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International Journal of Differential Equations
Volume 2012 (2012), Article ID 154085, 13 pages
Axisymmetric Solutions to Time-Fractional Heat Conduction Equation in a Half-Space under Robin Boundary Conditions
1Institute of Mathematics and Computer Science, Jan Dlugosz University, 42200 Czestochowa, Poland
2Department of Computer Science, European University of Informatics and Economics (EWSIE), 03741 Warsaw, Poland
Received 23 May 2012; Revised 18 July 2012; Accepted 23 July 2012
Academic Editor: Nikolai Leonenko
Copyright © 2012 Y. Z. Povstenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The time-fractional heat conduction equation with the Caputo derivative of the order is considered in a half-space in axisymmetric case under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of the values of temperature and the values of its normal derivative and the physical condition with the prescribed linear combination of the values of temperature and the values of the heat flux at the boundary.
The generalized Fourier law, the time-nonlocal dependence between the heat flux vector and the temperature gradient with the ####^~^~^~^~^~^####x201c;long-tail memory####^~^~^~^~^~^####x201d; power kernel [1, 2] (see also ) as where is the gamma function, can be interpreted in terms of the fractional calculus: and in combination with the law of conservation of energy as results in the time-fractional heat conduction equation with the Caputo fractional derivative Here, is the mass density, denotes the specific heat capacity, and is the thermal diffusivity coefficient.
Recall that the Riemann-Liouville fractional integral and derivative are defined as follows (see [4####^~^~^~^~^~^####x2013;6]): whereas the Caputo fractional derivative has the following form [5####^~^~^~^~^~^####x2013;7]:
A detailed explanation of derivation of time-fractional heat conduction equation (1.4) from the constitutive equations (1.2) and the law of conservation of energy (1.3) can be found in . Here we briefly present the main idea. In the case , as a consequence of (1.2), (1.3), and (1.6), we have or after integration with respect to time as Applying to both sides of (1.9) the Caputo derivative , and taking into account that for , we obtain the time-fractional heat conduction equation (1.4) for .
If the heat conduction equation is investigated in a bounded domain, the boundary conditions should be prescribed. 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 considered domain with some nonzero constants and , 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 condition of convective heat exchange between a body and the environment with the temperature where is the convective heat transfer coefficient, leads to
The literature on mathematical aspects concerning correctness of initial-boundary-value problems for time-fractional diffusion equation and form and properties of its solutions is quite extensive (see, e.g., [9####^~^~^~^~^~^####x2013;16], among others). Geometrical explanation of fractional calculus is given in [17####^~^~^~^~^~^####x2013;19].
There are only a few papers [20, 21] in which the fractional diffusion is investigated under the mathematical Robin boundary condition. In previous publications, problems for a cylinder  and a sphere  under mathematical and physical Neumann boundary conditions were considered. In the present paper, for the first time, the solutions to time-fractional heat conduction equation in a half-space are studied under both the mathematical and physical Robin boundary conditions. The Laplace integral transform with respect to time , the Hankel transform with respect to the spatial coordinate , and the sin-cos-Fourier transforms with respect to spatial coordinate are used. The solutions under the mathematical and physical Neumann boundary conditions are obtained as particular cases.
2. Mathematical Preliminaries
2.1. Laplace Transform
The Laplace transform is defined as where is the transform variable.
The inverse Laplace transfrom is carried out according to the Fourier-Mellin formula: where is a positive fixed number.
The Laplace transform of the Riemann-Liouville fractional integral of the order is carried out according to the formula similar to the Laplace transform of -fold primitive of a function as
The Caputo derivative for its Laplace transform requires the knowledge of the initial values of the function and its integer derivatives of the order whereas the Riemann-Liouville derivative for its Laplace transform rule requires the knowledge of the initial values of the fractional integral and its derivatives of the order The reader interested in applications of integral transforms in fractional calculus is referred to [4####^~^~^~^~^~^####x2013;7, 24].
2.2. Hankel Transform
The Hankel transform is used to solve problem in cylindrical coordinates in the domain and is defined as where is the Bessel function of the order .
The following formula is fulfilled:
2.3. Sin-Cos-Fourier Transform
In the case of boundary condition of the third kind with the prescribed boundary value of linear combination of a function and its normal derivative the following sin-cos-Fourier transform  is employed: with the kernel
Application of the sin-cos-Fourier transform to the second derivative of a function gives
3. Solution to the Problem under Mathematical Robin Boundary Condition
Consider the axisymmetric time-fractional heat conduction equation in cylindrical coordinates with zero initial conditions and the mathematical Robin boundary condition
The zero conditions at infinity are also assumed
The solution to the initial-boundary-value problem (3.1)####^~^~^~^~^~^####x2013;(3.4) can be written as where is the fundamental solution being the solution of the following problem: Here, is the Dirac delta function.
The Laplace transform with respect to time , the Hankel transform with respect to the spatial coordinate , and the sin-cos-Fourier transform with respect to the spatial coordinate result in where each of the integral transforms is denoted by the asterisk.
Invertion of the integral transforms leads to Here, is the Mittag-Leffler function in two parameters and defined by the series representation: The essential role of the Mittag-Leffler function in fractional calculus results from the following formula for the inverse Laplace transform Consider several particular cases of solution (3.8).
The case corresponds to the mathematical Neumann boundary condition with the prescribed boundary value of the normal derivative of temperature, and the solution reads
In the case of classical heat conduction (), the Mittag-Leffler function: and taking into account the following integrals [26, 27]: where is the modified Bessel function, is the complementary error function, we get
Of particular interest is also the case for which
4. Solution to the Problem under Physical Robin Boundary Condition
Consider the following axisymmetric time-fractional heat conduction equation: under zero initial conditions and the physical Robin boundary condition where .
The Laplace transform with respect to time leads to the following equation: and the following boundary condition:
In this case, the kernel of the sin-cos-Fourier transform with respect to the spatial coordinate depends on the Laplace transform variable and in the transform domain we obtain
Inversion of the Laplace transform in (4.10) depends on the value of . For , we have whereas for , we get The fundamental solution under physical Neumann boundary condition is obtained for as
We have derived the analytical solutions to time-fractional heat conduction equation in a half-space under mathematical and physical Robin boundary conditions. The integral transform technique has been used. It should be emphasized that in the case of physical Robin boundary condition, the order of integral transforms is important as the kernel of the sin-cos-Fourier transform depends on the Laplace transform variable. The limiting case corresponds to solutions of problems under mathematical and physical Neumann boundary conditions with the prescribed boundary value of the normal derivative and with the prescribed boundary value of the heat flux, respectively. The difference between mathematical and physical boundary conditions (as well as the difference between the solutions) disappears in the case of standard heat conduction equation ().
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