Identification of the Thermal Conductivity Coefficient in the Heat Conduction Model with Fractional Derivative

• Autorzy: Brociek R. , Słota D.

Identyfikatory

DOI

10.24425/afe.2019.127136

• Języki publikacji: EN

Abstrakty

EN

Main goal of the paper is to present the algorithm serving to solve the heat conduction inverse problem. Authors consider the heat conduction equation with the Riemann-Liouville fractional derivative and with the second and third kind boundary conditions. This type of model with fractional derivative can be used for modelling the heat conduction in porous media. Authors deal with the heat conduction inverse problem, which, in this case, consists in identifying an unknown thermal conductivity coefficient. Measurements of temperature, in selected point of the region, are the input data for investigated inverse problem. Basing on this information, a functional describing the error of approximate solution is created. Minimizing of this functional is necessary to solve the inverse problem. In the presented approach the Ant Colony Optimization (ACO) algorithm is used for minimization.

Słowa kluczowe

EN

heat conduction equation; thermal conductivity coefficient; inverse problem; fractional derivative; thermal conductivity

PL

model przewodzenia ciepła; współczynnik przewodzenia ciepła; problem odwrotny; pochodna ułamkowa; przewodność termiczna

• Wydawca: Komisja Odlewnictwa Polskiej Akademii Nauk Oddział w Katowicach
• Czasopismo: Archives of Foundry Engineering
• Rocznik: 2019
• Tom: iss. 3
• Strony: 38--42
• Opis fizyczny: Bibliogr. 19 poz., tab., wykr.

Twórcy

autor

Brociek R.

• Silesian University of Technology, Institute of Mathematics, Gliwice, Poland

autor

Słota D.

• Silesian University of Technology, Institute of Mathematics, Gliwice, Poland

Bibliografia

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• [3] Kosztołowicz, T. (2008). Application of differential equations with fractional derivatives to the description of subdiffusion. UJK, Kielce (in Polish).
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• [6] Sowa, M. (2018). A harmonic balance methodology for circuits with fractional and nonlinear elements. Circuits, System, and Signals Processing. 37(11), 4695-4727. DOI: 10.1007/s00034-018-0794-8.
• [7] Voller, V.R. (2016). Computations of anomalous phase change. International Journal of Numerical Methods for Heat Fluid Flow. 26, 624-638. DOI: 10.1108/HFF-08-2015-0326.
• [8] Voller, V.R. (2018). Anomalous heat transfer: examples, fundamentals, and fractional calculus models. Advances in Heat Transfer. 50, 333-380. DOI: 10.1016/bs.aiht. 2018.06.001.
• [9] Szymanek, E., Błaszczyk, T., Hall, M.R., Keikhaei Dehdezi, P. & Leszczyński, J.S. (2014). Modelling and analysis of heat transfer through 1D complex granular system. Granular Matter. 16, 687-694. DOI: 10.1007/s10035-014-0517-1.
• [10] Obrączka, A., Kowalski, J. (2012). Modeling of heat distribution in ceramic materials using differential equations of incomplete order. Materiały XV Jubileuszowego Sympozjum ,,Podstawowe Problemy Energoelektroniki, Elektromechaniki i Mechatroniki” (Ed. M. Szczygieł), Vol. 32 of Archiwum Konferencji PTETiS, Komitet Organizacyjny Sympozjum PPEE i Seminarium BSE, 133-135.
• [11] Zhuang, Q., Yu, B. & Jiang, X. (2015). An inverse problem of parameter estimation for time-fractional heat conduction in a composite medium using carbon–carbon experimental data. Physica B. 456, 9-15. DOI: 10.1016/j.physb. 2014.08.011.
• [12] Murio, D.A. (2007). Stable numerical solution of a fractional diffusion inverse heat conduction problem. Computers & Mathematics with Applications. 53, 1492-1501. DOI: 10.1016/j.camwa.2006.05.027.
• [13] Murio, D.A. (2008). Time fractional IHCP with Caputo fractional derivatives. Computers & Mathematics with Applications. 56, 2371-2381. DOI: 10.1016/j.camwa. 2008.05.015.
• [14] Murio, D.A. & Mejia, C.E. (2008). Generalized time fractional IHCP with Caputo fractional derivatives. Journal of Physics: Conference Series. 135:012074 (8pp). DOI: 10.1088/1742-6596/135/1/012074.
• [15] Murio, D.A. (2009). Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional IHCP. Inverse Problems in Science and Engineering. 17, 229-243. DOI: doi.org/10.1080/ 17415970802082872.
• [16] Zhang, Z. (2016). An undetermined coefficient problem for a fractional diffusion equation. Inverse Problems. 32(1): 015011. DOI: 10.1088/0266-5611/32/1/015011.
• [17] Brociek, R. & Słota, D. (2016). Implicit finite difference method for space fractional heat conduction equation with mixed boundary conditions. Silesian Journal of Pure and Applied Mathematics. 6, 125-136.
• [18] Socha, K. & Dorigo, M. (2008). Ant colony optimization for continuous domains. European Journal of Operational Research. 185, 1155-1173. DOI: 10.1016/j.ejor.2006.06.046.
• [19] Brociek, R. & Słota, D. (2017). Application of Real Ant Colony Optimization algorithm to solve space and time fractional heat conduction inverse problem. Information Technology and Control. 46(2), 171-182. DOI: 10.5755/j01.itc.46.2.17.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).