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Consider the following non-homogeneous parabolic partial differential equation (PDE)

\begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + \epsilon \frac{\partial^{2}}{\partial \psi^{2}} \right)u(r, \psi) = -1, \end{align}

where $u(r, \psi): [0,1]\times[0,2\pi] \to \mathbb{R}^+$ and $\epsilon,\gamma \in \mathbb{R}^+$ are constant parameters. The boundary conditions are Dirichlet $u(r, \psi)|_{r=1} = 0$ and periodic $u(r, \psi+2\pi) = u(r, \psi)$.

How to find a particular solution of the above non-homogeneous PDE?

Thank you for your time and consideration.

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  • $\begingroup$ The issue with this equation from a PDE perspective is that it's backward-parabolic on part of the region (where cos $\psi$ is negative). Usually that's bad and I would expect it to be ill-posed, but the question here is very specific with analytic "data." As such, maybe there are approaches leveraging the analyticity that would work, but to me it does not seem easy to analyze. $\endgroup$
    – user378654
    Oct 3, 2021 at 4:02

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