A multi-resolution collocation procedure for time-dependent inverse heat problems

In this paper, a Haar wavelet collocation method (HWCM) is developed for PDEs related to the framework of so-called inverse problem. These include PDEs with unknown time dependent heat source and unknown solution in interior of the domain. To this end, a transformation is used to eliminate the unknown heat source to obtain a PDE without a heat source. After elimination of unknown non-homogeneous term, an implicit finite-difference approximations is used to approximate the time derivative and Haar wavelets are used for approximation of the space derivatives. Several numerical experiments related to one- and two-dimensional heat sources are included to validate small condition number of coefficient matrix, accuracy and simple applicability of the proposed approach.