# 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 ?

Origins

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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