- #1
maverick280857
- 1,774
- 5
Hi,
After considerable simplification in a problem I'm working on, I end up with the following partial differential equation:
<br /> \partial_{\eta}\left(\frac{\sinh\eta}{\Delta}\partial_{\eta}g\right) + \partial_{\theta}\left(\frac{\sinh\eta}{\Delta}\partial_{\theta}g\right) + c^2\left[\frac{E_{p} - V(\eta,\theta)}{\Delta\sinh\eta}\right]g = 0<br />
where c is a positive constant, E_{p} is a constant, V_{0} is a constant and
V(\eta,\theta) = V_0\sqrt{\frac{\Delta}{\sinh\eta}}Q_{-1/2}(\coth\eta)
where Q is the Legendre function,
\Delta = \cosh\eta - \cos\theta
The d.e. actually comes from a separation of variables in toroidal coordinates. I am not sure how I should proceed to solve this differential equation for g(\eta,\theta).
Any inputs on how to solve this differential equation analytically will be appreciated.
Thanks in advance!
After considerable simplification in a problem I'm working on, I end up with the following partial differential equation:
<br /> \partial_{\eta}\left(\frac{\sinh\eta}{\Delta}\partial_{\eta}g\right) + \partial_{\theta}\left(\frac{\sinh\eta}{\Delta}\partial_{\theta}g\right) + c^2\left[\frac{E_{p} - V(\eta,\theta)}{\Delta\sinh\eta}\right]g = 0<br />
where c is a positive constant, E_{p} is a constant, V_{0} is a constant and
V(\eta,\theta) = V_0\sqrt{\frac{\Delta}{\sinh\eta}}Q_{-1/2}(\coth\eta)
where Q is the Legendre function,
\Delta = \cosh\eta - \cos\theta
The d.e. actually comes from a separation of variables in toroidal coordinates. I am not sure how I should proceed to solve this differential equation for g(\eta,\theta).
Any inputs on how to solve this differential equation analytically will be appreciated.
Thanks in advance!
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