The transport equation is a partial differential equation of the form

u _{t} + c u _{x} = 0 .
(1)

Here, u is a function of two variables (x,t) and the subscripts
denote partial derivatives. We will assume that c is a fixed constant. Given
an initial condition

**Applications:**

This equation can be used to model air pollution, dye dispersion, or even traffic flow with u representing the density of the pollutant (or dye or traffic) at position x and time t. For a discussion of the physical model, see [2]. For a discussion of the more general transport equation and its solutions, see [1].

**Solution:**

Using the gradient operator: = ( t , x) we may rewrite (1) as:

u(x,0)=f(x)
(2)

we would like to find a function of two variables that
satisfies both the transport equation (1) and the initial condition (2).

For discussion and simulation of more general conservation laws, including shock wave phenomena, see Scott Sarra's article The Method of Characteristics with Applications to Conservation Laws.

For discussion and simulation of more general conservation laws, including shock wave phenomena, see Scott Sarra's article The Method of Characteristics with Applications to Conservation Laws.

This equation can be used to model air pollution, dye dispersion, or even traffic flow with u representing the density of the pollutant (or dye or traffic) at position x and time t. For a discussion of the physical model, see [2]. For a discussion of the more general transport equation and its solutions, see [1].

Using the gradient operator: = ( t , x) we may rewrite (1) as:

(1,c) ·
u = 0 .

**Sample Problem:**

Suppose dye is spilled in a fast moving stream. Let u(x,t) represent the concentration of dye at x meters downstream from the initial spill at time t (measured in minutes). Suppose the inital concentration of dye at t=0 has the form u(x,0)=e^(-(x-2)^2) and c=.5.

**References:**

[1]* Introduction to Partial Differential Equations with MATLAB, * J. Cooper, Birkhauser,
1998.

[2]* An Introduction to the Mathematical Theory of Waves, * R. Knobel, Student Mathematical
Library of the AMS, 2000.

**Acknowledegment:**

Components for the applet are based on the Java Components for Mathematics at Hobart and William Smith Colleges.

Note that this last equation says that the directional
derivative in the (1,c) direction (in the t-x plane) is zero. So our solution
u(x,t) must be constant in this direction. In the t-x plane, the (1,c) direction
is along lines parallel to x=ct. Note: the lines parallel to x=ct are called
the **characteristics **of equation (1).

**Applet: **

Now, fix a point on the x-axis, say (x_{0}, 0).
The line through this point, parallel to x=ct is given by x=x_{0}
+ct. Since our solution is constant along this line we must have

u(x,t) = u(x_{0}+ct, t) = u (x_{0}, 0).

But from the initial data

u(x_{0},0) = f(x_{0})

where f is known. So for any (x,t):

u(x,t)= f(x_{0}) = f(x-ct)

The applet given below allows you to enter the initial
data u(x,0) and the slope of the characteristic lines, c. Pressing return in any of
the input boxes will update the graph. A Scrollbar allows
the user to control the time units on the graph of the solution (note that
changing c changes the "speed" of the solution). Scrollbars at the top and
bottom allow you to control the viewpoint (somewhat).

When entering the initial condition u(x,0), be sure to include * for
multiplication, / for division, ^ for a power, etc. The parser also recognizes
many standard math functions including sin(x), cos(x), ln(x), arctan(x), etc. ** Do
not leave any blank spaces in the middle of the input box. **

Suppose dye is spilled in a fast moving stream. Let u(x,t) represent the concentration of dye at x meters downstream from the initial spill at time t (measured in minutes). Suppose the inital concentration of dye at t=0 has the form u(x,0)=e^(-(x-2)^2) and c=.5.

- Use the applet to show what the concentration profile looks like when t=5.
- Now try changing the value of c (c=-1, -.5, 0, 1, 2). How does the profile at time t=5 change?
- What does c represent in this application? What would be proper units for c?

[1]

[2]

Components for the applet are based on the Java Components for Mathematics at Hobart and William Smith Colleges.