8.A. Separation of variable in elliptic and parabolic coordinates


## 8.A. Separation of variable in elliptic and parabolic coordinates

Recall that elliptic and parabolic coordinates, and also elliptic cylindrical and parabolic cylindrical coordinates are described in Subsection 6.3.4.

### Laplace equation in the ellipse

Consider Laplace equation in the elliptic coordinates $(\mu,\nu)$: $$\Delta u= \frac{1}{c^2\bigl(\sinh^2(\mu)+\sin^2(\nu) \bigr)} (\partial_\mu^2 +\partial_\nu^2 )u=0 \label{eq-8.A.1}$$ which is obviously equivalent to $$(\partial_\mu^2 +\partial_\nu^2 )u=0; \label{eq-8.A.2}$$ separating variables $u= M(\mu)N(\nu)$ we arrive to $M''= \alpha M$, $N''=-\alpha N$ with periodic boundary conditions for $N$; so $N=\cos (n\nu), \sin (n\nu)$, $\alpha = n^2$ and $N=A cosh (n\mu) + B\sinh (n\mu)$. So \begin{multline} u_n = A \cosh(n\mu)\cos (n\nu) + B \cosh(n\mu)\sin (n\nu) +\\ C \sinh(n\mu)\cos (n\nu) + D \sinh(n\mu)\sin (n\nu) \label{eq-8.A.3} \end{multline} as $n=1,2,\ldots$ and similarly $$u_0 = A + Bu. \label{eq-8.A.4}$$

### Laplace equation in the parabolic annulus

Consider Laplace equation in the parabolic coordinates $(\sigma,\tau)$: $$\Delta u = \frac{1}{\sigma^2+\tau^2} (\partial_\sigma^2 +\partial_\tau^2 )=0. \label{eq-8.A.5}$$ Then again formulae (\ref{eq-8.A.3}) and (\ref{eq-8.A.4}) work but with $(\mu,\nu)$ replaced by $(\sigma,\tau)$.

### Helmholtz equation in the ellipse

Consider Helmholtz equation in the elliptic coordinates $(\mu,\nu)$: $$\Delta u= \frac{1}{c^2\bigl(\sinh^2(\mu)+\sin^2(\nu) \bigr)} (\partial_\mu^2 +\partial_\nu^2 )u=-k^2u \label{eq-8.A.6}$$ which can be rewritten as $$\Bigl(\partial_\mu^2 +k^2 c^2 \sinh^2(\mu) + \partial_\nu^2 +\sin^2(\nu) \Bigr)u=0 \label{eq-8.A.7}$$ and separating variables we get \begin{gather} M''+k^2 c^2\bigl( \sinh^2(\mu) +\lambda\bigr)M=0, \label{eq-8.A.8}\\ N''+k^2 c^2\bigl( \sin^2(\nu) -\lambda\bigr)N=0. \label{eq-8.A.9} \end{gather}

### Helmholtz equation in the parabolic annulus

Consider Helmholtz equation in the parabolic coordinates $(\sigma,\tau)$: $$\Delta u = \frac{1}{\sigma^2+\tau^2} (\partial_\sigma^2 +\partial_\tau^2 )=-k^2u \label{eq-8.A.10}$$ which can be rewritten as $$\Bigl(\partial_\sigma^2 +k^2 \sigma^2 + \partial_\tau^2 +k^2\tau^2 \Bigr)u=0 \label{eq-8.A.11}$$ and separating variables we get \begin{gather} S''+k^2 \bigl( \sigma^2 +\lambda\bigr)S=0, \label{eq-8.A.12}\\ N''+k^2 \bigl( \tau ^2-\lambda\bigr)T=0. \label{eq-8.A.13} \end{gather}

### Other coordinates

Exercise 1. Consider Laplace and Helmholtz equations in elliptic cylindrical and parabolic cylindrical coordinates.