Create a Gallery of Eigenfunctions for the Laplacian in a Ball
Define a 3D Laplacian operator.
In[1]:=
![Click for copyable input](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_127.png)
\[ScriptCapitalL] = -Laplacian[u[x, y, z], {x, y, z}];
Specify homogeneous Dirichlet boundary conditions.
In[2]:=
![Click for copyable input](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_128.png)
\[ScriptCapitalB] = DirichletCondition[u[x, y, z] == 0, True];
Find the 16 smallest eigenvalues and eigenfunctions in a ball.
In[3]:=
![Click for copyable input](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_129.png)
\[CapitalOmega] = Ball[{0, 0, 0}, 2];
In[4]:=
![Click for copyable input](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_130.png)
{vals, funs} =
DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x, y, z], {x, y, z} \[Element] \[CapitalOmega], 16];
The eigenvalues are given in terms of BesselJZero.
In[5]:=
![Click for copyable input](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_131.png)
vals[[1]] // TraditionalForm
Out[5]//TraditionalForm=
![](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/O_63.png)
Generate a gallery of the eigenfunctions.
show complete Wolfram Language input
Out[6]=
![](assets.en/create-a-gallery-of-eigenfunctions-for-the-laplaci/O_64.png)