Two regularization strategies for an evolutional type inverse heat source problem

This paper investigates an evolutional type inverse problem of determining an unknown heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which often rely on only one variable, the unknown coefficient in this paper depends not only on the space variable x, but also on time t. Two regularization strategies which are called the time semi-discrete scheme (TSDS) and the integral reconstruction scheme (IRS), respectively, are proposed to deal with such a problem. By the TSDS the inverse problem is transformed into a sequence of stationary inverse problems and the unknown heat source is reconstructed layer by layer, while the IRS is to recover the source function from the situation as a whole. Both theoretical and numerical studies are provided. Two numerical algorithms on the basis of the Landweber iteration are designed, and some typical numerical experiments are performed in this paper. The numerical results show that the proposed methods are stable and the unknown heat source is recovered very well.