A solution to the problem of time-fractional heat conduction in a multi-layer slab

Abstract

Abstract. In this paper a solution of the time-fractional heat conduction problem in a multilayer slab is presented. The boundary conditions of the third kind and the perfect contact at the interfaces are assumed. A space-time dependent volumetric heat source in the slab and time dependent surroundings temperatures are taken into account in the formulation of the problem. The solution is obtained in the form of a series expansion with respect to eigenfunctions of an auxiliary problem. A numerical example shows temperature distribution in the slab for various values of the order of the Caputo fractional derivative in the heat conduction equation.

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

“…The values of parameter  will be chosen such that nonzero solutions of problem (18)(19)(20)(21)(22) The values of parameter  will be chosen such that non- …”

“…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 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].…”