# The Transport Equation and Directional Derivatives

Author(s):
Joan Remski

The transport equation is a partial differential equation of the form

 (1)

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

 (2)

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

This equation can be used to model pollution (Lehn and Scherer , undated), dye dispersion (Roychoudhury , undated), or even traffic flow (Jungel , 2002), with  u  representing the density of the pollutant (or dye or traffic, respectively) at position  x  and time  t.  For a discussion of the physical model, see Knobel (2000). For a discussion of the more general transport equation and its solutions, see Cooper (1998). For discussion and simulation of more general conservation laws, including shock wave phenomena, see Sarra  (2003).

Acknowledgment

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

#### Joan Remski is an Assistant Professor of Mathematics at University of Michigan-Dearborn .

##### Copyright 2004 by Joan Remski

Published June, 2004

Joan Remski, "The Transport Equation and Directional Derivatives," Convergence (August 2004)

## JOMA

Journal of Online Mathematics and its Applications