Nondimensionalization

Differential equations that show up in modeling “real world situations” usually have many constants in them. Often one can reduce the number of constants in a problem by choosing the right units for the various quantities in the problem.

Example

Suppose that a quantity $x(t)$ changes in time according to the ODE \begin{equation} \frac{dx}{dt} = a + bx +cx^2. \label{eq:nondimensionalize-this} \end{equation} The coefficients $a$, $b$, and $c$ all must have different units, otherwise we could not add the terms $a$, $bx$, and $cx^2$.

To simplify the equation we choose a constant value for $x$, let's say $x_0$, and we let this value be our unit. The ratio \[ \xi = \frac{x}{x_0} \] has no units. In the same way we can pick a unit of time $t_0$ and introduce the quantity \[ \tau = \frac{t}{t_0} \] which also has no units.

The quantities $\xi$ and $\tau$ are nondimensionalized versions of our original variables $x$ and $t$. The point of nondimensionalization is that we can now derive a differential equation for $\xi$ and $\tau$, and then afterwards figure out which choice of the units $x_0$ and $t_0$ simplifies things most.

In this example we substitute \[ x(t)=x_0\xi(\tau), \qquad \text{ and } \qquad t=t_0\tau \] which leads to \[ \frac{dx}{dt} = \frac{x_0}{t_0} \frac{d\xi}{d\tau} \] by the chain rule, and \[ a+bx+cx^2 = a + (bx_0)\, \xi + (cx_0^2)\, \xi^2 \] by direct substitution. The differential equation \eqref{eq:nondimensionalize-this} for $x$ and $t$ is therefore equivalent with \[ \frac{x_0}{t_0} \frac{d\xi}{d\tau} = a + (bx_0)\, \xi + (cx_0^2)\, \xi^2, \] and thus \[ \frac{d\xi}{d\tau} = \frac{t_0a}{x_0} + (t_0b) \, \xi + (t_0x_0c)\, \xi^2. \] At this point we choose $x_0$ and $t_0$. We can try to make the constant term and the coefficient of $\xi$ both equal to $1$. If $a\neq0$ and $b\neq0$ then this is possible provided we choose \[ t_0 = \frac{1}{b}, \qquad x_0 = at_0 = \frac{a}{b}. \] The coefficient of $\xi^2$ then becomes \[ \alpha = t_0x_0^2 c = \frac{a^2c}{b^3}, \] and we get the following differential equation for $\xi$ as a function of $\tau$: \begin{equation} \frac{d\xi}{d\tau} = 1+\xi+\alpha\, \xi^2. \label{eq:this-nondimensionalized} \end{equation} This is a nondimensionalized version of equation \eqref{eq:nondimensionalize-this}. Note that instead of three undetermined parameters ($a$, $b$, $c$) it only has one parameter, namely $\alpha$.

Problems

Assuming $x$ has units of length (say, meters) and $t$ has units of time (e.g. seconds), what units do the coefficients $a$, $b$, and $c$ have?

What units do the variables $\xi$ and $\tau$ have?

What units does the new parameter $\alpha = a^2c/b^3$ have?