I am trying to solve the 2D heat equation (or diffusion equation) in a disk:

    \*SubscriptBox[\(\[PartialD]\), \(t\)]\(f[x, y, t]\)\) == \!\(
    \*SubsuperscriptBox[\(\[Del]\), \({x, y}\), \(2\)]\(f[x, y, t]\)\), 
      f[x, y, 0] == E^(-x^2 - y^2)}, {t, 0, 1}, {x, y} \[Element] Disk[]]

but I get this error

NDSolve::underdet: There are more dependent variables, {f[x,y,0],f[x,y,t],(f^(0,0,1))[x,y,t]}, than equations, so the system is underdetermined.

I want to include the Neumann boundary condition such that the normal derivative is zero. How can I include that?

edit: user21 fixed my mistake, here is a density plot from the code

DensityPlot3D[f[x, y, t] /. sol, {x, -1, 1}, {y, -1, 1}, {t, 0, 1}, 
 PlotRange -> All, AxesLabel -> Automatic]

with sol the solution as found below.

enter image description here

  • $\begingroup$ Please, post the code of the 3D plot $\endgroup$ Commented Jun 30, 2017 at 17:43
  • $\begingroup$ done, see edit. $\endgroup$
    – Ruud3.1415
    Commented Jul 2, 2017 at 13:19

1 Answer 1


You need to add the dependent variable f like so:

NDSolve[{Derivative[0, 0, 1][f][x, y, t] == 
   Derivative[0, 2, 0][f][x, y, t] + Derivative[2, 0, 0][f][x, y, t], 
  f[x, y, 0] == E^(-x^2 - y^2)}, f, {t, 0, 1}, 
 Element[{x, y}, Disk[{0, 0}]]]

Concerning the Neumann zero boundary condition: Neumann zero boundary conditions are the default so nothing needs to be done.

  • $\begingroup$ yes... I should have done that.. Thanks user21! Also thanks for the info andre! $\endgroup$
    – Ruud3.1415
    Commented Jun 30, 2017 at 17:30
  • $\begingroup$ @andre, thanks added your comment to the post. $\endgroup$
    – user21
    Commented Jun 30, 2017 at 17:55

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