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Chapter 24
Visualizing the heat and wave equations on some 2-dimensional regions.
The heat equation in a rectangle.
Recall in previous chapters we studied the heat equation
$\dfrac{\partial u}{\partial t} = \dfrac{\partial^2 u}{\partial x^2}$
that describes the evolution of temperature along a rod. This is a time-evolution problem with 1 spatial dimension. It is of interest to study the evolution of the temperature distribution of a two-dimensional object, such as a plate or sheet of metal. In this case, the two-dimensional heat equation takes the form
\begin{equation}
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
\end{equation}
for a function $u(x,y,t)$ representing the temperature of an object at time $t$ and position $(x,y)$.
A new challenge with 2 spatial dimensions is that we need to take the shape of the object into consideration; this influences
the nature of the boundary conditions. (In the 1-dimensional case with a rod, this amounted to specifying
information at just two points, namely the endpoints $x = 0, L$).
Here we treat the case of a rectanglar shape. Consider a rectangle with sides having lengths $L_1$ and $L_2$. Set up coordinates
so it is the region $$R = [0, L_1]\times [0, L_2] = \{(x,y) \in \R^2: 0 \le x \le L_1, 0 \le y \le L_2\}.$$
We will consider the heat equation on $R$ with specified boundary conditions on all four sides.
In contrast with a rod, whose ``boundary'' consisted of two distinct points (so that boundary falls into two parts, each a point), the boundary of $R$ is a
connected curve (consisting of 4 line segments touching at endpoints).
Our focus is on cases with constant temperature along the boundary.
We call this a Dirichlet boundary condition. (The concept has a more general definition, but constancy is sufficient for our needs.)
Since constant functions satisfy the 2-dimensional heat equation, to find
all solutions that have a given constant temperature $\tau$ on the boundary
it is the same as solving the heat equation satisfying the vanishing Dirichlet boundary condition.
Indeed, if $u$ satisfies the heat equation then so does
$u \pm \tau$, and that allows us to pass between solutions with value 0
on the entire boundary and solutions with value $\tau$ on the entire boundary.
So we will focus on those solutions which vanish on the boundary (and then any desired constant can be added to this to get
the solutions with a specific constant value on the boundary).
Now, as we learned in this chapter, using separation of variables, we find that the general solution to the heat equation on a rectangle
$[0,L_1] \times [0, L_2]$ with vanishing Dirichlet boundary values is
\begin{equation}
u(t,x,y) = \sum_{n_1=1}^\infty \sum_{n_2 = 1}^\infty A_{n_1,n_2} e^{-((n_1^2/L_1^2) + (n_2^2/L_2^2))\pi^2 t} \sin((n_1 \pi/L_1) x) \sin((n_2 \pi/L_2) y).
\end{equation}
Below we visualize heat map plots of
$e^{-((n_1^2/L_1^2) + (n_2^2/L_2^2))\pi^2 t} \sin((n_1 \pi/L_1) x) \sin((n_2 \pi/L_2) y)$.
Pick $L_1, L_2, n_1, n_2$ and see this function evolves with time. Note that as you change the $L_j$'s, the scale along
the axes changes (to keep the box looking the same on the computer screen). Trying running with a slower speed (such as 1/20 or 1/50 times as slow) to better
see what is happening with the heat flow. Beware that if you want to change the speed of the animation, you need to begin from the start rather than pressing
Pause and trying to change the speed and then pressing Continue.