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V.Gnanaraj May 27, 2008 02:50

How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
I have an equation ∂^2U/∂X^2 +∂^2U/∂Y^2 = K^2U with the boundary conditions U = 0 on the boundary of the equilateral triangle. How to solve the equation

Rami May 28, 2008 09:15

Re: How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
If your LHS should be -k^2U, this is the Helmholtz equation. It has analytic solution by separation of variables for certain geometries and BC, and maybe yours falls into these. See

If this is not the case, please clarify - are you looking for analytic solution, or are you after a numerical one?

Rob Hong May 30, 2008 00:04

Re: How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
I thought the solution is simply U=0. U=0 satisfies the governing equation and also the boundary conditions at all points on the sides if the triangle. Could not be this simple?

Jed May 31, 2008 14:08

Re: How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
I would assume this is an eigenvalue problem (modulo sign confusion) so you must find K and corresponding U, |U|=1 which solve this equation. That is, find the first few eigenmodes of the Laplacian on this geometry. For this special case, you can compute the modes analytically. For very accurate numerical methods for more general planar regions, see the references in the paper below. Of course, you can always just discretize the problem and hand it to a solver package.

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