The wonder of the Lord's handiwork is all around us, to paraphrase Psalms 19:1. In Genesis 1:28 it is recorded that God commanded we who were created in His image should “fill the earth and subdue it”. One way that we can do this is to find appropriate mathematical models to express natural phenomena and use these models to predict, even affect, future behavior. Frequently scientists are best able to express models as differential equations — that is, in terms of one or more unknown quantities and their rates of change. Each of you has seen some of the most important ordinary differential equation (ODE) models. In this course we shall study partial differential equations (PDEs), and a number of techniques used to solve them. We will see that while, in general, these types of equations require more sophisticated solution methods, many interesting and practical problems require such models.
The course we will draw and expand upon such topics from earlier courses as infinite series, vector spaces, Laplace transforms, etc. Along with spending time on these (and new ones as well) theoretical constructs, a fair amount of our time will be devoted to the practical consideration of solving PDEs numerically. The course should provide a solid introduction to the vast, expanding subject of PDEs.
Our text is Introduction to Partial Differential Equations: A Computational Approach, by Tveito and Winther. Topics to be covered include modeling with PDEs, nondimensionalization, asymptotic analysis, solving 1st-order linear PDEs via the method of characteristics, pure initial value problems, 2nd-order initial boundary value problems, and numerical solutions using finite differences and finite elements. For those in this list not addressed in our text, handouts or other supplementary material will be provided when possible/appropriate.
I may be reached by phone at x66856, but a better way
to reach me is by email.
If you require my approval for something, do
not consider having left a message for me
as equivalent to having obtained that approval.
In addition, a number of assigned problems will require
the use of software. We will not necessarily stick to any
single software package for these problems, though whenever
appropriate, I will encourage/require the use of Octave, a GNU-license
(free!) package which is available for all major operating systems
and has the feel of Matlab. In general, whether the choice
of software is left open or not on a specific assignment, you will
be expected to hand in both nicely-formatted output (graphs, tables,
etc.) and the code (which must be well documented) used to generate it.
Your write-ups for all assignments must be in your own hand (or typed)
and words, except when otherwise directed.
Give as much attention to presenting your solutions in a coherent
manner (using mathematical symbols as part of your sentence
structure) as you give to actually solving problems, as it is the
explanation of each problem that is graded (not simply the answer
itself). While submission of work that incorporates another's writeup
will be considered academic dishonesty
(See
Section 4.2.8 of the Faculty Handbook), you may borrow someone's
idea for solving a problem, so long as significant steps which
were borrowed from that person (website, book, etc.) are attributed to
that person (website, book, etc.).
Projects (the term here is not meant to be equated with exercises in
the "project" sections at the end of chapters in your text) are
lengthier than problems on homework assignments, and constitute a
different category than homework exercises. A list of projects will
be maintained along with degree of difficulty and due dates. You are
not required to do every project—only enough that the degrees of
difficulty sum to 3 (or greater).
Please speak with me about problems or issues as they arise during
the semester. I am still growing as a teacher, and if you have
concerns, it is simply a matter of “building one another up”
that you should raise them in an appropriate moment, preferably
while adjustments may still be made that affect your class.
This page maintained by: Course objectives
Contacting the professor
My office is NH 281. The hours
I am intentionally in my office for student
questions are posted on my homepage,
and are subject to change during the semester. If we
cannot hook up at one of these times, feel free to
talk with me about an appointed time to meet, or swing
by my office in the hopes that I am available to help.
If you feel yourself falling behind in the class, it
is very important not to put things off, but to seek
help right away. Do not wait until a time close to an
exam before speaking with me.
Use of technology
Most course information including this syllabus,
homework assignments, a
calendar showing due dates,
test dates, etc. (updated as often as a couple of times each week),
and handouts are accessible via a web browser.
(Student's of mine using Microsoft's Internet Explorer have, in
the past, encountered problems with some of my course pages not
coming up in their browser when, correspondingly, they came up just
fine in Mozilla.) While
many announcements, hints, etc. may be given in class, things
that cannot wait until the next class period will be
sent to you as email messages. Thus, it is important that
you be checking your email frequently.
Your written work
Evaluations
Your grade will be determined by your performance on homework,
projects, in-class exams, and the cumulative final exam. Homework
will be collected, in general, once per week. Assignments themselves
may be viewed from the
homework page. Problems are placed on an assignment throughout
the week prior to when it is due, and the list should not be
considered complete until the end of that week (Saturday).
As these problem sets represent a significant part of your overall
grade (likely somewhere in the 20-25% range), you are advised to
give attention to content, neatness and organization.
Electronically-produced write-ups are most welcome, and TeXShop
(one of several LaTeX distributions
for typesetting mathematics) is installed on machines in the Mathematics
Department computer laboratory NH 067; the document
"The Not So Short Introduction to LaTeX2e" (subtitled "LaTeX2e in
133 minutes") does a very nice job of presenting generally (i.e., not
specific to any particular distribution) how to use LaTeX. I consider
an assignment late if I receive it after I have graded those
which were handed to me in class on the due date.
Accomodations
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Citizenship
The type of concentration required for mathematics/statistics calls
for a distraction-free environment. Please do your part to make
the classroom one conducive for learning by arriving on time, not
working on homework assignments during class, refraining from frivolous
talk, and actively participating in in-class discussions/activities.
Thomas L. Scofield
Department of Mathematics and Statistics
Calvin College