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

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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I replaced your math display using the built-in LaTeX parsing capabilities. See also the FAQ and… so you can also take advantage of it in the future. – Willie Wong Jan 14 '11 at 18:16
Thank you for that. I'll make sure to use LaTeX in the future. – McCoy hfdxvnhgdez Jan 14 '11 at 18:57
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. – Willie Wong Jan 15 '11 at 14:13
up vote 1 down vote accepted

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 exp(3it) you could easily solve it, but in the case exp(exp(3it)) I don't think you're going to be able to write down exact solutions.

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by the way, your equation is actually the Klein-Gordon equation not the wave equation. – ctw Jan 14 '11 at 18:04
I've looked up the Klein-Gordon equation and it appears to be a homogeneous wave equation. If there is no exact solution, do you have any idea as to a strategy of best solving this? – McCoy hfdxvnhgdez Jan 14 '11 at 19:00
in one time and one space dimension, the homogeneous wave equation is $u_{tt}-u_{xx}=0$ whereas the homogeneous Klein-Gordon equation is $u_{tt}-u_{xx}+u=0$. So no, the Klein-Gordon equation is not a homogeneous wave equation. – ctw Jan 14 '11 at 19:10
Is it possible to solve the homogeneous Klein-Gordon equation and then add to it a particular solution for the problem I've stated above? If so, do you have any guess on how to go about that particular solution? – McCoy hfdxvnhgdez Jan 14 '11 at 19:23
Fourier transform in space leads to a fine solution, but I don't see any double-exponential involved. Instead, one can find a particular solution in the form $\Psi=e^{3t}f(x)$, provided $f''=10 f + \delta(x)$. Then the unique solution bounded as $|x|\to\infty$ is – Bob Pego Jan 19 '11 at 21:41

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