I want to solve the problem:

$$ – frac {1} {1-t} partial_x ^ 2 phi + t ^ 4 (1-t) partial_t ^ 2 phi-t ^ 4 partial_t, phi = mu ^ 2 phi $$

with initial conditions $ phi (x, 0) = cos ( mu x) $ and $ dot { phi} (x, 0) = 0 $ for a certain time $ t in[0,0.9]$, for example. Missing analytical methods, I thought to try numerically. However, numerically means that I have to give a solver a spatial domain $[-a,a]$ on which to solve … and I do not know at all which boundary conditions to impose on $ x pm to $. In a sense, the value of $ phi $ on spatial slices like $ x = a $ are exactly what I am looking for in the first place!

How can we get around this?