Solve the Eigenproblem of a Constrained Laplacian
Find the four smallest eigenvalues and eigenfunctions of a Laplacian equation over a 1D region constrained by homogeneous Dirichlet boundary conditions.
Specify a Laplacian.
In[1]:=
![Click for copyable input](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/In_4.png)
\[ScriptCapitalL] = -Laplacian[u[x], {x}];
Set up a Dirichlet boundary condition.
In[2]:=
![Click for copyable input](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/In_5.png)
\[ScriptCapitalB] = DirichletCondition[u[x] == 0, True];
Numerically find the eigenvalues.
In[3]:=
![Click for copyable input](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/In_6.png)
NDEigenvalues[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x], {x, 0, \[Pi]}, 4]
Out[3]=
![](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/O_3.png)
Numerically find the eigenvalues and eigenfunctions.
In[4]:=
![Click for copyable input](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/In_7.png)
{vals, funs} =
NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x], {x, 0, \[Pi]}, 4];
Inspect the eigenvalues.
In[5]:=
![Click for copyable input](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/In_8.png)
vals
Out[5]=
![](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/O_4.png)
Visualize the eigenfunctions.
In[6]:=
![Click for copyable input](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/In_9.png)
Plot[Evaluate[funs], {x, 0, \[Pi]}]
Out[6]=
![](assets.en/solve-the-eigenproblem-of-a-constrained-laplacian/O_5.png)