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\centerline{Mathematics 373 -- Applied Mathematics (PDE)}
\centerline{Information on Final Projects}
\centerline{March 13, 2001}
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{\bf General Information}
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As announced in the course syllabus, one of your assignments for 
this course will be a group term project, leading to a paper of 
approximately 10 - 15 pages.  You may work individually, or in 
groups of 2 or 3 on the term project.  I will leave the selection 
of groups up to you, unless problems develop.  All members
of a group are expected to participate fully in 
the development of the project, and as a part of the final 
submission of materials, all members of each group will 
be asked to give evaluations of the contributions of the others.
I will be assigning individual grades for the 
project based on the paper, and the ``peer reviews''
from your fellow group members.  The following 
gives a timetable to help you structure your work on this assignment
and several suggested topics for you to consider.  Other topics of 
your choice are also possible, but you will need to consult with
me {\it on or before March 22} to make sure the idea you have in mind is 
acceptable.
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{\bf Timetable}
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\item{1)}  By Thursday, March 22 {\it at the latest\/}, notify me of the
group you have chosen to work with, and the topic you will be pursuing.
I will be happy to make  suggestions and/or help you locate 
materials.  Mr. Stankus. the Science Librarian and the other library staff
will also be happy to assist you if you need to find other presentations
of the material.
\item{2)}  The week of April 16 (the week after Easter Break), I will
schedule a short ``progress report'' session with each of the groups 
(at mutually convenient times outside of class).  By this time, 
 you should be prepared to discuss what you want
to put in the paper.  I will be more than 
happy to meet with you before and after these formal
progress report sessions as well to help you tackle your projects.
\item{3)}  The term project papers will be due Tuesday, May 1.  (This
is a {\it firm deadline\/}.  I will not be able to accept late work on 
this assignment!)
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\noindent
{\bf Suggested Topics}
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{\it 1. Numerical Methods for Solving PDE}
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(This topic will require some extensive Maple calculations
to solve big systems of linear equations.)
The analytic techniques we have studied in this class for solving
PDE (including
Fourier series and other eigenfunction expansions) are applicable
only in quite limited situations (i.e. for very simple PDE coming 
from rather unrealistic physical situations).  In fact, when applied mathematicians,
scientists, or engineers need to solve PDE ``in the real world'' it
is much more common for approximate numerical methods to be used.
For this project, you would learn the basic ideas behind the 
construction of {\it finite difference} numerical methods for
the heat equation, the wave equation, Poisson's equation, and so forth.
The first step in all of these cases is to {\it discretize} the 
space and time variables by subdividing the ranges of these variables
(as in the construction of Riemann sum approximations to integrals
over the same regions).  This introduces a collection of {\it grid
points} where we seek approximate values of the solution function $u$.

The partial derivatives in the PDE are replaced by corresponding
finite difference approximations.  For instance if we subdivide
in the $x$ and $y$ directions with $\Delta x = \Delta y = h$, then
with $x_i = x_0 + ih, y_j = y_0 + jh$, we might use
$$\cases{u_x(x_i,y_j)\doteq {u(x_{i+1},y_j) - u(x_i,y_j)\over h}&\cr
         u_y(x_i,y_j)\doteq {u(x_i,y_{j+1}) - u(x_i,y_j)\over h}&\cr}$$
(called {\it forward difference} approximations to the first derivatives), or
$$\cases{u_{xx}(x_i,y_j)\doteq {u(x_{i+1},y_j) - 2u(x_i,y_j) + u(x_{i-1},y_j)\over h^2}&\cr
         u_{yy}(x_i,y_j)\doteq {u(x_i,y_{j+1}) - 2u(x_i,y_j) + u(x_i,y_{j-1})\over h^2}&\cr}$$
(these are called {\it centered difference} approximations to 
second derivatives).  We substitute these approximations into the PDE
and derive a system of equations for the approximate values $u(x_i,y_j)$.
If the original PDE was a linear equation, then the corresponding
equations are linear algebraic equations, and these can be solved
by several different approaches.  The details of these methods, as
well as the linear algebra techniques that are often used
to solve the resulting systems of linear equations is 
described in Chapter 12 of the text {\it Numerical Analysis} by Burden and 
Faires, on reserve for our class in the Science Library.
For this project you would 
work through and present the details of this approach, and solve an applied
problem like the one described in Project D in Chapter 10 in our text.
