Hi, after rigorous trying and help from some in the forum here, I turned out with the given command for a complex PDE:
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 = CapitalPsi[r, Phi] /.
NDSolve[{op,
CapitalPsi[0, Phi] == 1,
Derivative[1, 0][CapitalPsi][0, Phi] == 0,
Derivative[2, 0][CapitalPsi][0, Phi] == 10,
Derivative[3, 0][CapitalPsi][0, Phi] == 0,
CapitalPsi[r, 0] == 1,
Derivative[0, 1][CapitalPsi][r, 0] == 0},
CapitalPsi, {r, 0, 3}, {Phi, 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}, {Phi, 0, 3}, AxesLabel -> Automatic]
NDSolve::deqn: Equation or list of equations expected instead of r \!(*SubscriptBox[([PartialD]), (r)]#1)+Tan[[Phi]] \!(*SubscriptBox[([PartialD]), ([Phi])]#1)& in the first argument {r \!(*SubscriptBox[([PartialD]), (r)]#1)+Tan[[Phi]] \!(*SubscriptBox[([PartialD]), ([Phi])]#1)&,CapitalPsi[0,Phi]==1,(CapitalPsi^(1,0))[0,Phi]==0,(CapitalPsi^(2,0))[0,Phi]==10,(CapitalPsi^(3,0))[0,Phi]==0,CapitalPsi[r,0]==1,(CapitalPsi^(0,1))[r,0]==0}.
ReplaceAll::reps: {NDSolve[{r \!(*SubscriptBox[([PartialD]), (r)](Slot[<<1>>]))+Tan[[Phi]] \!(*SubscriptBox[([PartialD]), ([Phi])](Slot[<<1>>]))&,CapitalPsi[0,Phi]==1,(CapitalPsi^(1,0))[0,Phi]==0,(CapitalPsi^(2,0))[0,Phi]==10,(CapitalPsi^(3,0))[0,Phi]==0,CapitalPsi[r,0]==1,(CapitalPsi^(0,1))[r,0]==0},<<6>>,Method->{MethodOfLines,SpatialDiscretization->{TensorProductGrid,MinPoints->32,[Ellipsis] ->32,DifferenceOrder->2},Method->{Adams,MaxDifferenceOrder->1}}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing.
It appears it is complaining about the tan function. What is the problem with this command?
Thanks