to a drastic amount of errors in the output coefficients. The finite difference method with the Crank-Nicolson scheme is adopted as a direct solver of the problem in a fixed domain. The inverse problem is solved subjected to both exact and noisy measurements by using the MATLAB optimization toolbox routine lsqnonlin , which is also applied to minimize the nonlinear Tikhonov regularization functional. The thermal conductivity and heat source coefficients are reconstructed using heat flux measurements. The root mean squares error is used to assess the accuracy of the approximate solutions of the problem. A couple of numerical examples are presented to verify the accuracy and stability of the solutions.