I've attempted a few different solutions to this math methods problem from an old qualifying exam, but I can't seem to hack it. The setup for the problem is that the temperature sand in the Australian outback obeys the usual heat equation, $$ \frac{\partial T}{\partial t} = \frac{\kappa}{\rho s} \frac{\partial^2 T}{\partial z^2},$$ where z is the distance into the sand, with the temperature at z=0 given by $$ T = T_0 \cos \left[2 \pi \frac{t-t_0}{t_d} \right],$$ where $t_0$ is the hottest time of day, and $t_d$ is the length of the day. We are asked to find how the temperature depends on z and the coolest part of the sand at $t=t_0$.
How is this possible? Don't we need at least two boundary conditions in order to find the solution? Is the best thing to assume the temperature goes to zero at infinity? Also, using separation of variables, one finds $$\frac{1}{T} \frac{\partial T}{\partial t} = \lambda = \text{const}.$$ And so the time dependence should be exponential. How is it that the solution is periodic? Lastly, I tried to use a Green's function solution (which I admittedly am not super confident about), and found that $$T(z,t) = \int_0^t \frac{z~ \cos \left[2 \pi \frac{\tau-t_0}{t_d} \right] }{\sqrt{ (t - \tau)}} \exp \left(\frac{z^2}{4 \alpha (t- \tau)} \right) d\tau.$$ If I set the z derivative of this to zero and solve for $z$, I get the only extremum is at $$z^2 \propto \int_0^t \frac{ \cos \left[2 \pi \frac{\tau-t_0}{t_d} \right] } {\sqrt{4 \pi \alpha (t - \tau)}} \exp \left(\frac{z^2}{4 \alpha (t- \tau)} \right) d\tau \\ \left( \int_0^t \frac{ \cos \left[2 \pi \frac{\tau-t_0}{t_d} \right] }{\sqrt{ (t - \tau)^3}} \exp \left(\frac{z^2}{4 \alpha (t- \tau)} \right) d\tau \right)^{-1} = 0.$$ I assume this is a maximum. This fits with my intuition, but not the instructions. Did I do this correctly?
To recap my specific questions:
Do we have to assume more boundary conditions to solve the problem?
How can there be non-exponential time dependence without some source term added to the heat equation?
Is my Green's function correct?