求解有约束的拉普拉斯算子的特征值问题
在齐次狄利克雷边界条件约束下求一维区域上拉普拉斯方程 最小的四个特征值和特征函数.
设定拉普拉斯算子.
In[1]:=
![Click for copyable input](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/In_4.png)
\[ScriptCapitalL] = -Laplacian[u[x], {x}];
设定狄利克雷边界条件.
In[2]:=
![Click for copyable input](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/In_5.png)
\[ScriptCapitalB] = DirichletCondition[u[x] == 0, True];
数值法求解特征值.
In[3]:=
![Click for copyable input](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/In_6.png)
NDEigenvalues[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x], {x, 0, \[Pi]}, 4]
Out[3]=
![](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/O_3.png)
数值法求解特征值和特征函数.
In[4]:=
![Click for copyable input](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/In_7.png)
{vals, funs} =
NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x], {x, 0, \[Pi]}, 4];
查看特征值.
In[5]:=
![Click for copyable input](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/In_8.png)
vals
Out[5]=
![](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/O_4.png)
可视化特征函数.
In[6]:=
![Click for copyable input](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/In_9.png)
Plot[Evaluate[funs], {x, 0, \[Pi]}]
Out[6]=
![](assets.zh/solve-the-eigenproblem-of-a-constrained-laplacian/O_5.png)