Temperature and heat flux solutions due to steady and non-steady periodic-type surface temperatures in a semi-infinite solid

An analytical solution for the temperature and heat flux distribution in the case of a semi-infinite solid of constant properties is investigated. The solutions are presented for time-dependent, surface temperatures of the forms: (i)T₁(t)=T₀(1+a cos ωt), and (ii)T₂(t)=T₀(1+b t cos ωt), where a and b are controlling factors of the periodic oscillations about the constant surface temperature T₀. The dimensionless (or reduced) temperature and heat flux solutions are presented in terms of decompositions C_Γ and S_Γ of the generalized representation of the incomplete Gamma function. It is demonstrated that the present analysis covers the limiting case for large times which is discussed in several textbooks, for the case of steady periodic-type surface temperatures.