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.
Highlights
The standard heat conduction equation for temperature TT a T t is obtained from the balance equation for energy (1) C div q, t (2)where ρ is the mass density, C is the specific heat capacity, q is the heat flux vector, and the classicalFourier law which states the linear dependence between the heat flux vector q and the temperature gradient q k gradT (3)with k being the thermal conductivity
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: T1
The Laplace transform with respect to time t applied to Equations (24) and (25) leads to two ordinary differential equations
Summary
Where ρ is the mass density, C is the specific heat capacity, q is the heat flux vector, and the classical. 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). 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. 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 g x S , t n s (15). Some authors [15,25] do not use a separate notation for the fractional integral I f t
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