A spectral method for solving the sideways heat equation

Abstract

We consider an inverse heat conduction problem, the sideways heat equation, which is the model of a problem where one wants to determine the temperature on the surface of a body, using interior measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, where the data are given along the line x = 1, and a 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. Continuous dependence of the data is restored by replacing the time derivative in the heat equation with a bounded spectral-based approximation. The cut-off level in the spectral approximation acts as a regularization parameter. Error estimates for the regularized solution are derived and a procedure for selecting an appropriate regularization parameter is given. The discretized problem is an initial value problem for an ordinary differential equation in the space variable, which can be solved using standard numerical methods, for example a Runge-Kutta method. As test problems we take equations with constant and variable coefficients.

Determination temperature of a backward heat equation with time-dependent coefficients

“…One of the major object of this paper is to provide a regularization method to establish the Hölder estimates for (1.1). The truncated regularization method is a very simple and effective method for solving some ill-posed problems and it has been successfully applied to some inverse heat conduction problems [2,7,11]. However, in many earlier works, we find that only logarithmic type estimates in L 2 -norm are available; and estimates of Hölder type on [0, T ] are very rare (see some remarks for more detail comparisons).…”

Numerical Methods for Reconstruction of the Source Term of Heat Equations From the Final Overdetermination

“…Due to the infinite span of the variable x in (2), when numerical methods can only deal with finite span problem, some techniques on transforming the function in the whole line to an approximate function in a limited span are needed. To solve this, we refer to [3,9,10,11].…”

Sequential Solution of the Sideways Heat Equation by Windowing of the Data

“…The most popular methods for solving the sideways heat equation cannot be used directly since they require that the temperature and heat-flux data, for the whole time domain, are collected before the solution can be computed. In particular this is true for the methods based on replacing the time derivative by a spectral or wavelet approximations [2,6,15]. Methods based on mollification [9,13,14,18] can either be considered as full-domain methods, or as sequential in time, depending on their implementation.…”

“…The appropriate choice for ξ c , based on a priori bounds on the solution and the noise level, was discussed in [2], see also [6]. More precisely, suppose that the heat equation is valid for 0 ≤ x ≤ L, where L ≥ 3, and that the bounds…”

“…In a series of papers [2,5,6,15] methods for solving the sideways heat equation numerically, where the time derivative ∂ t is replaced by a bounded operator, have been investigated. Thus, we discretize the problem in time, and solve it essentially as an initial value problem for a system ordinary differential equations.…”