NDLI: Wavelet and Fourier Methods for Solving the Sideways Heat Equation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Berntsson, Fredrik Eldn, Lars Reginska, Teresa
Copyright Year	2000
Abstract	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 $0 \leq x < 1$.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 onreplacing 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.
Starting Page	2187
Ending Page	2205
Page Count	19
File Format	PDF
ISSN	10648275
DOI	10.1137/S1064827597331394
e-ISSN	10957197
Issue Number	6
Volume Number	21
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	inverse problem Improperly posed problems Inverse problems ill-posed heat conduction wavelet Fourier analysis Cauchy problem
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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