A discrete sampling inversion scheme for the heat equation

Summary

We present a simple and extremely accurate procedure for approximating initial temperature for the heat equation on the line using a discrete time and spatial sampling. The procedure is based on the “sinc expansion” which for functions in a particular class yields a uniform exponential error bound with exponent depending on the number of spatial sample locations chosen. Further the temperature need only be sampled at one and the same temporal value for each of the spatial sampling points. ForN spatial sample points, the approximation is reduced to solving a linear system with a (2N+1)×(2N+1) coefficient matrix. This matrix is a symmetric centrosymmetric Toeplitz matrix and hence can be determined by computing only 2N+1 values using quadratures.