Heat conduction in a composite sphere-the effect of fractional derivative order on temperature distribution

The aim of the contribution is an analysis of time-fractional heat conduction in a sphere with an inner heat source. The object of the consideration is a solid sphere with a spherical layer. The heat conduction in the solid sphere and spherical layer is governed by fractional heat conduction equation with a Caputo timederivative. Mathematical (classical) or physical formulations of the Robin boundary condition and the perfect contact of the solid sphere and spherical layer is assumed. The boundary condition and the heat flux continuity condition at the interface are expressed by the Riemann-Liouville derivative. An exact solution of the problem under mathematical conditions is determined. A solution of the problem under physical boundary and continuity conditions using the Laplace transform method has been obtained. The inverse of the Laplace transform by using the Talbot method are numerically determined. Numerical results show the effect of the order of the Caputo and the Riemann-Liouville derivatives on the temperature distribution in the sphere.


Introduction
The fundamental of the classical heat transfer theory is the Fourier law which leads to the parabolic partial differential equation of the heat conduction [1]. A consequence of the Fourier's law is unrealistic speed of heat flow in the medium. This inconvenience can be avoided by a generalization of the Fourier law which leads to a fractional heat conduction equation [2]. The heat conduction governed by the fractional differential equation is the subject of papers [3][4][5][6][7][8][9]. Applications of fractional order calculus are presented in books [10][11][12] and papers [13][14][15].
If the heat transfer in a bounded medium is considered then the heat equation is complemented by boundary conditions. The Dirichlet, Neumann and Robin boundary conditions are often used in describing the heat transfer between the body and the surroundings. In the classical heat theory, the Neumann and Robin boundary conditions include the normal derivative at the boundary of the considered region. Introducing the time-fractional derivative in the Neumann and Robin boundary conditions, the physical formulation of these conditions is obtained [16].
A solution of the linear fractional differential equation under classical boundary conditions can be determined in an analytical form. To solve the fractional equation under physical boundary conditions, the Laplace technique can be applied. This approach leads to a solution of the problem in the Laplace domain. The temperature distribution in the time domain by using an algorithm for numerical inversion of the Laplace transform can be obtained. The methods of numerical inversion of the Laplace transforms used in classical analysis can be also applied to Laplace transforms obtained by solving the problems with fractional derivatives. Selected methods of numerical inversion of the Laplace transforms are presented in papers [17][18][19][20].
In this paper, we present the solution of the fractional heat conduction problem in a sphere consisting of an inner solid sphere and a spherical layer. The mathematical and physical formulation of the Robin boundary conditions is considered. The perfect thermal contact of the inner sphere and the spherical layer is assumed. The effect of the fractional order on the temperature distribution in the sphere has been numerically investigated.

Formulation of the problem
We consider the time-fractional radial heat conduction problem in a sphere. The two regions of the sphere are distinguished: 1 0 r r   -a solid inner sphere and 1 r r b   -a spherical layer, where r is the radial coordinate. The heat transfer in the regions is governed by the fractional heat conduction equation [3]: g r t is the volumetric rate of heat generation, i a is the thermal diffusivity, i  is the thermal conductivity and i  denotes the fractional order of the Caputo derivative with respect to time t . The Caputo derivative is defined by [21]       where  denotes the gamma function. The boundary condition and the continuity condition at interface are assumed in a form with the Riemann-Liouville fractional derivative On the outer surface of the sphere, the Robin boundary condition [16] is assumed where a  is the outer heat transfer coefficient and T  is the ambient temperature. The perfect thermal contact at the interface between the inner sphere and the spherical layer is described by conditions: (6) and the initial condition is The conditions (4) and (6) (4) and (6) means an identity operator and can be omitted. We further consider the case of

Solution to the problem
In order to transform the heat conduction equation (1) Taking into account equation (8) in the initial-boundary problem (1) and (4-7) , we obtain formulation of the problem for the function Moreover, the conditions (10-12) are complemented by a condition for 0 r  , which is obtained using equation (8) The functions   * , i g r t in equation (9) are given by the formula The solutions of the initial-boundary problem (9)(10)(11)(12)(13)(14) for mathematical and physical formulations will be presented below.

Mathematical formulation of boundary and continuity conditions
The heat conduction problem (9)(10)(11)(12)(13)(14) under mathematical conditions for 1 2      can be solved analytically. We search for the solution to this problem in the form of the series of orthogonal functions In the first step, we find the functions The functions These functions fulfil the orthogonality condition in the form The coefficients 1,k Finally, taking into account equation (8) and (16), we obtain the temperature distribution in the sphere under mathematical formulation of the boundary and continuity conditions in the form are given by (28) and   , i k r  are defined by (22) and (23).

Physical formulation of boundary and continuity conditions
A solution of the heat conduction problem (9)(10)(11)(12)(13)(14) under physical boundary and continuity conditions ( 1 (10) and (12)) will be obtained by using the Laplace technique. The Laplace transform is defined as where   f t for 0 t  is a given function of the exponential type and s is a complex parameter. After applying the Laplace transformation to the equations (9-12) and (14), and using the properties of the Laplace transform, we obtain     The solution of the equation (33)