Temperature distribution changes analysis based on Gr¨unwald-Letnikov space derivative
In this paper, the numerical approximation of the Dual-Phase-Lag (DPL) model has been presented. The approximation scheme is based on the Gr¨unwald-Letnikov (GL) definition of fractional derivatives. Moreover, that definition has been applied to the space derivative of the temperature in Fourier-Kirchhoff (FK) model. Then, the Dual-Phase-Lag model has been approximated based on the prepared modification of FK model, which has been called the space GL FK model. Furthermore, the finite difference method methodology for the approximation of the considered thermal model has also been employed. All mathematical formulas obtained during the determination of the approximation scheme are presented in the paper. Numerical examples of temperature distributions obtained in the case of new space GL FK model are also included in the paper. Moreover, the behavior of the model based on order parameter values has also been investigated.
J. B. J. Fourier, Th´eorie analytique de la chaleur, Firmin Didot, Paris, 1822.
A. Nabovati, et al, On the lattice boltzmann method for phonon transport, Journal of Computational Physics, 5864–5876, 2011.
T. Raszkowski, A. Samson, The Numerical Approaches to Heat Transfer Problem in Modern Electronic Structures, Computer Science, vol. 18 (1) , 71–93, 2017.
D. Y. Tzou, A Unified Field Approach for Heat Conduction From Macro- to Micro- Scales, Transactions of ASME J. Heat Transfer, 8–16, 1995.
T. Raszkowski, M. Zubert, M. Janicki, A. Napieralski, Numerical solution of 1-D DPL heat transfer equation, Proc. of 22nd International Conference Mixed Design of Integrated Circuits and Systems MIXDES 2015, 436–439, Torun, 2015.
B. A. Jacobs, A new Grunwald Letnikov derivative from second order scheme, Abstract and Applied Analysis, vol. 2015, 2015.
Copyright (c) 2020 Łódzkie Towarzystwo Naukowe
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.