The Convection Equation is [math]u_t=-Vu_x[/math] where [math]V[/math] is a constant.[br]For an initial condition of a line [math]u\left(x,0\right)=mx+b[/math] substitution of [math]u_x=m[/math] gives[br][math]u_t=-Vm[/math] which, since [math]V[/math] and [math]m[/math] are both constants, can be integrated from the initial time to give[br][br][size=150][math]u\left(x,t\right)=u\left(x,0\right)-Vmt[/math][/size][br][br]This applet graphs the initial condition and the value at a later time, [math]t[/math]. Two arrows indicate the change. One arrow indicates the change in [math]u[/math] at [math]x=1[/math] and the other indicates the line movement in the [math]x[/math] direction.[br][br]Several sliders are available to change all of the parameters.

Observe what happens as you adjust each slider.[br][br]Particularly interesting is what happens to the [math]Vt[/math] displacement as the slope [math]m[/math] is changed.[br][br]A curve can be considered as a set of short connected line segments so how would a curve 'move'?[br][br]