I need to solve the steady-state heat equation a.k.a. Laplace equation over a rectangle For $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$ defined on $x \in [0,a]$ and $y \in [0,b]$

with the BC as

$T(x,0) = T(x,b) = T_a$

$\frac{\partial T(0,y)}{\partial x} = k\bigg[e^{-sy}\bigg(T_1 + s\int_0^y e^{s t} T(x,t)\mathrm{d}t\bigg) - T(0,y)\bigg]$

$\frac{\partial T(a,y)}{\partial x} = 0$

The third Boundary condition seems like a Robin type. I applied separation of variables, but the resulting equation to calculate the Fourier coefficients was unsolvable for me.


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