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\centerline{Mathematics 173--Applied Mathematics I}
\centerline{Course Information--Fall 1999}
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\leftline{{\bf Professor:}  John Little}
\leftline{{\bf Office:} Swords 335}
\leftline{{\bf Office Phone:}  793-2274}
\leftline{{\bf Office Hours:}  MWF 10 - 12, TR 1 - 3, and by appointment}
\leftline{{\bf Course Homepage:} {\tt http://math.holycross.edu/\~{}little/AM199.html}}
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{\bf Course Description}
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The general term ``applied mathematics''
refers to the portions of the subject where mathematics is 
used to study phenomena of interest in the real world and to predict
and/or control their behavior.  Applied mathematics
deals with the construction of
{\it mathematical models} 
for real-world systems under consideration,
analysis of the behavior of the models through mathematical techniques
from calculus, linear algebra, and so forth, and comparison
of the behavior of the models and the behavior of the real-world 
systems.  Mathematics 173 is the first semester of a full-year course on 
the branch of applied mathematics dealing with problems in
{\it heat conduction, vibration of strings and membranes, and 
related physical phenomena} and mathematical models for these
systems involving {\it partial differential equations} or ``PDE'' --
relations between the partial derivatives of an unknown function
$u$ and other, known functions.  

For instance, one of the key examples that
we will study in great detail is the problem of describing the 
conduction of heat through a thin, straight rod of material of known 
thermal properties.  We will show that the temperature $u(x,t)$ at
location $x$ within the rod and time $t$ satisfies the {\it heat
equation:}
$$c \rho {\partial u \over \partial t} = 
{\partial \over \partial x}\left(K_0 {\partial u \over \partial x}\right) 
+ Q(x,t),\leqno(1)$$
a relation between the partial derivatives of the unknown function
$u = u(x,t)$ and a known function 
$Q(x,t)$ which describes heat sources within the rod ($Q = 0$
if there are no sources).  If we know the initial temperature
distribution $u(x,0) = f(x)$, and the ends of the rod are kept
at constant temperature $0$ (for instance), on physical
grounds we expect the function $u(x,t)$ to be determined for all $t$.
The {\it mathematical model} is then the heat equation (1), the
initial condition $u(x,0) = f(x)$, and the boundary conditions
$u(0,t) = 0$, $u(L,t) = 0$.  We will learn several different
techniques for {\it solving} partial differential equations like (1), 
that is for finding $u(x,t)$ which satisfy the equation and any 
side conditions present, 
either by producing analytical formulas for the functions $u(x,t)$
when this is possible, or when necessary, by deriving numerical 
approximations to their values at selected points. 

Since (partial) derivatives are involved here, it should be clear
that many topics from calculus, through Analysis 3 (Multivariable Calculus) 
will be needed here.  (Indeed, we will need to go several steps farther
and introduce additional topics from vector calculus in order
to deal with two- and three-dimensional physical problems.)  
We will see that several topics from linear algebra also make
appearances as we study the structure of solutions of various PDE.

The topics we will study this semester are as follows:
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\item{} Unit I:  Introduction to PDE -- The Heat Equation (Chapter 1 in 
text) -- about 6 class days, including two days on Green's Theorem,
surface integrals, and the Divergence Theorem from vector calculus.
\item{} Unit II:  Separation of Variables and Fourier Series
(Chapters 2 and 3 in text) -- about
12 class days, including one day on second order ordinary differential 
equations with constant coefficients.
\item{} Unit III:  The Wave Equation (Chapter 4 in text) -- about 3 class
days
\item{} Unit IV:  Sturm-Liouville Boundary Value Problems (Chapter 5, 
sections 1 - 5) -- about 4 class days.
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The remaining day will be devoted to an in-class midterm exam.

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{\bf Text}
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The text for the course is {\it Elementary Applied Partial Differential
Equations} by Richard Haberman.  We will cover most of the material
in Chapters 1 - 5 this semester and continue on with Chapters 6 - 11
next semester.
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{\bf Course Format}
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To get you more directly involved in the subject matter of this course,
about 9 times during the semester, the class will break down into groups
of 3 or 4 students for one or more days, and each group will work together
for a portion of those class periods on a group discussion exercise.
The exercises will be made up by me.  I will be present and available for 
questions and other help during these periods.  At the conclusion of some
of these discussions, at times the class as a whole may
reconvene to talk about what has been done, to sum up the results, 
to hear short oral reports from each group, etc.  Each group will be 
responsible for a write-up of solutions for the questions from each discussion 
day, and those will be graded and and returned with comments.
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Some of the other meetings of the class will be structured as 
lectures or computer laboratory days when that seems appropriate.
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{\bf Computer Work}
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We will be using Maple on the departmental Sun workstation network
quite extensively throughout the course
to implement the techniques we discuss and to generate 
solutions to problems.  Several class meetings will take place in 
the SW 219 computer lab and some of the individual problem sets will
include problems for which you will need to use Maple.  If you have
not used Maple previously in a course here, please see me as soon as
possible.  
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{\bf Grading}
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The assignments for the course will consist of:
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\item{1.} Two midterm exams (one in class, one take-home)  
together worth $40\%$ of the course grade.  Tentative dates:
\itemitem{a.}  Thursday, October 7.
\itemitem{b.}  Take-home out Thursday, November 11, due November 18.
\item{2.} Final exam worth $30\%$ of the course grade.  The final exam for 
this course will be held at 8:30 a.m. Friday, Dec. 17.
\item{3.} Individual problem sets and lab reports, worth $15\%$ of the course grade.
\item{4.} Group reports from discussion
days, worth $15\%$ of the course grade.  
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If you ever have a question about the grading policy, or about your 
standing in the course, please feel free to consult with me.
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