PDE with tangential derivative condition along boundary. NDSolve not working

Question

I'm considering the following PDE

$$ \begin{align} \left[\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\right]\left[\sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial\theta}\right]\alpha(r,\theta)=\alpha(r,\theta), \end{align} $$

on the quarter $0\leq r \leq 1$, $0\leq \theta\leq \pi/2$ with conditions

$$ \begin{align} \alpha(r,0)&=1\\ \frac{\partial \alpha(r,\pi/2)}{\partial r}&=1 \end{align} $$

This is not exactly a Neumann boundary value problem, since the imposed derivative is tangential (not normal) to the portion of the surface. Nor is it an elliptic equation but a hyperbolic one, so this is more like a Goursat problem.

When using NDSolve as

pde=-(1/(2 r^2))(2 Cos[2 \[Theta]] ((\[Alpha]^(0,1))[r,\[Theta]]-r (\[Alpha]^(1,1))[r,\[Theta]])+Sin[2 \[Theta]] ((\[Alpha]^(0,2))[r,\[Theta]]+r ((\[Alpha]^(1,0))[r,\[Theta]]-r (\[Alpha]^(2,0))[r,\[Theta]])))+\[Alpha][r,\[Theta]]==0



NDSolve[{pde,\[Alpha][r,0]==1,(\[Alpha]^(1,0))[r,\[Pi]/2]==1},\[Alpha][r,\[Theta]],{r,0,1},{\[Theta],0,\[Pi]/2}]

I get the error

 NDSolve::fembderiv: The expression (\[Alpha]^(1,0))[r,\[Pi]/2]==1 given as a spatial boundary condition for the possibly automatically chosen finite element method should not have explicit derivatives of the dependent variables. NeumannValue should be used to specify spatial derivatives at the boundary.

Answer 1

Does this approach give reasonable results? Rewrite the PDE

$$ \left( \cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta}\right) \beta(r,\theta) = \alpha(r,\theta),$$ where $$ \beta(r,\theta)=\left( \sin\theta\frac{\partial}{\partial r}+\frac{\cos\theta}{r}\frac{\partial}{\partial\theta}\right) \alpha(r,\theta), $$

so that the boundary conditions become

$$ \begin{align} \alpha(r,0)&=1\\ \beta(r,\pi/2)&=1. \end{align} $$

pde = {Cos[θ] Derivative[1, 0][β][r, θ] -
    Sin[θ]/r Derivative[0, 1][β][r, θ] == α[r, θ],
  β[r, θ] == Sin[θ] Derivative[1, 0][α][r, θ] +
    Cos[θ]/r Derivative[0, 1][α][r, θ]};

bc = {α[r, 0] == 1, β[r, π/2] == 1};

soln = NDSolveValue[{pde, bc}, {α, β}, {r, 0, 1}, {θ, 0, π/2}];

ParametricPlot3D[{r Cos[θ], r Sin[θ], soln[[1]][r, θ]}, {r, 0, 1}, {θ, 0, π/2}]

Is the warning message important? The solution is not as flat as it looks in this plot.