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?
How to determine units in a partial differential equation
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0You're right, I flipped the units in the ODE part of the post. It's now been fixed. – 2012-08-01
2 Answers
$k$ has units $[distance^2 / time]$, so $\frac{\partial}{\partial t}u - k\Delta u = \left[\frac{temp}{time} - \frac{distance^2}{time} \frac{temp}{distance^2}\right] = \left[\frac{temp}{time}\right] $
You are already performing the dimensional analysis correctly!
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.