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 concent ration 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, 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 as sume 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,
Start solving your Algebra Problems
in next 5 minutes!
Download (and optional CD)
Click to Buy Now:
2Checkout.com is an authorized reseller
of goods provided by Sofmath
Attention: We are
currently running a special promotional offer
for Algebra-Answer.com visitors -- if you order
Algebra Helper by midnight of
you will pay only $39.99
instead of our regular price of $74.99 -- this is $35 in
savings ! In order to take advantage of this
offer, you need to order by clicking on one of
the buttons on the left, not through our regular
If you order now you will also receive 30 minute live session from tutor.com for a 1$!
You Will Learn Algebra Better - Guaranteed!
Just take a look how incredibly simple Algebra Helper is:
: Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:
Step 2 :
Let Algebra Helper solve it:
Step 3 : Ask for an explanation for the steps you don't understand:
Algebra Helper can solve problems in all the following areas:
simplification of algebraic expressions (operations
with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots
(radicals), absolute values)