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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.

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    $\begingroup$ Did you perhaps change notation halfway through the question? Is the field to-be-solved-for $\psi$ or $u$? And is position $r$ or $x$? A bit murky. $\endgroup$ – rchilton1980 Mar 2 '18 at 19:58
  • $\begingroup$ @rchilton1980 That was really sloppy of me. Thanks a lot for pointing that out. I've corrected it now $\endgroup$ – pkg Mar 3 '18 at 8:13
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Yes. You can. The wave equation is time reversal.

Your PDE, namely $$\partial_{tt}u-\Delta u=0$$ is invariant under the transformation $t\to -t$. This means that if $u(x,t)$ is a solution, $u(x,-t)$ is also a solution.

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