# differential equations – numerical resolution of an initial value problem on an unbounded domain

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?