Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

Abstract

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.