Sequential solution of the sideways heat equation by windowing of the data

The sideways heat equation is a one-dimensional model of a problem, where one wants to determine the temperature on the surface of a body using interior measurements. More precisely, we consider a heat conduction problem, where temperature and heat-flux data are available along the line x = 1 and the solution is sought in the interval 0 = x < 1. The problem is ill-posed in the sense that the solution does not depend continuously on the data. Stability can be restored by replacing the time derivative in the heat equation with a bounded spectral approximation. The cut off level in the spectral approximation acts as a regularization parameter, that controls the degree of smoothness in the solution. In certain applications one wants to solve the sideways heat equation in real time, i.e. to constantly update the solution as new measurements are recorded. For this case sequential solution methods are required