One of the driving motivations for studying PDE’s is to describe the
physical world around us. We can use a flux argument to derive equations describing the evolution of a density, which is just a fancy
word describing the concentration of something (mass in a region, heat
in a metal bar, traffic on a highway) per unit volume.
Consider a one-dimensional freeway and let be
of cars per unit length on the freeway.
Figure 1.1: Flux argument for cars on a freeway.
( draw your own figure).
Then the mass of cars in the region a < x < b is given by
Now suppose we are measuring the flux, Q, of cars into
measured in mass/unit time. It can written in terms of the number of
cars crossing into the region at x = a, called q(a), minus the number
of cars that flow out of the region at x = b, called q(b),
Now, by conservation of mass, the rate of change of
the mass between
a and b is given by the flux into the region,
We can rewrite the flux by a clever application of the
theorem of calculus:
We can now rewrite the conservation of mass equation as
Since this is true for every interval a < x < b,
the integrand must
vanish identically. So
Equations of this form are called transport equations
or conservation laws – they are a very active area of study in PDE’s.
We can propose a simple model for the flux function q(x, t) – suppose
we assume the cars are all moving at a constant speed C. Then
we can argue that the flux is just equal to the product of the number
of cars time the speed they are moving at,
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