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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

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$\frac{\delta u}{\delta t}$ 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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The best example for this equation is a dragged clump. Imagine a gas on a one-dimensional lattice being dragged to the right by an external force. At each time, each particle is moved to the right by one unit. There are additional random motions of the particles, but they have this drift superposed on top.

If you have a gradient in the particle density, so that the density steadily goes down as you move to the right, if you look at any point, the particles will, on average, come to you, so the density goes up with time. The amount it goes up is proportional to the density gradient (it just comes from the steady motion of the particles).

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.

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