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

Article
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; in revised form: 22 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013

Abstract

:
The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion Entropy 15 04122 i001 and a matrix Entropy 15 04122 i002 being in perfect thermal contact at Entropy 15 04122 i003 is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional orde Entropy 15 04122 i004 and Entropy 15 04122 i005 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
PACS Codes:
02.30.Gp; 44.10+i; 66.30.-h

1. Introduction

The standard heat conduction (diffusion) equation for temperature T
Entropy 15 04122 i006
is obtained from the balance equation for energy
Entropy 15 04122 i007
where ρ is the mass density, C is the specific heat capacity, q is the heat flux vector, and the classical Fourier law which states the linear dependence between the heat flux vector q and the temperature gradient
Entropy 15 04122 i008
with k being the thermal conductivity. In the heat conduction Equation (1) Entropy 15 04122 i009 is the heat diffusivity coefficient.
To describe heat conduction in media with complex internal structure, the standard parabolic Equation (1) is no longer accurate enough. In nonclassical theories, the Fourier law Equation (3) and the parabolic heat conduction Equation (1) are replaced by more general equations (see [1,2,3,4,5,6]). The time-nonlocal dependence between the heat flux vector q and the temperature gradient [7,8]
Entropy 15 04122 i010
results in the heat conduction with memory [7,8]
Entropy 15 04122 i011
Several particular cases of choice of the memory kernel Entropy 15 04122 i012 were analyzed in [9,10,11,12]. The time-nonlocal dependence between the heat flux vector q and the temperature gradient with the long-tail power kernel [9,10,11,12]
Entropy 15 04122 i013
Entropy 15 04122 i014
where Entropy 15 04122 i015 is the gamma function, can be interpreted in terms of fractional calculus:
Entropy 15 04122 i016
Entropy 15 04122 i017
where Entropy 15 04122 i018 and Entropy 15 04122 i019 are the Riemann–Liouville fractional integral and derivative of the order respectively [13,14,15,16]:
Entropy 15 04122 i020
Entropy 15 04122 i021
The balance Equation (2) and the constitutive Equations (8) and (9) yield the time-fractional equation
Entropy 15 04122 i022
with the Caputo fractional derivative
Entropy 15 04122 i023
The details of obtaining the time-fractional heat conduction Equation (12) from the balance Equation (2) and the constitutive Equations (8) and (9) can be found in [17].
Equations with fractional derivatives, in particular the time-fractional heat conduction equation (diffusion-wave equation), describe many important physical phenomena in different media (see [9,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32], among many others). Fractional calculus plays a significant part in studies of entropy [33,34,35,36,37,38]. It should be noted that entropy is also used in analysis of anomalous diffusion processes and fractional diffusion equation [39,40,41,42,43,44,45].
Different kinds of boundary conditions for Equation (12) in a bounded domain were analyzed in [46,47]. It should be emphasized that due to the generalized constitutive equations for the heat flux (8) and (9) the boundary conditions for the time-fractional heat conduction equation have their traits in comparison with those for the standard heat conduction equation. The Dirichlet boundary condition specifies the temperature over the surface of a body
Entropy 15 04122 i024
For time-fractional heat conduction Equation (12) two types of Neumann boundary condition can be considered: the mathematical condition with the prescribed boundary value of the normal derivative of temperature
Entropy 15 04122 i025
and the physical condition with the prescribed boundary value of the heat flux
Entropy 15 04122 i026
Entropy 15 04122 i027
Here n is the outer unit normal the boundary surface. Similarly, the mathematical Robin boundary condition is a specification of a linear combination of the values of temperature and the values of its normal derivative at the boundary of the domain
Entropy 15 04122 i028
with some nonzero constants c1 and c2, while the physical Robin boundary condition specifies a linear combination of the values of temperature and the values of the heat flux at the boundary of the domain. For example, the Newton condition of convective heat exchange between a body and the environment with the temperature TE
Entropy 15 04122 i029
where h is the convective heat transfer coefficient, leads to
Entropy 15 04122 i030
Entropy 15 04122 i031
If the surfaces of two solids are in perfect thermal contact, the temperatures on the contact surface and the heat fluxes through the contact surface are the same for both solids, and the boundary conditions of the fourth kind are obtained:
Entropy 15 04122 i032
Entropy 15 04122 i033
where subscripts 1 and 2 refer to the first and second solid, respectively, and n is the common unit normal at the contact surface. In fractional calculus, where integrals and derivatives of arbitrary (not only integer) order are considered, there is no sharp boundary between integration and differentiation. For this reason, some authors [15,25] do not use a separate notation for the fractional integral Entropy 15 04122 i034 The fractional integral of the order Entropy 15 04122 i035 is denoted as Entropy 15 04122 i036. In the equation of perfect thermal contact (23) Entropy 15 04122 i037 and Entropy 15 04122 i038, are understood in this sense.
Starting from the pioneering papers [48,49,50,51,52], considerable interest has been shown in solutions to time-fractional heat conduction equation. In the literature, there are only a few papers in which the fractional heat conduction equation (fractional diffusion-wave equation) is studied in composite medium [47,53,54]. In the present paper, the problem of fractional heat conduction in a composite medium consisting of a spherical inclusion Entropy 15 04122 i039 and a matrix Entropy 15 04122 i040 being in perfect thermal contact at Entropy 15 04122 i041 is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional order Entropy 15 04122 i042 and Entropy 15 04122 i043, respectively.

