Assume that the asympotic solution has a time-dependent part, which is linear in time, and a positiondependent solution, which is an unknown function. Thus the solution should be of the following form:

\[\theta_{\text {asy }}=A \tau+F(\xi)\]

where \(A\) is a constant to be determined. The linear variation in time can be justified by physical arguments. Substitute into the equations, solve for \(F\), and use the boundary conditions to show that the constant \(A=1\).

Hence the solution is

\[\theta_{\text {asy }}=\tau+\frac{\xi^{2}}{2}+B\]

The constant \(B\) is the integration constant, which has to be determined from the initial conditions. However, the effect of the initial conditions is lost in the asymptotic solution, and hence an overall heat balance from time zero to time \(\tau\) is used as an alternative.

Find the average temperature of the solid at any time and the total energy content at this time. Equate this to the heat added from the start and show that \(B=-1 / 6\) for a starting temperature of zero.