Thanks for the hint. I tried your form in the previous command:
op = Function[r*D[#, r] + Tan[\[Phi]] D[#, \[Phi]]];I*Nest[op, \[CapitalPsi][r, \[Phi]], 3] == 2 \[CapitalPsi][r, \[Phi]]*r^2/Cos[\[Phi]]^5
sol = u[r, y] /.
NDSolve[{op,
u[0, y] == 1,
Derivative[1, 0][u][0, y] == 0,
Derivative[2, 0][u][0, y] == 10,
Derivative[3, 0][u][0, y] == 0,
u, {r, 0, 3}, {y, 0, 3},
MaxSteps -> Infinity, PrecisionGoal -> 1,
AccuracyGoal -> 1,
Method -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"MinPoints" -> 32, "MaxPoints" -> 32, "DifferenceOrder" -> 2},
Method -> {"Adams", "MaxDifferenceOrder" -> 1}}] //
Plot3D[sol, {r, 0, 3}, {y, 0, 3}, AxesLabel -> Automatic]
I got the same error.
How can I start with some basic and simple I.C. that are completely arbitary, and work from there?
Can you propose a model with some arbitrary conditions, so I see what is wrong here?