2. Statement of the Problem

Consider the time-fractional heat conduction equations in a spherical inclusion
Entropy 15 04122 i044
and in a matrix
Entropy 15 04122 i045
under the initial conditions
Entropy 15 04122 i046
Entropy 15 04122 i047
Entropy 15 04122 i048
Entropy 15 04122 i049
and the boundary condition of perfect thermal contact
Entropy 15 04122 i050
Entropy 15 04122 i051
The boundedness condition at the origin and the zero condition at infinity are also assumed:
Entropy 15 04122 i052
The limitations on α and β in Equations (26–29) express the fact that if Entropy 15 04122 i053 or Entropy 15 04122 i054, then the additional condition on the first time derivative should be also imposed.
In what follows we restrict ourselves to the particular case when a sphere Entropy 15 04122 i055 is at initial uniform temperature T0 and the matrix Entropy 15 04122 i056 is at initial zero temperature
Entropy 15 04122 i057
Entropy 15 04122 i058
Entropy 15 04122 i059
Entropy 15 04122 i060
The Laplace transform with respect to time t applied to Equations (24) and (25) leads to two ordinary differential equations
Entropy 15 04122 i061
Entropy 15 04122 i062
having the solutions
Entropy 15 04122 i063
Entropy 15 04122 i064
It follows from conditions at the origin and at infinity Equation (32) that
Entropy 15 04122 i065
The integration constants B1 and B2 are obtained from the perfect thermal contact boundary conditions Equations (30) and (31)
Entropy 15 04122 i066
Entropy 15 04122 i067
Hence, the solution is written as
Entropy 15 04122 i068
Entropy 15 04122 i069
Now we will investigate the approximate solution of the considered problem for small values of time. In the case of classical heat conduction this method was described in [55,56]. Based on Tauberian theorems for the Laplace transform (see, for example [57]), for small values of time t (the large values of the transform variable S) we can neglect the exponential term in comparison with 1,
Entropy 15 04122 i070
thus obtaining
Entropy 15 04122 i071
Entropy 15 04122 i072
In the following particular cases Entropy 15 04122 i073, Entropy 15 04122 i074; Entropy 15 04122 i075 the denominator in Equations (47) and (48) can be treated as a cubic equation and the decomposition into the sum of partial fractions can be obtained similar to that used in [58].
Now we will consider another particular case when Entropy 15 04122 i076
To invert the Laplace transform the following formula will be used [14,15,16]
Entropy 15 04122 i077
where Entropy 15 04122 i078 is the generalized Mittag-Leffler function in two parameters
Entropy 15 04122 i079
Additionally [51,52,59,60,61]
Entropy 15 04122 i080
Entropy 15 04122 i081
Entropy 15 04122 i082
Here Entropy 15 04122 i083 is the Wright function [1,51,52,62]
Entropy 15 04122 i084
whereas Entropy 15 04122 i085 is the Mainardi function [15,51,52]
Entropy 15 04122 i086
From Equations (47) and (48) we get:
Entropy 15 04122 i087
Entropy 15 04122 i088
where
Entropy 15 04122 i089
It should be emphasized that the solution is expressed in terms of the Mainardi function Entropy 15 04122 i090 and the Wright function Entropy 15 04122 i091. The limitation Entropy 15 04122 i092 in Equations (51–53) means that Entropy 15 04122 i093 in Equations (56) and (57).

4. Conclusions

We have obtained the approximate solution to the time-fractional heat conduction equations in a composite body consisting of a matrix and spherical inclusion with different thermophysical properties. The conditions of perfect thermal contact have been assumed: the temperatures at the boundary surface are equal and the heat fluxes through the contact surface are the same. The Laplace integral transform allows us to obtain the ordinary differential equations for temperatures. Inversion of the Laplace transform has been carried out analytically for small values of time.

Acknowledgments

The author thanks the anonymous reviewers for their helpful suggestions.

Conflicts of Interest

The author declares no conflict of interest.

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