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Numerical Solution of a Nonlinear Inverse Heat Conduction Problem
Linköping University, Department of Mathematics. Linköping University. (Scientific Computing)
2010 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
Abstract [en]

 The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, ux]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0.

The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense.

The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.

Place, publisher, year, edition, pages
2010. , 71 p.
Keyword [en]
inverse problem, ill-posed, Cauchy problem, heat conduction, well-posed, nonlinear problem, spline derivative, spectral method.
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-57486ISRN: LiTH - MAT - EX - 2010 / 10 - SEOAI: oai:DiVA.org:liu-57486DiVA: diva2:325988
Presentation
2010-06-10, Kompakta rummet, MAI, 13:15 (English)
Uppsok
Physics, Chemistry, Mathematics
Supervisors
Examiners
Available from: 2010-06-22 Created: 2010-06-21 Last updated: 2010-06-22Bibliographically approved

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Hussain, Muhammad Anwar
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