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]:=

\[ScriptCapitalL] = -Laplacian[u[x], {x}];

Numerically find the four smallest eigenvalues and eigenfunctions.

In[2]:=

NDEigensystem[\[ScriptCapitalL], u[x], {x, 0, \[Pi]}, 4]

Out[2]=

Visualize the eigenfunctions.

In[3]:=

NDEigensystem[\[ScriptCapitalL], u[x], {x, 0, \[Pi]}, 4]; Plot[Evaluate[%[[2]]], {x, 0, \[Pi]}]

Out[3]=