Wavelet and error estimation of surface heat flux

We consider the determination of the surface heat flux of a body from a measured temperature history at a fixed location inside the body. Mathematically, it is formulated as a problem for the one-dimensional heat equation in a quarter plane with data given along the line x=1, and the gradient of the solution is wanted for 0<x<1. This problem is severely ill-posed and there are few theoretic results. The results available in the literatures are mainly devoted to the case of determining surface temperature. The aim of this paper is to indicate a possibility of a new approach for solving this problem based on an application of wavelet basis decomposition of measured data. The error estimate with Hِlder type is given and the convergence of regularization approximation solution at zero is also obtained which is a difficult problem left over by some authors.