[Math] How to determine units in a partial differential equation

Question

How do we determine the units used in a differential equation? Yes, in theory a PDE has nothing to do with units, but I'm interested in this question from a modeling point of view. By units, I mean the following. In an ordinary differential equation, finding the correct units seems relative straightforward. For example, if we have a function $u : [0,T] (seconds) -> \mathbb{R} (meters)$, we know that taking the derivative with respect to time gives velocity, $(meters)/(seconds)$. Taking two derivatives with respect to time gives acceleration, $(meters)/(seconds)^2$. Therefore, when we write an ODE
$$
\frac{\partial^2}{\partial t^2} u = f,
$$
we know that $f$ should have units $(meters)/(seconds)^2$. In a PDE, this same trick doesn't seem to work. For example, say we have a function $u : [0,T] (seconds) \times \Omega (meters^2) \rightarrow \mathbb{R} (celsius)$ where $\Omega \subseteq \mathbb{R}^2$. Then, we write the heat equation
$$
\frac{\partial}{\partial t} u – k \frac{\partial^2}{\partial x^2} u – k\frac{\partial^2}{\partial y^2} u = f
$$
or more simply as
$$
\frac{\partial}{\partial t} u – k\Delta u = f.
$$
Now, using the above trick, the term $\frac{\partial}{\partial t} u$ has units $(celsius)/(seconds)$. However, the term $\Delta u$ has units $(celsius)/(meters^2)$. In this context, it doesn't make sense to add the two terms. It also doesn't give clear insight into what the units of the forcing function $f$ need to be. As such, what are the correct units for the heat equation and what's the general rule for establishing units for an arbitrary PDE?

Answer 1

I think that instead of units, you should study the scaling behaviour of PDEs. For example, what is the effect of simultaneously scaling $t$ and $x$ in the heat equation? And noting that if you scale $x$ with $a$ and $t$ with $a^2$, the equation is unchanged (except for the RHS, if nonzero). Scale invariant solutions are particular interesting and useful when they exist – one classic example being the fundamental solution of the heat equation.

The connection between scaling and the units of physics is via Buckingham's Π theorem.