MATH-6500 Partial Differential Equations
Instructor: Victor Roytburd (Amos Eaton 405, Ph.
276-6889, roytbv at rpi dot edu)
Office hours: Mo
2-3, Th 3-4 (or by appointment)
Course web-page: www.rpi.edu/~roytbv/pde/pde.html (it is linked to the
math department COURSE MATERIALS page).
Course meets: Carnegie 106
(Tu) (note room change) and Carnegie 205 (Fr).
Course Description
The course is a first graduate course in partial
differential equations. It is devoted to the basic theory of PDEs.
Tentative Course Outline
- PDEs of
the first order (GL: 2-1, 2-2, 2-4; J: 1.4-1.7).
- 2nd order linear PDEs, characteristics
(GL: 2-6; J: 2.1-2.3).
- Primer
of Fourier series and integrals (GL: Chap. 3).
- The
wave equation in one dimension: D’Alembert’s solution (GL: 4-1);
Separation of variables (GL: 4-2, 4-3); Hyperbolic systems (GL: 4-4, 4-5,
4-9; J: 2.5-2.6).
- Primer
of distributions: definitions; δ-function; Green's functions for
ode's; weak solutions; fundamental solutions (J: 3.6; GL: 10-5).
- The
heat equation on a finite interval (GL: 5-1, 5-3; J: 7.1). Maximum
principle (GL: 5.2; J: 7.1). Ill-posedness for the backward heat equation.
- The
heat equation on the infinite interval: the fundamental solution; energy;
uniqueness classes (GL: 5-4; J: 7.1).
- The
linear operator approach to evolutionary problems: integral operators (GL:
7-3—7-7) [Green’s functions, eigenfunction expansions – maybe later].
- Laplace’s
equation: local regularity, Green’s functions, the maximum
principle.
- Elliptic
potential theory.
- Hilbert
space methods and weak solutions of boundary value problems for Laplace’s
and other elliptic equations.
- Applications.
Textbooks:
R. B.
Guenther and J. W. Lee, Partial Differential Equations of Mathematical
Physics.
F. John,
Partial Differential Equations, 4th edition.
Fritz John’s book is a set of lecture
notes. It is extremely brief, but contains deep insights into many important
issues. Guenther and Lee’s book is a traditional textbook. It is very well
written and discusses in detail some questions just touched upon in John’s
book.
Books on reserve in the library:
Courant
and Hilbert, Methods of Mathematical Physics, vol. II.
L. C.
Evans, Partial Differential Equations
M. J.
Lighthill, Introduction to Fourier Analysis and Generalized Functions.
Grading policy (tentative):
Homework 50% of course grade
Midterm(s) and a final 50% of course grade
Homework
Homework
will play a very important role in this course. Many problems are chosen to
supplement the in-class material, not to duplicate it. Homework assignments are
selected from the list below.
Homework assignment #1, due September 9: problems 1.1-1.4, 2.1.
Warning: Problem 2.1 is erroneous as stated (thanks to Zhang Ning,
who noticed the mistake). The curve should be given as
. Correspondingly the characteristic is given by 
, and
the tangential derivative, by 
. Sorry
for the confusion.
Homework assignment #2, due September 16: problems 2.2, 3.1-3.4.
The seminar will be in Amos Eaton 402, Wednesday 1-2 pm.
Homework assignment #3, due September 25: problems 4.2, 4.4, 4.7.
Homework assignment #4, due October 7: problems 5.2-5.3, 5.5; 6.1, 6.3
Homework assignment #5, due November 4: problems 6.4-6.6, 7.1-7.3
There will be no lectures on October 28 and 31
Homework assignment #6, due November 14: problems 8.1-8.4 (from 4th
installment)
Note:
To do Problem 8.1(a) you need to know more about the Fourier transform than
just the definition. Namely, that Fourier transform of a product is a
convolution of the Fourier transforms. If you don’t understand the last
statement you can skip Problem 8.1(a).
There will be no lecture on Tuesday, November 18
There will be makeup lectures on Wednesdays November 12, 4-6 and on
December 3, 4-6 both in Sage 2112
Homework assignment #7, due Friday, December 5: problems 8.5, 9.?-? (from
4th installment)
There will be a makeup lecture on Wednesday December 3, 4-6 in Sage
2112.
The take-home final exam will be handed out on Friday. It will be
due on or before the scheduled day of the final.
List of homework problems
List of homework problems (2nd installment)
List of homework problems (3rd installment)
List of homework problems
(4th installment)
Instructor's
Web Page: Roytburd,
Victor
Email: roytbv
