Since Mathematica 10 there's been an option to solve PDEs over a region which I'm currently playing around with to solve Poisson-like PDEs in cartesian coordinates. By changing the region from a disk to an ellipse, I've been able to solve 2D problems on an ellipse with Dirchlet conditions quite easily, like the following achieves;

      usol = NDSolveValue[{\!\( \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), 
        \(2\)]\(u[x, y]\)\) == 10, DirichletCondition[u[x, y] == 100, True]}, 
   u, {x, y} \[Element] Ellipsoid[{0, 0}, {3, 5}]];

Plot3D[usol[x, y], {x, y} \[Element] Ellipsoid[{0, 0}, {3, 5}],  BoxRatios -> {3, 5, 2}]

which gives an output as depicted in the figure. What I really want to do however is recast this in spherical co-ordinates to solve for ellipsoid with axial symmetry, so that the Lapacian is two-termed, and the equation to be solved is

$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial u}{\partial r} \right) + \frac{1}{r^2 \sin \theta}\frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial u}{\partial \theta} \right) = c$

on an ellipsoid with outer boundary condition $u = d$ and where $c$ and $d$ are known constants. I've been trying to get this working by modifying the example given, but I'm getting ridiculous answers. Is there a clever way to do this and solve over a region, or do I need to take a different approach?

enter image description here

  • $\begingroup$ Please include your code in the question. $\endgroup$
    – bbgodfrey
    Commented Oct 4, 2017 at 16:59
  • $\begingroup$ Done - similar to linked example! $\endgroup$
    – DRG
    Commented Oct 4, 2017 at 17:20


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