On various approaches to solving the coefficient inverse problem of heat conductivity for a inhomogeneous rod

Authors

  • Nesterov S.A. Southern Mathematical Institute, a branch of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russian Federation ORCID 0000-0003-3780-5104

UDC

563.24

DOI:

https://doi.org/10.31429/vestnik-21-3-32-44

Abstract

In recent years, the theoretical basis of methods for non-destructive testing of materials with variable thermophysical properties, based on the apparatus of coefficient inverse problems of thermal conductivity, has been intensively improved. Two formulations of the inverse heat conduction problem for an inhomogeneous rod are presented. It is required to restore the thermophysical characteristics using additional information about the temperature measured at any point of the rod. In the case of the first setting, the temperature is measured at the end of the rod, at the location of the thermal load. The solution to the nonlinear inverse problem is based on an iterative approach of the Newtonian type, where at each iteration the Fredholm integral equation of the 1st kind is solved. Operator relations are obtained that characterize the sensitivity of additional information to changes in thermophysical characteristics. A numerical study was carried out of the influence of the time interval for collecting additional information on the accuracy of separate reconstruction of the thermal conductivity coefficient and specific heat capacity. In the case of the second setting, the temperature is measured at the internal point, and the loading is implemented at the end of the rod. The solution to the inverse problem is based on an iterative approach, where corrections are determined in the class of polynomial functions. After finding the initial approximation, at the first stage an iterative search for corrections is carried out in the class of linear functions, using additional information measured at two time points. At the second stage, an iterative search for corrections is carried out in the class of quadratic functions, with additional information being measured at three time points. Computational experiments were carried out on separate reconstruction of thermophysical characteristics, both monotonic and non-monotonic.

Keywords:

shooting method, polynomials, heat conduction, coefficient inverse problem, thermophysical characteristics, rod, iteration process

Acknowledgement

The study did not have sponsorship.

Author Info

Sergey A. Nesterov

д-р физ.-мат. наук, старший научный сотрудник отдела дифференциальных уравнений Южного математического института – филиала ВНЦ РАН

e-mail: 1079@list.ru

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Issue

Section

Mechanics

Pages

32-44

Submitted

2024-07-04

Published

2024-09-24

How to Cite

Nesterov S.A. On various approaches to solving the coefficient inverse problem of heat conductivity for a inhomogeneous rod. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2024, vol. 21, no. 3, pp. 32-44. DOI: https://doi.org/10.31429/vestnik-21-3-32-44 (In Russian)