Fourier Analysis and Differential Equations II
Math 4/522, CRN 23746/23754
Lecture: MWF, 10:00-10:50, 340 Tykeson Hall
Instructor: Yuan Xu, Office:
Deady 101, Telephone: 346-5619,
Office Hours: MF 11:00-11:50, W 1:00-1:50 or by appointment.
Asmar: Partial Differential Equations with Fourier Series and Boundary Value Problems,
2nd ed. Prentice Hall 2005 = Dover Books on Mathematics, 2016.
This is the secnd term of a two term introductory course in Partial
Differential Equations (PDEs). Standard processes in nature such as propagation of heat,
diffusion through a porous solid, vibrating strings, membranes, and solids, lead
to various differential equations involving one to three spatial variables and
one time variable. In the above examples, these equations are known as the
"heat equation" and the "wave equation" though there are many versions of
these equations depending on the setup of the situation. Typically in applications,
the solutions of these equations that are of interest also subject to conditions
at the boundary of the domain (say, of the spatial variables) or
conditions at the initial point (say, in time variable). An example is the equation
of a vibrating string (a version of the wave equation), subject to the
constraint that the endpoints of the string remain fixed in place. The goal
of this course is to learn how to solve such equations subject to constraints
at the boundary of some region. Because so many physical phenomena are governed
by equations like this, considerable effort has gone into developing techniques
of solution. One essential tool for this purpose is the Fourier series, which
has its origin in solving the heat equation.
The students are expect to understand basics on (1) Sturm-Lioville Theory:
using Bessel functions and Legendre polynomials as examples. (2)PDE in spherical
coordinates: PDE, Dirichlet problems, spherical harmonics. (3) Fourier transform
and its applications: Fourier transform, Fourier cosine and sine transforms,
applications in PDE. (4) Select topics in PDE: Schordinger's equaitons, Green's functions
There will be a homework assignment for each week. You can check
ASSIGNMENTS here. Homework will be collected each Wedensday in class
on the material of the previous week. Late homework will not be accepted.
Your lowest homework score will be dropped. It is very important to keep up
with your homework. Start it early, do not wait until the night before
you have to turn it in.
There will be one midterm in class on Monday, Feb. 10. If you must miss a
test due to extraordinary circumstances, you must ask for my permission and
schedule a make-up exam in advance.
The final exam is scheduled on 10:15 Tuesday, March 17 in 340 Tykeson Hall.
Your course grade will be based on your homework, midterm and final.
|Final Exam: ||40%|
Incomplete are only awarded in the Mathematics Department when two criteria
have been satisfied: first, a student must have a passing grade at the time
the I is assigned; secondly, some work could not be completed due to
extenuating circumstances (illness, auto accident, etc.). Under no
circumstances will an I be awarded as a substitute for a
W, D or
F/N . If you find yourself in trouble, drop the course!
If you are a student with a documented disability please meet
with me soon to discuss your needs. If you have not already
requested a notification letter from Disability Services outlining
recommended accommodations, please do so soon.
The University Student Conduct Code (available at
http://dos.uoregon.edu/conduct) defines academic misconduct. Students are
prohibited from committing or attempting to commit any act that constitutes
academic misconduct. By way of example, students should not give or receive
(or attempt to give or receive) unauthorized help on assignments or examinations
without express permission from the instructor. Students should properly
acknowledge and document all sources of information (e.g. quotations,
paraphrases, ideas) and use only the sources and resources authorized by
the instructor. If there is any question about whether an act constitutes
academic misconduct, it is the students' obligation to clarify the question
with the instructor before committing or attempting to commit the act.