# Using periodic boundary conditions for disk region in polar coordinates

I am trying to find the eigen values/functions of laplace operator in polar coordinates r, fi in disk region. In cartesian coordinates: (NDEigensystem[{-Laplacian[u[x, y], {x, y}], DirichletCondition[u[x, y] == 0, True]}, u[x, y], {x, y} \[Element] Disk[], 6]). In polar coordinates angle variabe should sutisfy periodic b.c. and radial shoud sutisfy dirichlet. The code I am working on looks like:

NDEigensystem[
{
-(D[y[r, fi], {fi, 2}]/r^2) - 1/r D[r*D[y[r, fi], r], r], (* equation *)
DirichletCondition[y[r, fi] == 0, r == 1.],
PeriodicBoundaryCondition[y[r, fi], fi == 2 Pi, Function[fi, fi - 2 Pi]
},

y[r, fi],
{r, 0, 1}, {fi, 0, 2 Pi},
3]


The problem is with setting following periodic condition y(r,fi+2 Pi) = y(r,fi) the way I did: PeriodicBoundaryCondition[y[r, fi], fi == 2 Pi, Function[fi, fi - 2 Pi]]. Is it possibe to do such "mixing" of boundary condition, and if yes how exacly should I specify them?

I will appreciate any help. Thank you very much in advance.

• Please add the definition of rmax to the question. – user21 Feb 1 '18 at 11:11
• Can you clarify what you mean with 'mixing'? – user21 Feb 1 '18 at 11:12
• rmax was a typo , its the same 1. "Mixing" is to use Dirichlet b.c. for r and periodic for fi varibles. – David Baghdasaryan Feb 1 '18 at 11:48

## 2 Answers

After some trial and error process I finally made this work:

{vals, funs} =
NDEigensystem[{-(D[y[r, fi], {fi, 2}]/r^2) -
1/r D[r*D[y[r, fi], r], r],
DirichletCondition[y[r, fi] == 0, r == 1 && 0 < fi <= 2 * Pi],
PeriodicBoundaryCondition[y[r, fi], fi == 0,
TranslationTransform[{0, 2 *Pi}]]
},
y[r, fi], {r, 0, 1}, {fi, 0, 2 * Pi},
5
];
vals


(* {5.78319, 14.684, 14.684, 26.3764, 30.4756} *)

which are the eigenvalues of the Laplace operator in a unit disk:

Sort@Flatten@Table[(N@BesselJZero[i, j])^2, {i, 0, 2}, {j, 1, 2}]

• Great. Would you mind if I added this example to the documentation of PeriodicBoundaryCondition? – user21 Feb 2 '18 at 6:56
• Sure with great pleasure. – David Baghdasaryan Feb 2 '18 at 7:01

The DirichletCondition overlaps with the periodic boundary condition (See the Issues Section)

You can use:

{vals, funs} =
NDEigensystem[{-(D[y[r, fi], {fi, 2}]/r^2) -
1/r D[r*D[y[r, fi], r], r],
DirichletCondition[y[r, fi] == 0,
r == 1 && (fi != 0 && fi != 2 \[Pi])],
PeriodicBoundaryCondition[y[r, fi], fi == 2 \[Pi],
Function[fi, fi + 2 Pi]]}, y[r, fi], {r, 0, 1}, {fi, 0, 2 Pi}, 3];
vals
{5.9417694867356845, 10.845294000815311, 15.731501290360747}

• Im sorry that I forgot to mention, but I would like to obtain the results the same as here NDEigensystem[{-Laplacian[u[x, y], {x, y}], DirichletCondition[u[x, y] == 0, True]}, u[x, y], {x, y} \[Element] Disk[], 6] the eigenvalues shuld be expressed with BesselJZero like Sort@Flatten@Table[(N@BesselJZero[i, j])^2, {i, 0, 2}, {j, 1, 3}] – David Baghdasaryan Feb 1 '18 at 12:33
• @DavidBaghdasaryan, How does that relate to the question? Could you update the question? Why would you expect that a region with a periodic bc has the same eigenvalues as one with all Dirichlet conditions? – user21 Feb 1 '18 at 12:38
• Sure I will update it, I just want to solve the disk problem (NDEigensystem[{-Laplacian[u[x, y], {x, y}], DirichletCondition[u[x, y] == 0, True]}, u[x, y], {x, y} \[Element] Disk[], 6]) in polar coordinates in polar coorfinates angle variabe should sutisfy periodic b.c. and radial shoud sutisfy dirichlet. – David Baghdasaryan Feb 1 '18 at 12:43
• @DavidBaghdasaryan, I see. If you think it helps I can delete my answer. – user21 Feb 1 '18 at 13:17
• No, no there is no need. Thank you @user21, your answer was helpful for finding the solution. – David Baghdasaryan Feb 2 '18 at 6:49