Haar wavelets multi-resolution collocation analysis of unsteady inverse heat problems

In this paper, two different Haar wavelet collocation multi-resolution procedures are proposed for linear partial differential equations (PDEs) with an unknown space-dependent heat source and an unknown solution. An appropriate transformation is used to convert a non-homogeneous PDE into a homogeneous form. Two techniques based on multi-resolution Haar wavelets collocation methods are proposed for numerical evaluation of the unknown space-dependent heat source. In homogeneous form, first-order finite-difference approximation is used to discretize the time derivative and finite Haar wavelets series is used for approximation of the space derivatives. Unlike other numerical methods, the proposed methods have well-conditioned Haar coefficient matrices and need not be supplemented by any regularization technique. Several numerical experiments are carried out to validate accuracy, simple applicability and well-conditioned behaviour of the Haar system coefficient matrices of the proposed algorithms.