创建球内的拉普拉斯算子特征函数图集
定义一个三维拉普拉斯算子.
In[1]:=
![Click for copyable input](assets.zh/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_127.png)
\[ScriptCapitalL] = -Laplacian[u[x, y, z], {x, y, z}];
设定齐次狄利克雷边界条件.
In[2]:=
![Click for copyable input](assets.zh/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_128.png)
\[ScriptCapitalB] = DirichletCondition[u[x, y, z] == 0, True];
求球内最小的十六个特征值和特征函数.
In[3]:=
![Click for copyable input](assets.zh/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_129.png)
\[CapitalOmega] = Ball[{0, 0, 0}, 2];
In[4]:=
![Click for copyable input](assets.zh/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];
特征值以 BesselJZero 的形式给出.
In[5]:=
![Click for copyable input](assets.zh/create-a-gallery-of-eigenfunctions-for-the-laplaci/In_131.png)
vals[[1]] // TraditionalForm
Out[5]//TraditionalForm=
![](assets.zh/create-a-gallery-of-eigenfunctions-for-the-laplaci/O_63.png)
生成特征函数图集.
显示完整的 Wolfram 语言输入
Out[6]=
![](assets.zh/create-a-gallery-of-eigenfunctions-for-the-laplaci/O_64.png)