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{\it 2. Laplace Transform Method for Solving ODE and PDE}
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(Some portions of this project use ideas from the Complex Analysis course
that Prof. Anderson taught last year.)  The Laplace transform
of a function of $t$ is the function of a new variable $s$
defined by 
$${\cal L}[f](s) = F(s) = \int_0^\infty f(t)e^{-st}\ dt$$
The importance of the Laplace transform really comes from  
its properties, especially
$${\cal L}\left[{df\over dt}\right](s) = sF(s) - f(0)$$
This says the Laplace transform ``turns differentiation of $f$ with
respect to $t$ into multiplication by of the transform $F$ by $s$''.
For ODE's, the Laplace transform gets rid of derivatives
entirely -- it turns a linear ODE with constant
coefficients into an {\it algebraic equation} that can be solved
by elementary means.  Then the solution of the ODE can be reconstructed
by the ``inverse Laplace transform''.  This is described in Chapter 7
of Nagle, Saff, and Snider, which would be the starting point for this
project.  For PDE's in two independent variables $x,t$
where one variable is time, the Laplace transform with respect to $t$
takes the PDE to an ODE with respect the the space variable $x$.  This
is discussed briefly in Project B. of Chapter 9 of Nagle, Saff, and
Snider, and in more detail in Haberman's {\it Elementary Applied
Partial Differential Equations}, which is on reserve in the Science
Library for our class.  For this project, you would learn the 
details of this technique, work through proofs of the key properties
of the Laplace transform, and present solutions of several PDE
by these methods.   (Note: The connection with complex analysis
comes because it is sometimes convenient to evaluate Laplace
transform integrals via complex contour integration and the 
calculus of residues.)
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{\it 3.  Green's Functions}
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The solutions to boundary value problems for ODE of the 
Sturm-Liouville form 
$$\cases{{d\over dx}(p(x){dy\over dx}) + q(x)y = -f(x) &\cr
          c_1y(a) + c_2y'(a) = 0&\cr
          c_3y(b) + c_4y'(b) = 0&\cr}$$
(and also to many
PDE can often be written via integral formulas involving 
``Green's functions'' for the appropriate problem).  The
Green's function $G(x,x_0)$ itself is the solution to a problem of this
form where $f$ is a Dirac delta (unit impulse) ``function''
at $x_0$.
Given this one solution, we get all solutions via the 
formula 
$$y = \int_a^b G(x,s)f(x)\ ds$$
Green's functions are a very important theoretical and practical
tool in more advanced PDE.  For this project, you would learn
about the basics of this idea, starting with Green's functions for
ODE boundary value problems.  (See the later part of Chapter 11
in Nagle, Saff, and Snider).  Then you would see how the 
same ideas lead to solutions of PDE such as the heat equation.
Chapter 9 in Haberman's {\it Elementary Applied
Partial Differential Equations}, which is on reserve in the Science
Library for our class, is a good source.
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{\it 4.  Numerical Methods for ODE Boundary Value Problems}
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(This topic will require some extensive Maple calculations
to implement the shooting method and solve big systems of linear equations.)
As mentioned under topic 1 above, the analytic techniques that
we have learned in this course are too limited to be applicable
in all cases of interest coming from problems in heat conduction,  
wave motion, etc.  This is also the case for ODE boundary
value problems.  For this project, you would learn two of the
basic numerical techniques that people use to derive approximate
solutions of  problems of this type -- the ``shooting method''
and the finite difference method.  The finite difference method
is actually very close to the idea described under project
1 above, except that we have only one independent variable. 
But in the same way, we subdivide the interval where the 
solution of the BVP is sought, and discretize by replacing the 
derivatives by finite difference approximations.  This 
leads to a system of algebraic linear equations for the 
approximate solution values at the grid points.  Depending
on the boundary conditions, we may have a unique solution
that can be found by linear algebra techniques.  Or in the Sturm-Liouville
situation, the finite differences can lead to an eigenvalue problem.
Both of these methods are discussed in Projects D and E in Chapter
11 of our text.  They are also covered in first sections of 
Chapter 11 of {\it Numerical Analysis} by Burden and 
Faires, on reserve for our class in the Science Library.  For
this project, you would write up an exposition of both of these
methods, together with several worked examples.
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{\it 5.  Further Steps in the Theory of ODE Boundary Value Problems}
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(This is the most theoretical of the suggested topics.)
In class we talked about the basic points of Sturm-Liouville
theory for ODE boundary value problems.  For this project, you
would take several further steps in the underlying theory.
The topics would come from Section 11.8 in our text (oscillation
and comparison theory for solutions of Sturm-Liouville problems).
The main goal would be to work through and present the proof of the
Picone Comparison Theorem (Theorem 17 on page 747) outlined in 
Project C of Chapter 11.  
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