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.)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Non-separable, linear pde

**Physics Forums | Science Articles, Homework Help, Discussion**