How to Solve a Complex 2D PDE in Toroidal Coordinates?

[maverick280857]

Hi,

After considerable simplification in a problem I'm working on, I end up with the following partial differential equation:

$$\partial_{\eta}\left(\frac{\sinh\eta}{\Delta}\partial_{\eta}g\right) + \partial_{\theta}\left(\frac{\sinh\eta}{\Delta}\partial_{\theta}g\right) + c^2\left[\frac{E_{p} - V(\eta,\theta)}{\Delta\sinh\eta}\right]g = 0$$

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!

 

[maverick280857]

Is there some substitution which will simplify this equation maybe?