16. Separation of variables in higher dimensions

Corrections

At 16:00, Y_n(x) should obviously be Y_n(y).

At 18:01, λ₅ should be 1²+3²=9 rather than 2²+3²=13, and similarly λ₆=3²+1². And then, at 18:23, they should of course be labelled λ₁₃ and λ₃₁. (The sequence continues with λ₇=λ₈=2²+3²=13, λ₉=λ₁₀=1²+4²=17, λ₁₁=3²+3²=18, λ₁₂=λ₁₃=2²+4²=20, …)

At 58:00 and 58:34, u(x,y) should be u(x,y,t).

At 1:06:28, λ²=(am/π)²+(bn/π)² should be λ=(πm/a)²+(πn/b)². (This is corrected a little later in the video, but maybe not in a perfectly clear way.)

Mathematica code

The plot at 55:25 is produced by the following Mathematica code:

With[{n = 4, m = 2}, With[{z = BesselJZero[n, m]},
  ParametricPlot3D[{r Cos[v], r Sin[v], Cos[n v] BesselJ[n, r z]}, {r, 0, 1}, {v, 0, 2 Pi},
    PlotPoints -> 50]]]

And the first row of plots at 1:11:40 is produced as follows:

region = Polygon[{{0, 0}, {1, 1}, {2, 0}, {1, -1}, {2, -2}, {2, -4}, {1, -3}, {0, -4}}];

{eigenvalues, eigenfunctions} =
  NDEigensystem[{-Laplacian[u[x, y], {x, y}], DirichletCondition[u[x, y] == 0, True]},
   u[x, y], {x, y} \[Element] region, 4];
  
Table[Plot3D[eigenfunctions[[i]] // N // Evaluate, {x, y} \[Element] region,
  PlotRange -> All, BoxRatios -> Automatic, PlotLabel -> eigenvalues[[i]],
  PlotTheme -> "Web", ImageSize -> Medium, PlotPoints -> 50], {i, Length[eigenvalues]}]

For the second row, do the same but with a another region:

region = Polygon[{{0, 0}, {2, 0}, {3, -1}, {1, -3}, {2, -4}, {0, -4}, {0, -2}, {1, -1}}];

Sidansvarig: Hans Lundmark
Senast uppdaterad: 2023-04-26