Summary
We study boundary movement identification for a parabolic partial differential equation describing a dynamic diffusion process, on basis of internally recorded data. Formulated as a sideways diffusion equation, the problem is treated by a spatial continuation technique to extend the solution to a known boundary condition at the desired boundary position. Recording the positions traversed in the continuation for each time instant yields the boundary position trajectory and hence the solution of the identification problem. As the problem is ill-posed, a hyperbolic approximation approach is used to regularize the computation and recast the equations into a form amenable to analysis.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alifanov, O. M. (1994): Inverse Heat Transfer Problems. International Series in Heat and Mass Transfer, Springer-Verlag, Berlin
Berntsson, F. (2001): Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation. Technical Report LiTH-MAT-R-2001-17, Department of Mathematics, Linköping University, Linköping, Sweden
Berntsson, F., Eldén, L., Loyd, D., Garcia-Padrón, R. (1997): A comparison of three numerical methods for an inverse heat conduction problem and an industrial application. In: Proc. Tenth Int. Conf. for Numerical Methdods in Thermal Problems, volume X, Institute for Numerical Methods in Engineering, University of Wales Swansea, Pineridge Press, Swansea, UK
Cannon, J. R., Rundell, W. (1991): Recovering a time dependent coefficient in a parabolic differential equation. Journal of Mathematical Analysis and Applications, 572–582
Carasso, A. (1982): Determining surface temperatures from interior observations. SIAM J. Appl. Math., 42, 558–574
Carasso, A. S. (1992): Space marching difference schemes in the nonlinear inverse heat conduction problem. Inverse Problems, 8, 25–43
Eldén, L. (1988): Hyperbolic approximations for a Cauchy problem for the heat equation. Inverse Problems, 4, 59–70
Eldén, L. (1995): Numerical solution of the sideways heat equation. In: Engl, H. W., Rundell, W. (eds.) Proc. GAMM-SIAM Symp., Gesellschaft für Angewandte Mathematik and Mechanik, GAMM-SIAM, Regensburg, Germany
Eldén, L. (1995): Numerical solution of the sideways heat equation by difference approximation in time. Inverse Problems, 11, 913–923
Eldén, L. (1997): Solving an inverse heat conduction problem by a “method of lines”. Transactions of the ASME, 119, 406–412
Eldén, L., Berntsson, F., Regińska, T. (2000): Wavelet and Fourier methods for solving the sideways heat equation. SIAM J. Sci. Comput., 21, 2187–2205
Jones, Jr, B. F. (1963): Various methods for finding unknown coefficients in parabolic differential equations. Communications on Pure and Applied Mathematics, XVI, 33–44
Kato, T. (1984): Perturbation Theory for Linear Operators, 2nd edition. A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin
Mejía, C. E., Murio, D. A. (1993): Mollified hyperbolic method for coefficient identification problems. Computers Math. Aplic., 26, 1–12
Mejía, C. E., Murio, D. A. (1996): Numerical solution of generalized IHCP by discrete mollification. Computers Math. Aplic., 32, 33–50
Morse, P. M., Feshbach, H. (1953): Methods of Theoretical Physics, volume I of International Series in Pure and Applied Physics. McGraw-Hill Book Company, Inc., New York
Richtmyer, R. D., Morton, K. W. (1967): Difference Methods for Initial-Value Problems, 2nd edition. Tracts in Pure and Applied Mathematics, Wiley Interscience, New York
Seidman, T. I., Eldén, L. (1990): An ‘optimal filtering’ method for the sideways heat equation. Inverse Problems, 6, 681–696
Sobolev, S. L. (1989): Partial Differential Equations of Mathematical Physics. Dover Publications Inc., New York
Taler, J., Duda, P. (2001): Solution of non-linear inverse heat conduction problems using the method of lines. Heat and Mass Transfer, 37, 147–155
Tikhonov, A. N., Samarskii, A. A. (1990): Equations of Mathematical Physics. Dover Publications Inc., New York
Weber, C. F. (1981): Analysis and solution of the ill-posed inverse heat conduction problem. Int. J. Heat Mass Transfer, 24, 1783–1792
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fredman, T.P. (2004). A Boundary Movement Identification Method for a Parabolic Partial Differential Equation. In: Feistauer, M., Dolejší, V., Knobloch, P., Najzar, K. (eds) Numerical Mathematics and Advanced Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18775-9_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-18775-9_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62288-5
Online ISBN: 978-3-642-18775-9
eBook Packages: Springer Book Archive