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

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


  1. Laplace equation in the ellipse
  2. Laplace equation in the parabolic annulus
  3. Helmholtz equation in the ellipse and parabolic annulus
  4. Helmholtz equation in the parabolic annulus
  5. Other 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 $(\mu,\nu)$: \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.


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