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Entropy 2013, 15(10), 4122-4133; doi:10.3390/e15104122

Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion

1
Institute of Mathematics and Computer Science, Jan Długosz University in Częstochowa, Armii Krajowej 13/15, Częstochowa 42-200, Poland
2
Department of Computer Science, European University of Informatics and Economics (EWSIE) Białostocka 22, Warsaw 03-741, Poland 
Received: 27 August 2013 / Revised: 22 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
(This article belongs to the Special Issue Dynamical Systems)
View Full-Text   |   Download PDF [230 KB, 24 February 2015; original version 24 February 2015]

Abstract

The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R)  and a matrix (R <  r < ∞)  being in perfect thermal contact at r = R  is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional order 0 < a ≤ 2 and 0 < β ≤ 2,  respectively. The Laplace transform with respect to time is used. The approximate solution valid for small values of time is obtained in terms of the Mittag-Leffler, Wright, and Mainardi functions.
Keywords: fractional calculus; non-Fourier heat conduction; fractional diffusion-wave equation; perfect thermal contact; Laplace transform; Mittag-Leffler function; Wright function; Mainardi function fractional calculus; non-Fourier heat conduction; fractional diffusion-wave equation; perfect thermal contact; Laplace transform; Mittag-Leffler function; Wright function; Mainardi function
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Povstenko, Y. Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion. Entropy 2013, 15, 4122-4133.

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