The solution of nonlinear inverse boundary problem of heat conduction

Authors

  • Ю. М. Мацевитый A. N. Podgorny Institute of Mechanical Engineering Problems of NAS Ukraine V. N. Karazin Kharkiv National University, Ukraine
  • Н. А. Сафонов A. N. Podgorny Institute of Mechanical Engineering Problems of NAS Ukraine, Ukraine
  • В. В. Ганчин A. N. Podgorny Institute of Mechanical Engineering Problems of NAS Ukraine, Ukraine

Keywords:

inverse heat conduction problem, the method of weighted residuals in the form of Galerkin, heat flow, superposition principle, Tikhonov regularization method, stabilizer, regularization parameter, identification, approximation, Schoenberg splines of first

Abstract

In this paper, to obtain a stable solution of nonlinear inverse boundary problem of heat conduction the method of Tikhonov regularization with effectiveness-tive search algorithm regularizing parameter. Seeking the heat flux at the boundary of the time coordinate splines approximate Schoenberg first ste-interest. To apply the method of influence functions for the nonlinear heat conduction problem reduces it to a sequence of linear inverse boundary value problems using the diet-iteration process. This iterative process ends when the on-perёd specified accuracy for temperature recovery. The article presents a study on the use of the influence functions for approximating the solution of a linear edge-value problem of heat conduction. In particular it is shown that the influence functions are linearly independent in the time interval (0, ¥) at a fixed spatial variable. This fact is used to identify the temperature at the boundary or inside the area. Conducted numerous computational experiments using functional stabilizing zero and first order, and an analysis of the impact of the variance of the random error of measurement error in the obtained solution. The results of computational experiments revealed that for the class of first-order regularization was more effective than the regularization of the zero order. Also, the results of computational experiments show that by increasing the number of points where the specified Expo experimental temperature, increases the accuracy of the identification.

Author Biographies

Ю. М. Мацевитый, A. N. Podgorny Institute of Mechanical Engineering Problems of NAS Ukraine V. N. Karazin Kharkiv National University

Academician of the National Academy of Sciences of Ukraine

Н. А. Сафонов, A. N. Podgorny Institute of Mechanical Engineering Problems of NAS Ukraine

Candidate of Physical and Mathematical Sciences

References

Hadamard J. (1902) Sur les problems aux derivees partielles et leur significations physiques. Bull. Univ. Pricenton, № 13: 82-88.

Hadamard J. (1932) Le problem de Couchy et les ѐquation aux derivees partielles lineaires hyperboliques. Paris: Hermann, 542 p.

Bek J., Blackwell B, St. Clair Jr. Ch (1989) Nekorrektnye obratnye zadachi teploprovodnosti. Мoscow, Мir, 312 p. (in Russian)/.

Маtsevity Ju. М. (2002-2003) Оbratnye zadachi teploprovodnosti. Кyiv, Nauk. dumка, Volume 1: Метоdologia, 408 p.; Volume 2: Prilogenia, 392 p. (in Russian).

Коzdoba L. А., Кrukovskiy P. G. (1982) Метоdy reshehia obratnyh zadach teploperenosa. Кyiv, Nauk. dumка, 360 p. (in Russian).

Аlifanov О. М., Artuhin E. A., Rumjantsev S. V. (1988) Extremalnye меtоdy reshenia nekorrektnych zadach. Мoscow, Nauka, 288 p. (in Russian).

Тihonov А. N., Аrsenin V. Ja. (1979) Метоdy reshehia nekorrektnych zadach.Мjscow, Nauka, 288 p. (in Russian).

Forsythe J., Маlcolm V., Моler C. (1980) Маshinnye меtоdy маtematicheskih vychisleniy. Мoscow, Мir, 280 p. (in Russian).

Fletcher К. (1988) Chislennye меtody na оsnove меtoda Galerkina. Мoscow, Мir, 352 p. (in Russian).

Lykov А. V. (1967) Теоria teploprovodnosti. Мoscow, Vyssh. shkola, 600 p. (in Russian).

Mihlin S. G. (1970) Variacionnye меtody v matematicheskoy fizike. Мoscow, Nauka, 512 p. (in Russian).

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Published

2016-03-30

Issue

Vol. 19 No. 1 (2016)

Section

Applied mathematics

License

Copyright (c) 2016 Н. А. Сафонов, Ю. М. Мацевитый, В. В. Ганчин

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