Iraqi Journal of Science | 2021
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 \xa0heat source coefficients \xa0in the one-dimensional \xa0parabolic heat equation.\xa0 \xa0This \xa0mathematical \xa0formulation \xa0ensures that the inverse problem \xa0has a unique \xa0solution.\xa0 \xa0However, the problem \xa0is still \xa0ill-posed since small errors \xa0in the input data lead to a drastic \xa0amount \xa0of errors in the output coefficients. \xa0The \xa0finite \xa0difference method \xa0with \xa0the Crank-Nicolson \xa0scheme is adopted \xa0as a direct \xa0solver of the problem in a fixed domain.\xa0 \xa0The inverse problem is solved subjected to both exact and noisy measurements \xa0by using the MATLAB \xa0optimization \xa0toolbox \xa0routine \xa0lsqnonlin , which is also applied to minimize the nonlinear \xa0Tikhonov \xa0regularization functional. \xa0The 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 \xa0numerical \xa0examples are presented to verify the accuracy and stability of the solutions.