# What does transport equation represent in terms of physical quantities?

In my math course we're taught to solve PDE (partial derivative equations) like transport equation: $$c\frac{\partial u}{\partial x} +\frac{\partial u}{\partial t}~=~0.$$

If $u(x,t)$ is the quantity transported and $c$ has speed dimension (according to my book), $\frac{\partial u}{\partial t}$ must be speed too. What does $\frac{\partial u}{\partial x}$ represent? Does anybody have a good physical example to help me understand?

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It is a gradient of $u$. If there is no gradient, there is no transport. –  Vladimir Kalitvianski Nov 11 '11 at 16:14

$\frac{\delta u}{\delta t}$ does not always have the dimension of speed. It is the change rate of physical quantities respect to time, $u$ can be mass or concentration of electric charge (density) or probability density $\rho$ in quantum physics.

So if we only consider the classical physics (i.e. heat conduction can be described using this function), the $\frac{\delta u}{\delta x}$ can be looked as gradient of physical quantity (in one dimension in this equation). Since there is such a gradient, therefore, we can think that this gradient will produce a "force"(not quite an actual force usually) to drive the transport. Therefore, this quantity will have a change rate respect to time. According to the conservation law, this change rate must be equal to the gradient times a constant $c$, and this $c$ has a dimension of speed.

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So the result is that you have a density profile moving to the right at a steady speed. The solution to the equation is f(x-ct), which is a steady clump. If you want a more realistic equation, you can add a $d^2x\over dt^2$ term, which represents diffusion (the diffusion can also absorb a steady drift by a Galilean transformation). When you are only dealing with the first derivative terms, there is no diffusion.