可视化特征函数

定义一个三维拉普拉斯算子.

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

设定齐次狄利克雷边界条件.

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\[ScriptCapitalB] = DirichletCondition[u[x, y, z] == 0, True];

找出一个球内的最小特征值和特征函数.

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\[CapitalOmega] = Ball[{0, 0, 0}, 2]; {vals, funs} = DEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x, y, z], {x, y, z} \[Element] \[CapitalOmega], 2];

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funs

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用三维密度图绘制每一本征函数.

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Table[DensityPlot3D[ Evaluate[N[f]], {x, y, z} \[Element] \[CapitalOmega], PlotTheme -> "NoAxes", PlotLegends -> Placed[Automatic, Below]], {f, funs}]

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用坐标平面上绘制的密度.

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Table[SliceDensityPlot3D[ Evaluate[N[f]], {x, y, z} \[Element] \[CapitalOmega], PlotLegends -> Placed[Automatic, Below]], {f, funs}]

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