I wish to solve the pde: $$-\frac{1}{1-t}\partial_x^2\phi+t^4(1-t)\partial_t^2\phi-t^4\partial_t\phi=\mu^2 \phi,$$ with initial conditions $\phi(x,0)=\cos(\mu x)$ and $\dot{\phi}(x,0)=0$ for some time period $t\in[0,0.9]$, for example. Lacking analytical methods (separation of variables was not fruitful), I thought I'd try numerically. However, numerically means I have to give a solver a spacial domain $[-a,a]$ on which to solve... and I have no idea what boundary conditions to impose on $x=\pm a$. In a sense, the value of $\phi$ on spacial slices like $x=a$ are exactly what I'm trying to find in the first place!
How does one treat such problems numerically?