You are here

Home » MAA Publications » Periodicals » Loci/JOMA » The Transport Equation and Directional Derivatives - Solution of the Transport Equation

The Transport Equation and Directional Derivatives - Solution of the Transport Equation

Author(s): 
Joan Remski

static graphUsing the gradient operator 

Transport-png3.png 

we may rewrite equation (1) as

Transport-png4.png

This 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,  which are called the characteristics of equation (1).


static graph2

Now, fix a point on the  x-axis, say  (x0, 0). The line through this point parallel to  x = ct 
is given by  x = x0  + ct.  Since our solution is constant along this line, we must have
 
Transport-png5.png

But from the initial data,

Transport-png6.png

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

Transport-png7.png

 

Joan Remski, "The Transport Equation and Directional Derivatives - Solution of the Transport Equation," Convergence (August 2004)

JOMA