Modeling a Heat Source with the 2D Heat Equation

[CoolDude420]

Homework Statement

Hi,

So I have a perfectly insulated rectangular plate and I trying to use the 2D heat equation in conjunction with numerical finite diference methods and MATLAB to see how the temperature changes throughout the plate. My issue is with the heat source. I am supposed to decide on how to model this heat source. I have chosen it to be at the centre as a point heat source. Now I'm not sure how to set my Q value in the heat equation to mimic this point source? Apparently some people say I need to multiply by the dirac delta function but I feel that would make solving the PDE much harder.

I thought maybe I could set Q =0 and make an intiail conditions that sets the temperature at the centre, but didn't make much sense.

If I choose a constant Q, it would mean that this Q exists at all points in the plate. Any ideas on what I should do? How do I choose Q to mimic this point source?

Homework Equations

The Attempt at a Solution

 

[Orodruin]

Apparently some people say I need to multiply by the dirac delta function but I feel that would make solving the PDE much harder.

On the contrary, this would make the PDE much easier to solve. In fact, it is so much easier to work with delta functions that the entire idea behind using Green's functions is to first find the solution to the PDE when the inhomogeneity is a delta function and then making a superposition (you have likely already encountered this in relation to electric fields - the full field being the sum of all contributions). Besides, if what you really want to do is to model a point source, then you must use a delta function. The source distribution ##\kappa(\vec x)## has the property that
$$
\int_V \kappa(\vec x) \, dV
$$
is the heat produced in the volume ##V## per unit of time. Now, a point source at ##\vec x_0## is a source such that
$$
\int_V \kappa(\vec x) \, dV = \begin{cases} Q, & \vec x_0 \in V \\ 0, & \vec x_0 \notin V\end{cases}
$$
This is exactly what defines (##Q## times) the delta distribution ##\delta^{(3)}(\vec x - \vec x_0)##.

That being said, you will have problems implementing a delta function in a numerical finite element approach. If your elements are small enough, you can make a reasonable approximation by making the element that the point belongs to have a constant heat production within it.

 

[CoolDude420]

On the contrary, this would make the PDE much easier to solve. In fact, it is so much easier to work with delta functions that the entire idea behind using Green's functions is to first find the solution to the PDE when the inhomogeneity is a delta function and then making a superposition (you have likely already encountered this in relation to electric fields - the full field being the sum of all contributions). Besides, if what you really want to do is to model a point source, then you must use a delta function. The source distribution ##\kappa(\vec x)## has the property that
$$
\int_V \kappa(\vec x) \, dV
$$
is the heat produced in the volume ##V## per unit of time. Now, a point source at ##\vec x_0## is a source such that
$$
\int_V \kappa(\vec x) \, dV = \begin{cases} Q, & \vec x_0 \in V \\ 0, & \vec x_0 \notin V\end{cases}
$$
This is exactly what defines (##Q## times) the delta distribution ##\delta^{(3)}(\vec x - \vec x_0)##.

That being said, you will have problems implementing a delta function in a numerical finite element approach. If your elements are small enough, you can make a reasonable approximation by making the element that the point belongs to have a constant heat production within it.

Ah. I see. I'm planning on using the Crank-Nicholson method to solve the PDE. Do you have any suggestions on how I could actually convert that PDE(well the Q*diracDelta part) into a difference equation? I'm planning on using MATLAB.

 

[Orodruin]

If you have to solve it numerically, you can most likely introduce it as a source in a single node. Be careful to get the normalisation correctly. Note that you can easily get an analytic series solution in terms of the spatial eigenfunctions of the Laplace operator.