This web page http://www-math.cudenver.edu/~jmandel/classes/57330f01
serves as the class syllabus and source of further information. Check it
regularly during the semester for further information.
Important notes will be put here during the semester - please check
frequently
Office hours: CU Bldg 646, Monday 3:00-3:45 Wednesday 2:00-3:45
Prerequisites: Ordinary Differential Equations (Math 3200), Advanced Calculus (MA 4320) *or* Graduate standing. Generally what students have the most difficulty with is recalling material from third semester calculus (Math 2422/2423). Some review is presented in the course.
Textbook: Introduction to Partial Differenctial Equations with Applications by E. C. Zachmanoglou and D. W. Thoe, 1986, Dover: This book is a good book from the mathematical point of view - it is a good overview of the overall mathematical framework, i.e. looking at a pde, what can one say about the solution? Does it exist? Is it unique? What properties does the solution have? However it is not strong on linking the equations with physics. Partial Differential Equations of Mathematical Physics and Integral Equations by R. B. Guenther and J. W. Lee, 1988, Dover: This book motivates by physical problems, but consequently it is more difficult to see the mathematical coherence.
Class format: The class will follow quite closely the selected sections of the textbooks. This is both senior undergraduate and graduate class. Graduate students will be expected to understand and to be able to reproduce the proofs with the help of the text and solve (with a hint) problems that require the use of a part of a proof from the text. All students should be able to solve problems from the covered chapters of the texts.
Objective: Gain working knowledge of the basics of classical PDE theory (i.e. using the classical definition of function and derivalive, known from Calculus), and selected topics from modern (variational) theory. Understand the differences and links between classical and modern theory.
Tentative list of topics:
Grading: There will be two in-class written tests on October 17
and December 10, and homework
every week or two. Homeworks will be given Wednesday and due the following
Wednesday. All work will be graded on the scale 4=perfect, 3=a minor problem,
2=good progress, 1=some progress, 0=no progress or did not understand
the assignment. The grade will be computed by scaling the total number
of points from test 1, test 2, and all homeworks between 0 and 100 (100
is the best in the class) and averaging the best two of the three.
Letter gradesA >= 90, A- >= 85, B+ >= 80, B >= 75, B- >=70,
C+ >= 65, C>= 60, C- >= 55, D+ >= 50, D >= 45, D- >= 40.
How to study: Question and distrust all material in the book
and in the class, discover things for yourself, do not stop until you understand.
Study in a group. Bring your questions to the class and to your study group.
Do not just `go over the material,' that is a waste of time. Solve as many
problems from the text as possible, the homeworks can cover only a small
part of them.
Review for Test 1 The problems will be picked from the
following problem types:
- solve a given linear equation of first order by the method of characteristics
- solve a given nonlinear equation of first order by the method of characteristics
- compute the Fourier expansion, Fourier sine expansion, or Fourier transform
of a given function (smooth, discontinous, or involving the delta function)
- apply Fourier expansion or Fourier transform to a differential equation
and derive a formula for the solution
- verify by M-test that for a given function it Fourier transform can be differentiated and write the integral for the derivative
- write the integral of |f'|^2 from -pi to pi.
in terms of Fourier coefficients of f
- solve the wave equation for all x and t>0 given u and u_x for all x and t=0
- solve the wave equation by separation of variables,
given u and u_x for 0<=x<=L and t=0
- what can you say about the behavior of the solution of the wave equation
as t -> infinity? Is it bounded, can it go to zero? Why?
- given differential equation p(x,D)=0, find its characteristic
surfaces
Review for Test 2 Most problems and questions will be picked from the
following types:
- show that an integral operator with degenerate kernel satisfies Fredholm alternative
- show how an integral operator with continuous kernel is approximate by operators with
degenerate kernels
- what is the difference in the behavior of the wave equation and the heat equation as
t -> infinity?
- let f, g be continuous on [0,1], kernel k continuous on [0,1]^2, K the integral operator
with kernel k, (.,.) the L^2 inner product, show that (f, (I- lambda*K)^{-1} g) is an analytic
function of lambda at lambda=0 (need to show that the power series has positive radius of
convergence)
- solve the wave equation u_tt=(c u')' u(0,t)=u(1,t)=1, u(x,0)=f(x) formally using
expansion in eigenfunctions of the Sturm-Liouville problem