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How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2

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Old   May 27, 2008, 02:50
Default How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
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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
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Old   May 28, 2008, 09:15
Default Re: How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
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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?
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Old   May 30, 2008, 00:04
Default Re: How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
Rob Hong
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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?
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Old   May 31, 2008, 14:08
Default Re: How to solve equation ∂^2U/∂X^2 +∂^2U/∂Y^2
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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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