# Laplace's problem in Mathematica

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.

• Works for me in 12.2 on Windows 10Pro, outputting {{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]}}. Mar 11, 2021 at 8:10
• @user64494, sorry did not see your comment. I can delete my answer and you can have a go, if you want. Mar 11, 2021 at 8:19
• @user21: A trifle. Don't worry about it. Mar 11, 2021 at 8:38

## 1 Answer

Works just fine in 12.2:

\[CapitalOmega] = Polygon[{{1, 1}, {1, 2}, {-1, 2}, {-1, 1}}];
Clear@\[ScriptCapitalL]
Clear@\[ScriptCapitalB]
\[ScriptCapitalL][x_, y_] =
y^2*D[u[x, y], {x, 2}] + y^2*D[u[x, y], {y, 2}];
\[ScriptCapitalB][x_, y_] = DirichletCondition[u[x, y] == 0., True];
{vals, funs} =
NDEigensystem[{-\[ScriptCapitalL][x, y], \[ScriptCapitalB][x, y]},
u[x, y], {x, y} \[Element] \[CapitalOmega], 5];
vals

{25.8582, 40.7532, 64.6432, 87.71, 96.4144}

Plot3D[funs[[2]], {x, y} \[Element] \[CapitalOmega]]


• which method for 4th order PDE problems has been used in Mathemtica12.2.0, Finite Difference? Mar 23, 2021 at 10:44
• @ABCDEMMM, I am not sure I understand the question. The method to solve 4th order PDE has not changed in version 12.2 - I hope that's what you are looking for. Mar 23, 2021 at 12:21