First order partial derivatives

Advection equation

As an example of this general kind of PDE, consider a simple wave equation for a one-dimensional scalar field ($A(x,t)$), of the form:

$$\frac{\partial }{\partial t} A(x,t) = \mathcal{D}\cdot A(x,t) = -v\frac{\partial }{\partial x} A(x,t)$$ (2.2)

Note that for this equation the operator $\mathcal{D} = -v \frac{\partial }{\partial x}$. This is called the advection equation, and has the elementary solution:

$$A(x,t) = f(x-vt)$$ (2.3)

This corresponds to an arbitrary initial wave-form $f(x)$, traveling at fixed velocity $v$. This is the simplest possible equation that has wave-like solutions traveling at a fixed velocity.

Burger's equation

A simple nonlinear extension of this, is the inviscid Burger's equation, which is a nonlinear wave-equation used in nonlinear acoustics, shock-wave theory, and modern studies of traffic flow. It has an equation of form:^2.2

$$\frac{\partial }{\partial t} A(x,t) = -v \frac{\partial }{\partial x} [A - A^2/A_0]$$ (2.4)

These equations are sometimes used to describe traffic flow, where $A(x,t)$ would represent the density of cars on a road at a certain time and place, traveling at a velocity $v$ that depends on the density of cars, up to a maximum density, $A_{0}$. This equation models the fact that cars travel more slowly when the traffic density increases. Clearly, this is a rather oversimplified picture of the real situation, but it has useful real-world applications in modeling the behaviour of traffic at intersections, or the development of traffic jams after an accident has happened! Note that this is not a quasi-linear equation, and requires slightly modified techniques from the ones used for the other equations studied here. To model cars stopped at a red traffic-light, we can suppose that the initial distribution at $t=0$ is $A(x,t) = A_{0}$ for $x<0$, and $A(x,t) = 0$ for $x>0$. The solution at times $t>0$ is then:

$$A(x,t) = \begin{cases}A_{0} & \text{for } x<-vt \\ \frac{A_{0}}{... ...\frac{x}{vt}\right) & \text{for } -vt<x<vt \\ 0 & \text{for } x>vt.\end{cases}$$ (2.5)

Maxwell's equations

Another example of this general form is Maxwell's equations, which can be summarised as:

$$\begin{aligned}\frac{\partial }{\partial t}\mbf{d}& = -\mbf{j... ...l }{\partial t}\mbf{b}& = -\nabla \times \mbf{e}, \end{aligned}$$

together with subsidiary conditions of:

$$\begin{aligned}\nabla \cdot \mbf{d}& = \rho, \\ \nabla \cdot... ...on^{-1} \mbf{d}, \\ \mbf{h}& = \mu^{-1} \mbf{b}. \end{aligned}$$

Here the subsidiary conditions can be applied to the initial values of $\mbf{d}$ and $\mbf{b}$, to obtain the overall equations. The physical application in this case is to electromagnetic fields in free space, as in a radio transmitter. One particularly simple type of solution, in the case of free space propagation, is just the plane-wave solution with velocity $c = 1/\sqrt{\epsilon \mu}$, of form:

$$\begin{aligned}\mbf{d}(t,x)= & \mbf{d}^{(+)}(x-ct)+\mbf{d}^{(... ...(t,x)= & \mbf{b}^{(+)}(x-ct)+\mbf{b}^{(-)}(x+ct). \end{aligned}$$

With some modifications, the same equations can be used to describe integrated optics devices used in communications and photonics applications. Note that this example shows how a wave equation can be regarded as being first order, but with a larger number of fields!

Footnotes

... form:^2.2

For an good discussion of this equation and its numerical simulation see Numerical Methods for Physics by A. L. Garcia.