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Problem:

$ \Psi_{xx} - \Psi_{tt} - \Psi = \exp(3t) \cdot \delta(x)$

No boundary conditions specified.

  • I solved the homogeneous portion, $\Psi_\mathrm{homogeneous}$, of this equation via separation of variables but my solution is just for some random case of $k^2$ where: F''/F = G''/G + 4 = k^2. I chose the case where $k^2 = 0$ which gave me solutions for $\Psi_\mathrm{homogeneous}$ like $x\sin(2t)$ and $x\cos((2t)$.

  • With the guess method for $\Psi_\mathrm{particular}$, I have no idea what to guess on a general form of $\exp(3t)\delta(x)$ to plug back in to the PDE.

  • I have read about Green's Functions but, man, I'm having a hard time understanding the guides I have seen because they skip so many of the intermediate steps. I understand that these Green's Functions can provide a general solution and that seems like what I'm looking for. I likely spent a lot of time for nothing on my first attempt with separation of variables for $\Psi_\mathrm{homogeneous}$ and then looking for $\Psi_\mathrm{particular}$ using the guess method...

  • I'm curious if there is a general set of IC/BCs that I should be assuming as well?

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    in the future please do not do what you just did: close a question by removing the question text. It makes the previous answers not understandable. I've restored the question text.2011-01-15

1 Answers 1

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Taking the Fourier transform in the spatial variable only reduces the PDE to a nice ODE except for your "double" exponential term. If it were just $e^{3it}$ you could easily solve it, but in the case $e^{e^{3it}}$ I don't think you're going to be able to write down exact solutions.

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    $f(x)=-\frac1{2\sqrt{10}} e^{-\sqrt{10}|x|}$2011-01-19