here's the problem: the equation $\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial x^2}$ should be transformed using $\xi=\frac{x}{\delta(t)}$ and $T(x,t)=\phi(t)F(\xi,t)$. The result of the transformation should be $\phi\frac{\partial^2 F}{\partial \xi^2}=\delta\left( \delta \phi_t F+\delta \phi\frac{\partial F}{\partial t}-\xi\delta_t\phi \frac{\partial F}{\partial \xi}\right)$.
Can you please explain how to get this result?