Numerical identification of source terms for a two dimensional heat conduction problem in polar coordinate system

Abstract

This work investigates the inverse problem of reconstructing a spacewise dependent heat source in a two-dimensional heat conduction equation using a final temperature measurement. Problems of this type have important applications in several fields of applied science. Under certain assumptions, this problem can be transformed into a one-dimensional problem where the heat source only depends on the variable r. However, being different from other one-dimensional inverse heat source problems, there exists singularity on the coefficient of our model, which may make the analysis more difficult, regardless of theoretical or numerical. The inverse problem is reduced to an operator equation of the first kind and the corresponding adjoint operator is deuced. For the two dimensional case, i.e., f=f(r,θ), theoretical analysis can be done by similar derivation. Based on the landweber regularization framework, an iterative algorithm is proposed to obtain the numerical solution. Some typical numerical examples are presented to show the validity of the inversion method.