矩形のラプラス演算子の厳密な固有関数を計算する
同次ディリクレ境界条件を持つ2Dラプラス演算子を指定する.
In[1]:=
![Click for copyable input](assets.ja/calculate-exact-eigenfunctions-for-the-laplacian-i/In_115.png)
{\[ScriptCapitalL], \[ScriptCapitalB]} = {-Laplacian[u[x, y], {x, y}],
DirichletCondition[u[x, y] == 0, True]};
矩形の中の固有値と固有関数を小さい方から4個求める.
In[2]:=
![Click for copyable input](assets.ja/calculate-exact-eigenfunctions-for-the-laplacian-i/In_116.png)
{vals, funs} =
DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x, y], {x, 0, \[Pi]}, {y, 0, \[Pi]}, 4];
固有関数は三角関数である.
In[3]:=
![Click for copyable input](assets.ja/calculate-exact-eigenfunctions-for-the-laplacian-i/In_117.png)
funs
Out[3]=
![](assets.ja/calculate-exact-eigenfunctions-for-the-laplacian-i/O_57.png)
固有関数を可視化する.
In[4]:=
![Click for copyable input](assets.ja/calculate-exact-eigenfunctions-for-the-laplacian-i/In_118.png)
Plot3D[#, {x, 0, \[Pi]}, {y, 0, \[Pi]}] & /@ funs
Out[4]=
![](assets.ja/calculate-exact-eigenfunctions-for-the-laplacian-i/O_58.png)