Asymptotic solution of the problem of unsteady heat transfer into a stratified medium

An asymptotic expression is derived for the heat flux at the boundary of a semi-infinite region at long times under the assumption that the temperatur e at this boundary is a function of time. The solution is represented as a series expansion in positive powers of a fractional-order differentiation o perator of this function. Thermal diffusivity a is assumed to be dependent on a single coordinate x. Two practically important cases were scrutinized, namely, when (1) the function a(x) approaches a constant value as x is rai sed and (2) a(x) = a(x + l) (the function is periodic). In the latter case, the principal term of the asymptotic expansion at t --> infinity has the s ame form as that for a medium with a constant effective thermal diffusivity a*, which is the harmonic mean of the function a(x) integrated over its pe riod l. The principal terms of the asymptotic expansions are also obtained for the case in which thermal conductivity lambda and specific heat at a co nstant volume, C, depend on the coordinate x.