I am computing the Laplace's eigenvalue problem on the region $\Omega$ formed by the four vertices $(1,1),(1,2),(−1,2),(−1,1)$.
Consider the Laplace problem, $$-L u = \lambda u$$ where $$L = y^2 \Big( \frac{\partial ^2} {\partial x^2} + \frac{\partial ^2} {\partial y^2}\Big)$$
with the Dirichlet boundary condition that $u = 0$ at $\partial \Omega$.
I have attempted the following
First I have specified the region in Mathematica.
\[CapitalOmega]= Polygon[{{1, 1}, {1, 2}, {-1, 2}, {-1, 1}}]
Then I have written the main part.
Clear@ℒ
Clear@ℬ
ℒ[x_,y_]= y^2* D[u[x,y],{x,2}]+ y^2* D[u[x,y],{y,2}];
ℬ[x_, y_] = DirichletCondition[u[x, y] == 0., True];
{vals, funs} =
NDEigensystem[{-ℒ[x, y], ℬ[x, y]},
u[x, y], {x, y} \[Element] \[CapitalOmega], 5];
But it gives some errors.
Please help me. Thanking in advanced.
{{25.8582,40.7532,64.6432,87.71,96.4144},{InterpolatingFunction[Domain: {{-1.,1.},{1.,2.}} Output: scalar ][x,y],InterpolatingFunction[Domain: {{-1.,1.},{1.,2.}} Output: scalar ][x,y],InterpolatingFunction[Domain: {{-1.,1.},{1.,2.}} Output: scalar ][x,y],InterpolatingFunction[Domain: {{-1.,1.},{1.,2.}} Output: scalar ][x,y],InterpolatingFunction[Domain: {{-1.,1.},{1.,2.}} Output: scalar ][x,y]}}
. $\endgroup$