Solving the Heat Equation Problem under Periodic Non-Stationary Conditions

The one-dimensional parabolic heat equation problem has been widely studied under many of different conditions. When boundary conditions are periodic time-varying, two issues are traditionally assumed: a) the time-dependent component is given, which greatly simplifies the mathematical problem, b) mostly the semi-infinite solid bar has been reviewed. In this work, we present a solution for the one-dimensional parabolic heat equation in a finite solid bar when a periodic time-varying heat flux rules. The physical problem takes place for t >= 0 and 0 <= x <= l, and its resolved using the Variables Separation Method. A time-depending sinusoidal heat flux is set up at boundary x=l, while at x=0 the temperature remains constant. The temperature distribution T (x, t), is analytically calculated. It is verified, for a finite bar, that the temperature inside oscillates at the same frequency as the external heat source, however with a time-phase difference that depends on the thermal properties of the material.