Abstract
This paper describes reconstruction of the thermal conductivity coefficient in the time fractional diffusion equation. Additional information for the considered inverse problem was given by the temperature measurements at selected points of the domain. The direct problem was solved by using the finite difference method. To minimize functional defining the error of approximate solution the Fibonacci search algorithm was used.
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Ardakani, M., Khodadad, M.: Identification of thermal conductivity and the shape of an inclusion using the boundary elements method and the particle swarm optimization algorithm. Inverse Probl. Sci. Eng. 17, 855–870 (2009)
Battaglia, J.L., Cois, O., Puigsegur, L., Oustaloup, A.: Solving an inverse heat conduction problem using a non-integer identified model. Int. J. Heat Mass Transfer 44, 2671–2680 (2001)
Borukhov, V.T., Tsurko, V.A., Zayats, G.M.: The functional identification approach for numerical reconstruction of the temperature-dependent thermal-conductivity coefficient. Int. J. Heat Mass Transfer 52, 232–238 (2009)
Brociek, R.: Implicite finite difference metod for time fractional diffusion equations with mixed boundary conditions. Zesz. Nauk. PŚ., Mat. Stosow. 4 (in press 2014)
Carpinteri, A., Mainardi, F.: Fractal and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)
Chang, C.L., Chang, M.: Inverse estimation of the thermal conductivity in a one-dimensional domain by Taylor series approach. Heat Transfer Engineering 29, 830–838 (2008)
Czél, B., Gróf, G.: Inverse identification of temperature-dependent thermal conductivity via genetic algorithm with cost function-based rearrangement of genes. Int. J. Heat Mass Transfer 55, 4254–4263 (2012)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Divo, E., Kassab, A., Rodriguez, F.: Characterization of space dependent thermal conductivity with a BEM-based genetic algorithm. Numer. Heat Transfer A(37), 845–875 (2000)
Dou, F., Hon, Y.: Numerical computation for backward time-fractional diffusion equation. Eng. Anal. Bound. Elem. 40, 138–146 (2014)
Gabano, J.D., Poinot, T.: Fractional modelling and identification of thermal systems. Signal Processing 91, 531–541 (2011)
Gabano, J.D., Poinot, T.: Estimation of thermal parameters using fractional modelling. Signal Processing 91, 938–948 (2011)
Imani, A., Ranjbar, A.A., Esmkhani, M.: Simultaneous estimation of temperature-dependent thermal conductivity and heat capacity based on modified genetic algorithm. Inverse Probl. Sci. Eng. 14, 767–783 (2006)
Kanca, F., Ismailov, M.: The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data. Inverse Probl. Sci. Eng. 20, 463–476 (2012)
Klafter, J., Lim, S., Metzler, R.: Fractional dynamics. Resent advances. World Scientific, New Jersey (2012)
Kusiak, J., Danielewska-Tułecka, A., Oprocha, P.: Optimization. PWN, Warszawa (2009) (in Polish)
Majchrzak, E., Mendakiewicz, J., Piasecka-Belkhayat, A.: Algorithm of the mould thermal parameters identification in the system casting–mould–environment. J. Mater. Proc. Tech. 164-165, 1544–1549 (2008)
Mitkowski, W., Kacprzyk, J., Baranowski, J.: Advances in the Theory and Applications of Non-integer Order Systems. Springer Inter. Publ., Cham (2013)
Mitkowski, W., Obrączka, A.: Simple identification of fractional differential equation. Solid State Phenomena 180, 331–338 (2012)
Murio, D.A.: Stable numerical solution of a fractional-diffusion inverse heat conduction problem. Comput. Math. Appl. 53, 1492–1501 (2007)
Murio, D.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56, 1138–1145 (2008)
Murio, D.: Time fractional IHCP with Caputo fractional derivatives. Comput. Math. Appl. 56, 2371–2381 (2008)
Murio, D.: Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional IHCP. Inverse Probl. Sci. Eng. 17, 229–243 (2009)
Murio, D., Mejia, C.: Generalized time fractional IHCP with Caputo fractional derivatives. J. Phys. Conf. Ser. 135, 12074 (2008)
Obrączka, A., Kowalski, J.: Modeling of the heat distribution in ceramic materials by using the fractional differential equations. In: Szczygieł, M. (ed.) Proceedings of the XV Jubilee Symposium, Basic Problems in Power Electronics, Electromechanics and Mechatronics, PPEEm 2012, Gliwice, pp. 132–133 (2012) (in Polish)
Obrączka, A., Mitkowski, W.: The comparison of parameter identification methods for fractional partial differential equation. Solid State Phenomena 210, 265–270 (2014)
Orain, S., Scudeller, Y., Garcia, S., Brousse, T.: Use of genetic algorithms for the simultaneous estimation of thin film thermal conductivity and contact resistances. Int. J. Heat Mass Transfer 44, 3973–3984 (2001)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Poinot, T., Trigeassou, J.C.: Identification of fractional systems using an output-error technique. Nonlinear Dynamics 38, 133–154 (2004)
Rabsztyn, S., Słota, D., Wituła, R.: Gamma and Beta Functions, vol. 1. Wyd. Pol. Śl, Gliwice (2012) (in Polish)
Szczygieł, I.: Analysis of Inverse Heat Convection Problems. Zesz. Nauk. PŚ., Ener. 140, 1–141 (2005) (in Polish)
Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49, 926–935 (2013)
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Brociek, R., Słota, D., Wituła, R. (2015). Reconstruction of the Thermal Conductivity Coefficient in the Time Fractional Diffusion Equation. In: Latawiec, K., Łukaniszyn, M., Stanisławski, R. (eds) Advances in Modelling and Control of Non-integer-Order Systems. Lecture Notes in Electrical Engineering, vol 320. Springer, Cham. https://doi.org/10.1007/978-3-319-09900-2_22
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DOI: https://doi.org/10.1007/978-3-319-09900-2_22
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09899-9
Online ISBN: 978-3-319-09900-2
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