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I am attempting to solve a PDE as prescribed in a paper I am reading; however, Mathematica doesn't seem to be happy with the way I am posing the problem.

Im my case, r(t,100) is an approximation of r(t,inf).

d = 27;
k = 10;
s = 1;

sol = NDSolve[
  {D[u[t, r], t] == d*(1/r^2) D[r^2*D[u[t, r], r], r], 
   u[0, r] == 1, 
   Derivative[0, 1][u][t, s] == (k/d)*u[t, s], u[t, 100] == 1}, 

  u, {t, 0, 10}, {r, 0, 100}]

I am receiving an error stating that my boundary condition u(t,100)==1 is not specified on a single edge of the boundary of the computational domain. The answer is probably very simple, however I am just not seeing it.

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1 Answer

That means the boundary conditions defined should fit the the range of t and r in NDSolve command. You have 4 values: {t,0,10} and {r,0,100}, so you have to define the conditions at these 4 values. Here you define at r = s = 1 but you run r from 0 to 100. You need to define the boundary condition at r = 0 and at r = 100. But I don't know which boundary condition you need for r = 0, so I just run the NDSolve from s to 100 where at theses limits, the boundary conditions are well defined.

So this is the correct way to define the boundary conditions:

d = 27;
k = 10;
s = 1;

sol = NDSolve[{D[u[t, r], t] == d*(1/r^2) D[r^2*D[u[t, r], r], r],
   u[0, r] == 1,
   Derivative[0, 1][u][t, s] == (k/d)*u[t, s],
   u[t, 100] == 1},
  u,
  {t, 0, 10}, {r, s, 100}]


Plot[u[10, r] /. sol, {r, s, 100}]

enter image description here

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