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

Author(s):
Joan Remski

Using the gradient operator

we may rewrite equation (1) as

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

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

But from the initial data,

where  f  is known. So, for any  (xt),

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

## JOMA

Journal of Online Mathematics and its Applications