An analytical solution to the problem of time-fractional heat conduction in a composite sphere

Abstract

Abstract. An analytical solution to the problem of time-fractional heat conduction in a sphere consisting of an inner solid sphere and concentric spherical layers is presented. In the heat conduction equation, the Caputo time-derivative of fractional order and the Robin boundary condition at the outer surface of the sphere are assumed. The spherical layers are characterized by different material properties and perfect thermal contact is assumed between the layers. The analytical solution to the problem of heat conduction in the sphere for time-dependent surrounding temperature and for time-space-dependent volumetric heat source is derived. Numerical examples are presented to show the effect of the harmonically varying intensity of the heat source and the harmonically varying surrounding temperature on the temperature in the sphere for different orders of the Caputo time-derivative. analytical solution to the problem for thermal and mechanical properties of the sphere was obtained in the form of power functions of the radial direction. Thermal stresses in a sphere of a functionally graded material were also considered by Pawar et al. [9]. Transient temperature distribution was determined by assuming that the material properties of the sphere were exponential functions of the radial direction. The parabolic differential equation of heat conduction, derived under the framework of the classical theory of heat conduction is based on the local Fourier law. Non-local generalizations of the Fourier law lead to non-classical theories, in which the parabolic equation is replaced by a time-fractional and/or space-fractional heat conduction differential equation [10]. In these fractional differential equations, different kinds of derivatives of fractional order are used (the Riemann-Liouville derivative, the Caputo derivative and the Grűnwald-Letnikov derivative). Moreover, the boundary conditions may also include the fractional derivatives. The fundamentals of fractional calculus and of the theory of fractional differential equations are given in [11][12][13][14][15]. Some applications of fractional order calculus to modelling of real-world phenomena are presented in [16][17][18].Heat conduction problems formulated under the framework of the non-classical theories in the spherical coordinates with fractional Caputo or Riemann-Liouville derivatives were studied in [19,20]. An approximate analytical solution of time-fractional heat conduction in a composite medium consisting of an infinite matrix and a spherical inclusion is presented by Povstenko in [19]. The perfect thermal contact was realized by the conditions of equality of temperatures and heat fluxes at the boundary surfaces, wherein the heat fluxes are expressed by a Riemann-Liouville fractional derivative. An analytical solution to the problem of time-fractional heat conduction in a multilayered slab was presented by Siedlecka and Kukla in [20]. Ning and Jiang [21] use the Laplace transform and the method of variable separation to determine an analytical

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

“…Taking into account equation (8) in the initial-boundary problem (1) and (4-7) , we obtain formulation of the problem for the function…”

“…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…”

“…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].…”

A numerical‐analytical solution of multi‐term fractional‐order differential equations

“…3 The initial-boundary problems for time-fractional linear partial differential equations, for instance, the diffusion and heat conduction equations, can be solved using the Fourier method. [7][8][9] This approach also leads to the initial value problems for the fractional ordinary differential equations.…”

Fractional heat conduction in multilayer spherical bodies

“…A method of solution of a time-fractional heat conduction equation in a solid sphere has been discussed by Ning and Jiang in paper [7]. The time-fractional heat conduction in a multi-layered solid sphere assuming spherical symmetry was the subject of paper [8]. Heat conduction modelling using fractional order derivatives is presented by Žecová and J. Terpák in paper [9].…”