$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ##8.A. Separation of variable in elliptic and parabolic coordinates ------------------- > 1. [Laplace equation in the ellipse](#sect-8.A.1) > 2. [Laplace equation in the parabolic annulus](#sect-8.A.2) > 3. [Helmholtz equation in the ellipse and parabolic annulus](#sect-8.A.3) > 4. [Helmholtz equation in the parabolic annulus](#sect-8.A.4) > 5. [Other coordinates](#sect-8.A.5) Recall that elliptic and parabolic coordinates, and also elliptic cylindrical and parabolic cylindrical coordinates are described in [Subsection 6.3.4](../Chapter6/S6.3.html#sect-6.3.4). ###Laplace equation in the ellipse Consider Laplace equation in the elliptic coordinates $(\mu,\nu)$: \begin{equation} \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} \end{equation} which is obviously equivalent to \begin{equation} (\partial\_\mu^2 +\partial\_\nu^2 )u=0; \label{eq-8.A.2} \end{equation} 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 \begin{equation} u\_0 = A + Bu. \label{eq-8.A.4} \end{equation} ###Laplace equation in the parabolic annulus Consider Laplace equation in the parabolic coordinates $(\sigma,\tau)$: \begin{equation} \Delta u = \frac{1}{\sigma^2+\tau^2} (\partial\_\sigma^2 +\partial\_\tau^2 )=0. \label{eq-8.A.5} \end{equation} 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 $(\mu,\nu)$: \begin{equation} \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} \end{equation} which can be rewritten as \begin{equation} \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} \end{equation} 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 $(\sigma,\tau)$: \begin{equation} \Delta u = \frac{1}{\sigma^2+\tau^2} (\partial\_\sigma^2 +\partial\_\tau^2 )=-k^2u \label{eq-8.A.10} \end{equation} which can be rewritten as \begin{equation} \Bigl(\partial\_\sigma^2 +k^2 \sigma^2 + \partial\_\tau^2 +k^2\tau^2 \Bigr)u=0 \label{eq-8.A.11} \end{equation} 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. --------------- [$\Leftarrow$](./S8.2.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](../Chapter9/S9.1.html)