Steady state heat equation in a rectangle with a punkt heat source

[fluidistic]

Homework Statement

Solve the steady state heat equation in a rectangle whose bottom surface is kept at a fixed temperature, left and right sides are insulated and top side too, except for a point in a corner where heat is generated constantly through time.

Relevant Equations

$\kappa \nabla ^2 T + g =0$

I have checked several textbooks about the heat equation in a rectangle and I have found none that deals with my exact problem. I have though to use separation of variables first (to no avail), then Green's function (to no avail), then simplifying the problem for example by defining a new function in terms of $T(x,y)$ such that it would satisfy a homogeneous problem instead, but to no avail. (is a problem even called homogeneous when $dT/dx|_{x_0} = 0$ rather than $T(x=x_0)=0$? I guess not.)

Out of memory, when I went with separation of variables to tackle $\kappa \left( \frac{\partial ^2 T}{\partial x^2}+ \frac{\partial ^2 T}{\partial y^2}\right) = 0$, I obtained solutions of the form $X(x)Y(y)$ with $X(x)=A\cosh(\alpha x)+B\sinh(\alpha x)$ and $Y(y)=C\cos(\alpha y)+D\sin(\alpha y)$ where $\alpha$ is the separation constant. The boundary conditions are of the type Dirichlet for the bottom surface: $T(x,y=0)=T_0$. And Neumann elsewhere: $\frac{\partial T}{\partial x}|_{x=0, y=0}$ for $y\in [0,b)$, $\frac{\partial T}{\partial x}|_{x=a, y=0}$ for $y\in [0,b]$ and $\frac{\partial T}{\partial y}|_{x, y=b}$ for $x\in (0,a]$. The power generated translates as the Neumann boundary condition $\nabla T \cdot \hat n$ and so $\frac{\partial T}{\partial x}|_{x=0, y=b}+ \frac{\partial T}{\partial y}|_{x=0, y=b}=p$ where $p$ is the power density of the heat source.

I have been stuck there, I could not get to apply and know the constants $A$, $B$, $C$ and $D$, nor $\alpha$. All of these constants are in fact depending on $n$, natural numbers, because the separable solutions are eigenfunctions, etc.

Any pointer would be appreciated. Thank you!

 

[Chestermiller]

I’m thinking of an array of sources and sinks.