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This is related to: Numerical solution of PDE using the Shooting Method; implementation and errors

which I asked earlier today. I realised what I did wrong, I fixed it, but I still get errors.

This is what I want to do.

I want to solve numerically using the shooting method the following PDE (I am using the shooting method because from everything I read I think it fits better to the problem, however, I haven't done much work on numerical solutions of PDE's and if you have alternative suggestions, please let me know.)

$$\partial_r(r^5 \sin^3 \theta_B \partial_r \theta) + \partial_x(r \sin^3 \theta_B \partial_x \theta)- r^3(3 \cos^2\theta_B \sin \theta_B - \frac{3}{2} \sin^2 \theta_B) \theta = 0$$

where of course $\theta=\theta(r,x)$

with

$$\theta_B = \cos^{-1}(\frac{m}{r})$$

and with the conditions

$$\theta(m,x)= 1$$ $$\partial_{r}\theta(m,x) =0$$

and then plot the solution.

This is the code.

\[Theta]B[r_, m_] := ArcCos[m/r]

nsolone = NDSolve[\!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\((
\*SuperscriptBox[\(r\), \(5\)]\ 
\*SuperscriptBox[\(Sin[\[Theta]B[r, 1]]\), \(3\)] 
\*SubscriptBox[\(\[PartialD]\), \(r\)]\[Theta][r, x])\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \(x\)]\((r\ 
\*SuperscriptBox[\(Sin[\[Theta]B[r, 1]]\), \(3\)]\ 
\*SubscriptBox[\(\[PartialD]\), \(x\)]\ \[Theta][r, x])\)\) - 
    r^3 (3 Cos[\[Theta]B[r, 1]]^2 Sin[\[Theta]B[r, 1]] - 
       3/2 Sin[\[Theta]B[r, 1]]^2) \[Theta][r, x] == 0, \[Theta][r, 
   x] , {r, 1, 10^7}, {x, 0, 10^7}, 
  Method -> {"Shooting", 
    "StartingInitialConditions" -> {\[Theta][1, 0] == 1,  \!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\((\[Theta][1, 0])\)\) == 0}}] 

This is the error that I am getting.

NDSolve::femibcnd: No DirichletCondition or Robin-type NeumannValue was specified for {\[Theta]}; the result is not unique up to a constant.

When I try to study the error in order to fix it, this opens with the very generic info on D.E's. http://reference.wolfram.com/language/ref/NDSolve.html

And I do not have any idea how to plot this.

Thanks!!!

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  • 1
    $\begingroup$ What you call an error is only a hint result is not unique up to a constant. You can plot the solution with Plot3D[\[Theta][r, x] /. nsolone[[1]], {r, 1, 10^7}, {x, 0, 10^7}] but solution unfotunately appears to be zero. Perhaps with additional boundary conditions you can force unique solution? $\endgroup$ – Ulrich Neumann Feb 6 '18 at 8:55
  • $\begingroup$ Thanks for the hint. I will give it a go. Cheers!!! $\endgroup$ – Konstantinos Feb 6 '18 at 8:59

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