Fractional heat conduction in a sphere under mathematical and physical Robin conditions

In this paper, the effect of a fractional order of time-derivatives occurring in fractional heat
conduction models on the temperature distribution in a composite sphere is investigated.
The research concerns heat conduction in a sphere consisting of a solid sphere and a spherical
layer which are in perfect thermal contact. The solution of the problem with a classical Robin
boundary condition and continuity conditions at the interface in an analytical form has
been derived. The fractional heat conduction is governed by the heat conduction equation
with the Caputo time-derivative, a Robin boundary condition and a heat flux continuity
condition with the Riemann-Liouville derivative. The solution of the problem of non-local
heat conduction by using the Laplace transform technique has been determined, and the
temperature distribution in the sphere by using a method of numerical inversion of the
Laplace transforms has been obtained.