I wish to find exact solutions of Laplace's equation in cylindrical coordinates on (a subset of) the 3-sphere.(adsbygoogle = window.adsbygoogle || []).push({});

This pde is linear but not separable. The potential [itex]{\Phi}(x,z)[/itex] must fulfil the following pde:

[tex]

(1-{\frac{x^2}{a^2}}){\frac{{\partial}^2}{{\partial}x^2}}{\Phi}(x,z)+

(1-{\frac{z^2}{a^2}}){\frac{{\partial}^2}{{\partial}z^2}}{\Phi}(x,z)+

{\frac{1}{x}}(1-{\frac{3x^2}{a^2}}){\frac{{\partial}}{{\partial}x}}{\Phi}(x,z)-

{\frac{2xz}{a^2}}{\frac{{\partial}^2}{{\partial}x{\partial}z}}{\Phi}(x,z)-

{\frac{3z}{a^2}}{\frac{{\partial}}{{\partial}z}}{\Phi}(x,z)=0

[/tex]

Here a is a constant (and [tex]x,z<a[/tex], [tex]z{\neq}0[/tex]). Does anyone know how to solve this equation?

(I'm aware that a transformation of this equation to spherical coordinates yields a separable pde, but

this gives a bunch of useless solutions blowing up near the origin.)

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# Non-separable, linear pde

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