If a plate of thickness a (0 ≤ x ≤ a) is prepared at temperature T0 and left alone, how does its temperature distribution evolve? In ① we treat the inside of the plate only, solving it as in the lecture notes by separation of variables and a Fourier series, and we check the result by overlaying the series solution on the numerical solution. In ② we solve the surroundings too (air, water, or firebrick) — we write the heat conduction equation outside the plate as well, and by requiring temperature and heat flux to be continuous at the interface we can follow the time evolution of the whole system. What matters there is not only the thermal diffusivity κ². The thermal effusivity b = √(Kρc) determines the interface temperature. In either tab you can switch the boundary to Newton’s law of cooling −K∂T/∂x = h(T−T∞) and include convection and radiation as well — what happens inside and outside an electric furnace emerges directly.
The thermal conductivity K is the value from the table in the lecture notes; the density ρ and specific heat c are standard literature values. The numerical solution uses the finite volume method (face K by the harmonic mean) with backward Euler, so energy is conserved exactly in an insulated box. Convection and radiation enter only as boundary conditions (Newton’s law of cooling) — inside a material the transport is pure heat conduction.