Math 513, Fall 2008 at PSU


Motivation.

We derive the wave equation 

u_{tt} - c^2 u_{xx} = 0 

or the vibrating string equation with small amplitude
in the first lecture. It describes to the leading order 
the motion of waves on a taut string. Here u_{tt}
denotes the second order derivative of u with respect to
t. And c^2 is c squared.

Then this wave equation can be factorized into two 
1st order equations

u_t - c u_x = v;    v_t + c v_x =0.

So we need to study both first order and second order 
equations. Mathematics and general science have 
accumulated quite a lot of such equations. 
They include the Maxwell electro-magnectic
system of equations, Einstein equations for 
general relativity, Yang-Mills equations, Navier-Stokes, 
Euler equations for gas dynamics, Boltzman equation for
kinectics, Schrodinger equation in quantum mechanics, 
Reaction-Diffusions in chmistry and life sciences, 
Bellman equation and Black-Shole in economics, 
etc., etc., etc. 

Do we just study these equations individually and 
no more general theory?

Examples for supporting theory.

1. For a fourth-grader, there are these exercises: 
Find two numbers such that its sum is 11, but its 
product is 30. She (or he) can use the trial-and-error 
method to find the answer. A similar problem is 
to find two numbers such that its sum is 11 but 
product is 10. She initially got perplexed, and 
hesitated to make any attempt. So I suggested 
that she imagine that the two numers are x and y 
and form 

x + y =11,
xy =10.

Then try to solve for x and y. So you see, 
Algebra is needed here to handle the operations 
of arbitrary numbers x and y. Although in real life 
we will need to handle only a finite number of 
numbers and their additions and multiplications 
and they are all concrete numbers, but the theory 
of algebra is helpful in this inverse process 
because we do not know a priori these numbers. 
The theory covers them all a priori. 

2. The road construction company HRI is paving 
the inner loop road --the Blue Course Drive (Fall 2003). 
To me I feel we will only 
need a portion of the pavement, and why does the 
company pave every bit of the whole wide road?

Thus, we will study PDEs in quite general forms;
i.e., not just the few that are really applicable 
in science. We will however not try to be too fancy. 
For example we will not try to find two numbers 
whose sum is 4 but product is 5.

So we will study equations like this

u_t + a(t) u_x = f(t, x),

where a(t) and f(t, x) are given functions, 
which contains both of the above two equations 
of 1st order. 

What do we mean by study the equation? 
We mean that we give a ``formula'' for all its solutions.
The formula might be a procedure.

In a nutshell, we solve important equations
from sciences and learn a bit of theory.