Wavelet and Fourier methods for solving the sideways heat equation
2000 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 21, no 6, 2187-2205 p.Article in journal (Refereed) Published
We consider an inverse heat conduction problem, the sideways heat equation, which is a model of 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 data given along the line x = 1, where the solution is wanted for 0 ≤ x < 1.
The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge–Kutta method.
We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.
Place, publisher, year, edition, pages
2000. Vol. 21, no 6, 2187-2205 p.
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-47660DOI: 10.1137/S1064827597331394OAI: oai:DiVA.org:liu-47660DiVA: diva2:268556