Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem

Abstract

In this article, we consider an inverse heat conduction problem of determining a time-dependent unknown heat source and unknown initial temperature by means of observations of the temperature at the final time and temperature profile at one fixed point over the time interval. We prove that the heat source and initial temperature can be determined uniquely from two kinds of data. Numerical solutions are obtained by solving a backward heat conduction problem (BHCP) and two numerical derivative problems. The method of fundamental solutions combined with a discrete Tikhonov regularization is adopted to solve the BHCP and a radial basis functions approximation method is used to obtain stable numerical derivatives. Four numerical examples are tested to show the efficiency and accuracy of the proposed method.