The Transport Equation and Directional Derivatives - The Mathlet

Author(s):
Joan Remski

The applet on this page allows you to enter the initial data  u(x, 0)  and the slope of the characteristic lines,  c.  Press return in any of the input boxes to update the graph. Use the scroll bar beneath the graph allows to control time on the graph of the solution. Changing  c  changes the "speed" of the solution. Use the scroll bars at the top and right-hand side to control the viewpoint.

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.

Sample Problem

Suppose dye is spilled in a fast moving stream. Let  u(xt)  represent the concentration of dye at  x  meters downstream from the initial spill at time  t  (measured in minutes). Suppose the initial concentration of dye at  t = 0  has the form

and  c = 0.5.

1. Use the applet to show what the concentration profile looks like when  t = 5.
2. Now try changing the value of  c  (c = -1, -0.5, 0, 1, 2). How does the profile at time  t = 5  change?
3. What does  c  represent in this application? What would be proper units for  c?

JOMA

Journal of Online Mathematics and its Applications