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

• fluidistic
In summary, the person has been trying to solve a heat equation in a rectangle but has not been successful. They have tried using separation of variables and Green's function, but have not found a solution. They then considered simplifying the problem by defining a new function, but that also did not work. When using separation of variables, they obtained solutions of the form X(x)Y(y), but were unable to determine the constants. The person is seeking assistance with this problem.

#### fluidistic

Gold Member
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!

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