I'm trying to solve an equation whose solutions I know are plane waves but there are a few nuances.
First, the equation is of the form $$ \partial^2_t \psi + A(r)\partial^2_r \psi +B(r) \partial_r \psi + C(r)\psi =0 $$
Second, my boundary conditions are $ \psi(a)=\psi(b)=0 $
Third, my "initial" conditions are: $$ \psi(t,r)=f(r) $$ and $$\frac{\partial \psi}{\partial t}(t,r) =g(r)$$
And finally, I'm integrating the equation backwards in time. I have a solution at a certain time $t$ and I need the solution at a much earlier time.
Given these conditions, can I just convert the partial derivatives into finite differences and proceed with the usual method applied to wave equations? Will this yield the correct results?
I apologize if this is a trivial question but I just want to confirm that I'm on solid footing before I begin developing the tedious algorithm.