Laplacian coupled with another equation over a two-dimensional rectangular region

I have the two-dimensional Laplacian $(nabla^2 T(x,y)=0)$ coupled with another equation. The Laplacian is defined over $xin[0,L], yin[0,l]$. On manipulating the second equation (which I have described in the Origins section of my question) I have managed to reduce the problem to a boundary value problem on the Laplacian subjected to the following boundary conditions

$$frac{partial T(0,y)}{partial x}=frac{partial T(L,y)}{partial x}=0 tag 1$$

$$frac{partial T(x,0)}{partial y}=gamma tag 2$$

$$frac{partial T(x,l)}{partial y}=zeta Bigg[T(x,l)-Bigg{alpha e^{-alpha x}Bigg(int_0^x e^{alpha s }T(s,y)mathrm{d}s+frac{t_{i}}{alpha}Bigg)Bigg}Bigg] tag 3$$

$gamma, alpha, zeta, t_i$ are all constants $>0$. Can anyone suggest a way to solve this problem ?


The 3rd boundary condition is actually of the following form:

$$frac{partial T(x,l)}{partial y}=zeta Bigg[T(x,l)-tBigg] tag 4$$
The $t$ in $(4)$ is governed by the following equation (this is the other equation I mentioned earlier):

$$frac{partial t}{partial x}+alpha(t-T)=0 tag 5$$

where it is known that $t(x=0)=t_i$. To derive $(3)$, I solved $(5)$ using the method of integrating factor and substituted in $(4)$.

My original problem is the Laplacian coupled with $(5)$.

Physical meaning

The problem describes the flow of a fluid (with temperature $t$ and described by $(5)$) over a rectangular plate (at $y=l$) heated from the bottom (at $y=0$). The fluid is thermally coupled to the plate temperature $T$ through boundary condition $(3)$ which is the convection or Robin type condition.

Mathematics Asked on November 21, 2021

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