A Laplacian's Eigenvalues & Eigenfunctions
Find the four smallest eigenvalues and eigenfunctions of a Laplacian operator, i.e. solutions to , over a 1D region.
Specify a Laplacian.
In[1]:=
![Click for copyable input](assets.en/a-laplacians-eigenvalues-and-eigenfunctions/In_1.png)
\[ScriptCapitalL] = -Laplacian[u[x], {x}];
Numerically find the four smallest eigenvalues and eigenfunctions.
In[2]:=
![Click for copyable input](assets.en/a-laplacians-eigenvalues-and-eigenfunctions/In_2.png)
NDEigensystem[\[ScriptCapitalL], u[x], {x, 0, \[Pi]}, 4]
Out[2]=
![](assets.en/a-laplacians-eigenvalues-and-eigenfunctions/O_1.png)
Visualize the eigenfunctions.
In[3]:=
![Click for copyable input](assets.en/a-laplacians-eigenvalues-and-eigenfunctions/In_3.png)
NDEigensystem[\[ScriptCapitalL], u[x], {x, 0, \[Pi]}, 4];
Plot[Evaluate[%[[2]]], {x, 0, \[Pi]}]
Out[3]=
![](assets.en/a-laplacians-eigenvalues-and-eigenfunctions/O_2.png)