Consider the following parabolic partial differential equation (PDE)

begin{align}

label{eq:42}

left(cospsi frac{partial}{partial r} + frac{gamma}{r} sinpsi frac{partial}{partial psi} + epsilon frac{partial^{2}}{partial psi^{2}} right)u(r, psi) = -1,

end{align}

where $u(r, psi): (0,1)times(0,2pi) to mathbb{R}^+$ and $epsilon,gamma in mathbb{R}^+$ are constant parameters.

The boundary conditions are Dirichlet $u(r, psi)|_{r=1} = 0$ and periodic $u(r, psi+2pi) = u(r, psi)$.

When $epsilonto 0$, the above second order PDE reduces to first order PDE, the boundary conditions generally cannot be fulfilled anymore, entailing singular perturbation method to handle the equation.

**How to solve the above the equation with $epsilonto 0$ using singular perturbation method?**

I mainly refer to the book written by J. Kevorkian and J.D. Cole titled ‘‘Multiple Scale and Singular Perturbation Methods’’. Singular perturbation methods for elliptic and parabolic PDEs are introduced there. However, I still have no clue of how to handle the above parabolic PDE.

In case it helps, the (projected) characteristic curve of the reduced PDE when $epsilon=0$ is

begin{align}

label{eq:62}

r = |sinpsi|^{frac{1}{gamma}} C,

end{align}

where $C$ is constant, which can be visualized in the following plot ($gamma=1$).

My background is theoretical physics. Please let me know if there is something mathematically inaccurate in the above problem formulation. Any suggestion or recommendation of references would be greatly appreciated. Thanks.