Heat conduction in a finite medium using the fractional single-phase-lag model

• Autorzy: Siedlecka Urszula

Identyfikatory

DOI

10.24425/bpas.2019.128599

• Języki publikacji: EN

Abstrakty

EN

In the paper, a solution of the time-fractional single-phase-lagging heat conduction problem in finite regions is presented. The heat conduction equation with the Caputo time-derivative is complemented by the Robin boundary conditions. The Laplace transform with respect to the time variable and an expansion in the eigenfunctions series with respect to the space variable was applied. A method for the numerical inversion of the Laplace transforms was used. Formulation and solution of the problem cover the heat conduction in a finite slab, hollow cylinder and hollow sphere. The effect of the fractional order of the Caputo derivative and the phase-lag parameter on the temperature distribution in a slab has been numerically investigated.

Słowa kluczowe

EN

single-phase-lagging heat conduction; Caputo derivative; fractional heat conduction

PL

przewodzenie ciepła; pochodna Caputo

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2019
• Tom: Vol. 67, nr 2
• Strony: 401--407
• Opis fizyczny: Bibliogr. 22 poz., wykr., tab.

Twórcy

autor

Siedlecka Urszula

• Institute of Mathematics, Czestochowa University of Technology, 21 Armii Krajowej Ave., 42-201 Częstochowa, Poland

Bibliografia

• [1] M.N. Özişik, Heat conduction, Wiley, New York, 1993.
• [2] D.Y. Tzou, “Thermal shock phenomena under high-rate response in solids”, Annual Review of Heat Transfer 4, 111‒185 (1992).
• [3] S. Momani and Z. Odibat, “A novel method for nonlinear fractional partial differential equations : Combination of DTM and generalized Taylor’s formula”, Journal of Computational and Applied Mathematics 220, 85‒95 (2008).
• [4] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
• [5] M. Ciesielski, M. Klimek and T. Blaszczyk, “The fractional Sturm-Liouville problem – Numerical approximation and application in fractional diffusion”, Journal of Computational and Applied Mathematics 317, 573‒588, (2017).
• [6] T. Blaszczyk and M. Ciesielski, “Numerical solution of Euler-Lagrange equation with Caputo derivatives”, Advances in Applied Mathematics and Mechanics 9 (1), 173‒185, (2017).
• [7] Y. Povstenko, “Time-fractional heat conduction in a two-layer composite slab”, Fractional Calculus and Applied Analysis 19 (4), 940‒953, (2016).
• [8] Y. Povstenko and J. Klekot, “The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space”, Journal of Applied Mathematics and Computational Mechanics 14 (2), 73‒83, (2015).
• [9] S. Kukla and U. Siedlecka, “An analytical solution to the problem of time-fractional heat conduction in a composite sphere”, Bull. Pol. Ac.: Tech. 65 (2), 179‒186, (2017).
• [10] S. Kukla and U. Siedlecka, “Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab”, Journal of Applied Mathematics and Computational Mechanics 14 (4), 105‒113, (2015).
• [11] K.L. Kuhlman, “Review of inverse Laplace transform algorithms for Laplace-space numerical approaches”, Numerical Algorithms 63 (2), 339‒355, (2013).
• [12] H. Sheng, Y. Li and Y.Q. Chen, “Application of numerical inverse Laplace transform algorithms in fractional calculus”, Journal of the Franklin Institute 348, 315‒330, (2011).
• [13] E. Majchrzak and B. Mochnacki, “Implicit scheme of the finite difference method for 1D dual-phase lag equation”, Journal of Applied Mathematics and Computational Mechanics 16 (3), 37‒46, (2017).
• [14] H.-Y. Xu, X.-Y. Jiang, “Time fractional dual-phase-lag heat conduction equation”, Chinese Physics B 24 (3), 034401‒1, (2015),
• [15] T.N. Mishra and K.N. Rai, “Fractional single-phase-lagging heat conduction model for describing anomalous diffusion”, Propulsion and Power Research 5 (1), 45‒54, (2016).
• [16] A. Compte and R. Metzler, “The generalized Cattaneo equation for the description of anomalous transport processes”, Journal of Physics A: Mathematical and General 30, 7277‒7289, (1997).
• [17] W. Sumelka, “Thermoelasticity in the framework of the fractional continuum mechanics”, Journal of Thermal stresses 37 (6), 678‒706, (2014).
• [18] K. Diethelm, The analysis of fractional differential equations, Springer-Verlag Berlin Heidelberg, 2010.
• [19] M. Lewandowska and L. Malinowski, “An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides”, International Communications in Heat and Mass Transfer 33, 61‒69, (2006).
• [20] S.C. Mishra, N. Gullipalli, A. Jain, A. Barurah and A. Mukherjee, “Analysis of hyperbolic heat conduction in 1-D planar, cylindrical, and spherical geometry using the lattice Boltzmann method”, International Communications in Heat and Mass Transfer 74, 48‒54, (2016).
• [21] D.P. Gaver Jr., “Observing stochastic processes and approximate transform inversion”, Operational Research 14, 444‒459, (1966).
• [22] Wolfram Research, Inc., Mathematica, Version 5.2, Champaign, IL, 2005.

Uwagi

PL

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