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NDSolve::bcedge: Boundary condition c[t,5]==Cout is not specified on a single edge of the boundary of the computational domain. >>

I'd like to plot $\frac{\partial}{\partial t}c=\frac{d}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r}c) \equiv\Delta c $ with the initial condition $c(0,r)=c_{0}$ and the boundary conditions $\frac{\partial}{\partial r}c(t,0)=0$ and $c(t,R\in\mathbb{R})=c_{out}$ where $R$ is the radius of a circle. I know the analytical solution and I know how the profile looks. I'd like to use/learn Mathematica because it often helps if you can make a quick plot of unknown shapes.

NDSolve[{D[c[t, r], t] == d/(r^2) D[((r^2) D[c[t, r], r]), r],
Derivative[0, 1][c][t, 0] == 0, c[t, 5] == cout,c[0, r] == c0}, 
c, {t, 0, 10}, {r, -5, 5}]
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    – bbgodfrey
    Apr 8, 2015 at 21:50

1 Answer 1

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The specific error occurs because the range in r is given as {r, -5, 5} but the boundary condition Derivative[0, 1][c][t, 0] == 0 is given at r = 0, which is not a boundary. I imagine that {r, 0, 5} is meant, which eliminates the error. However, the equation is singular at r = 0, which creates other errors. This is a common issue in spherical coordinates. An easy work-around is to displace the inner radial boundary slightly, say to r = 0.01. Finally, all constants need to be specified. In all, I modified the code to

cout = 1; c0 = 2; d = 1;
ans = NDSolveValue[{D[c[t, r], t] == d/(r^2) D[((r^2) D[c[t, r], r]), r], 
   Derivative[0, 1][c][t, 0.01] == 0, c[t, 5] == cout, c[0, r] == c0},
   c, {t, 0, 10}, {r, 0.1, 5}];

and then plotted the solution

Plot3D[ans[t, r], {t, 0, 10}, {r, 0.01, 5}, AxesLabel -> {t, r, c}]

Mathematica graphics

This may get you started.

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  • $\begingroup$ Thanks for the example code and explanation! Actually my version does not support NDSolveValue and NDSolve returns NDSolve::ibcinc: Warning: Boundary and initial conditions are inconsistent and the plot is empty. Your guess on the BC was right. My attempt by choosing {r,-5,5} was to get a symmetrically continued function plot over r<0 (mirroring at the c-t-plane). Thanks a lot for your welcoming comment. $\endgroup$
    – dkeck
    Apr 9, 2015 at 6:58
  • $\begingroup$ NDSolve works just as well, and you usually can ignore the warning. See the documentation of NDSolve on how to extract and plot the results it produces. $\endgroup$
    – bbgodfrey
    Apr 9, 2015 at 12:16
  • $\begingroup$ Specifically, replace NDSolveValue[... by First@NDSolve[..., and ans[t, r] by (c /. ans)[t, r] in Plot3D. $\endgroup$
    – bbgodfrey
    Apr 9, 2015 at 12:44
  • $\begingroup$ I already tried ans = NDSolve[{D[c[t, r], t] == d/(r^2) D[((r^2) D[c[t, r], r]), r], Derivative[0, 1][c][t, 0.01] == 0.01, c[t, 5] == cout, c[0, r] == c0}, c, {t, 0, 10}, {r, 0.01, 5}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 1000}}] with Plot3D[Evaluate[c[t, r] /. ans], {t, 0, 10}, {r, 0.01, 5}, AxesLabel -> {t, r, c}, PlotRange -> {1, 3}] In both cases I only see a 2D plane constant to 2. Where MethodOfLines option was used to overcome the inconsistent boundary warning. $\endgroup$
    – dkeck
    Apr 9, 2015 at 20:26
  • $\begingroup$ What values are you using for your constants?. If cout=c0=2, the answer is 2 everywhere. $\endgroup$
    – bbgodfrey
    Apr 10, 2015 at 1:49

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