Numerical procedures for solving an inverse heat conduction problem are discussed.  More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with temperature data given along the line x = 1, where the solution is wanted for 0 £ x < 1. This problem is often referred to as the sideways heat equation.

The problem is analyzed, using both Fourier analysis and the singular value decomposition and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the problem. In this paper we attempt to give an overview that covers the most popular methods that have been considered. Numerical examples that illustrate the algorithms are given.