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A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

Published online by Cambridge University Press:  09 January 2015

Jingjun Zhao ,
Songshu Liu  and
Tao Liu
Jingjun Zhao*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Songshu Liu
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Tao Liu
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
*
*Email:hit_zjj@hit.edu.cn(J. J. Zhao)
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Abstract

In this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.

Keywords

47A5265J22Ill-posed problemCauchy problemmodified kernel method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 1 , February 2015 , pp. 31 - 42
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Beck, J. V., Blackwell, B. and Clair, S. R., Inverse Heat Conduction: Ill-posed Problems, Wiley, New York, 1985.Google Scholar
[2]Berntsson, F., A spectral method for solving the sideways heat equation, Inverse Probl., 15 (1999), pp. 891–906.Google Scholar
[3]Cannon, J. R., A Cauchy problem for the heat equation, Ann. Mat. Pura Appl., 66 (1964), pp. 155–166.Google Scholar
[4]Carasso, A. S., Determining surface temperature from interior observations, Siam J. Appl. Math., 42 (1982), pp. 558–574.Google Scholar
[5]Eldén, L., Approximations for a Cauchy problem for the heat equation, Inverse Probl., 3 (1987), pp. 263–273.Google Scholar
[6]Eldén, L., Berntsson, F. and Regińska, T., Wavelet and Fourier methods for solving the sideways heat equation, Siam J. Sci. Comput., 21 (2000), pp. 2187–2205.Google Scholar
[7]Eldén, L., Numerical solution of the sideways heat equation by difference approximation in time, Inverse Probl., 11 (1995), pp. 913–923.Google Scholar
[8]Hadamard, J., Lectures on Cauchy Problem in Linear Partial Differential Equations, Oxford University Press, 1923.Google Scholar
[9]Li, J., and Wang, F., Simplified Tikhonov regularization for two kinds of parabolic equations, J. Korean Math. Soc., 48 (2011), pp. 311–327.Google Scholar
[10]Manselli, P. and Miller, K., Cauculation of the surface temperature and heat flux on one side of a wall from measurements on the opposite side, Ann. Mat. Pura Appl., 123 (1980), pp. 161–183.Google Scholar
[11]Murio, D. A., The Mollification Method and the Numerical Solution of Ill-Posed Problems, New York: Wiley, 1993.CrossRefGoogle Scholar
[12]Qian, Z. and Fu, C. L., Regularization strategies for a two-dimensional inverse heat conduction problem, Inverse Probl., 23 (2007), pp. 1053–1068.Google Scholar
[13]Qian, Z., Regularization methods for a Cauchy problem for a parabolic equation in multiple dimensions, J. Inv. Ill-Posed Problems, 17 (2009), pp. 891-911.CrossRefGoogle Scholar