Time-dependent boundary conditions

[quZz]

Though this question arose in quantum mechanics, i think it should be posted here.
Consider a particle in a well with infinite walls:
$$i i \frac{\partial \Psi}{\partial t} = -\frac12 \frac{\partial^2 \Psi}{ \partial x^2},\:0<x<a$$
but the wall start to squeeze
$$\Psi(x=0,t) \equiv 0$$
$$\Psi(x=a-t,t) = 0$$
In the beginning the state function is known
$$\Psi(x,t=0) = \varphi(x)$$

What is the method for solving this type of PDE?

 

[matematikawan]

Have you try taking Laplace transform?

No success? Try another method, separating the variables.