2
$\begingroup$

I'm studying the transport equation

$$u_t + cu_x = 0$$

Where c is a constant.

What does the solution, $u(x,t)$, describe? Is it ... the height of the wave? The location of the wave? Does the transport equation only describe waves?

The source that I am studying from only says, "the wave, u(x,t) ... "

I know in the heat equation, the solution $u$ is temperature (dependent on position x and time t) and in the wave equation, the solution $u$ gives the height of the wave.

So, I'm just not sure of what the solution to the transport equation describes.

Thanks,

$\endgroup$
  • $\begingroup$ I thought that, at position $x$, time $t$, the solution $u(x,t)$ gives the concentration. $\endgroup$ – Tom Ultramelonman May 30 '16 at 22:19
2
$\begingroup$

The quantity $u(x,t)$ can describe:

  1. The height of a wave at point $x$ at time $t$
  2. The temperature at point $x$ at time $t$
  3. The concentration of some substance at point $x$ at time $t$
  4. The number of cars on a 50-meter stretch of a road...

PDEs are not tied to physical quantities; they describe physical processes. The PDE $u_t+cu_x$ describes the process of propagation (of whatever) with constant velocity. The PDE $u_t = ku_{xx}$ describes the process of diffusion, of whatever is able to diffuse. It is not really correct to say that "in $u_t =k u_{xx}$, the function $u$ is temperature". It could be that, or it could be radioactivity level, etc. Conversely, the temperature could be modeled by $u_t=ku_{xx}$, or by $u_t = cu_x$, or by $u_t = cu_x + ku_{xx}$, or by many other equations. It depends on what processes one chooses to focus on.

$\endgroup$
  • $\begingroup$ Ok, got it - thanks so much @sandwich :) $\endgroup$ – User001 May 31 '16 at 4:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.