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November 20th, 2014, 05:55 AM   #1
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Poisson equation with three boundary conditions

I have the following 2D Poisson equation (which can also be transformed
to Laplace) defined on a triangular region (refer to plot):

\begin{equation}
\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=C\end{equation}

with the following three boundary conditions:

\begin{equation}
\frac{\partial u}{\partial y}=0\,\,\,\,\,\,\,\mathrm{at}\, y=0\end{equation}

\begin{equation}
u=0\,\,\,\,\,\,\,\mathrm{at}\, y=ax+b\end{equation}

\begin{equation}
c\frac{\partial u}{\partial x}+d\frac{\partial u}{\partial y}=0\,\,\,\,\,\,\,\mathrm{at}\, y=ex+f\end{equation}

where $C,a,b,c,d,e,f$ are constants.

What is the easiest way to solve this problem (preferably analytically)?
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