# 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 $$x\in[0,L], y\in[0,l]$$. 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:

$$\frac{\partial T(x,l)}{\partial y}=\zeta \Bigg[T(x,l)-t\Bigg] \tag 4$$ 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.

• When you say two dimensional Laplacian, do you mean two dimensional Laplace's equation? Jul 19, 2020 at 3:20
• Yes you are right. $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}=0$. Jul 19, 2020 at 4:24
• @K.defaoite Can you provide some inputs on how to approach this problem ? Jul 21, 2020 at 16:05
• Start with the general solution of Laplace's equation in Cartesian coordinates then apply the boundary conditions. It might get messy. Other than that there isn't really anything else I can say. The third boundary condition will likely make things very complicated. Jul 21, 2020 at 16:08