I'm looking for a way to get analytical solution of diffusion with time-varying diffusivity, i.e.
$$\frac{d^2 T}{ d x^2} = \frac{d \frac{T}{\alpha(t)} }{d t}$$
Is there a possible way to get the analytical solution without using numerical methods? (assume the simplist $0$ initial condition and semi infinite boundary condition ($T(0,t)=T_0$)
Thanks in advance
Let $U=\frac{T}{\alpha(t)}$ Then $\alpha(t) U_{xx} =U_t$ and the change of variables $\alpha(t) dt=d\tau$ converts this to $U_{xx}= U_{\tau}$.
P.S. You might ask yourself if the more realistic initial model might actually already be $\alpha(t) T_{xx}= T_t$. That subtle difference depends upon what sub-components in the diffusion model are subject to time-dependence.