Spring 2012 -- Methods of
Applied Mathematics II
M
383D (56070) and CAM 385D (64425)
Instructor:
Prof. Irene M.
Gamba
Office: RLM 10.166, Phone: 471-7150
E-Mail: gamba@math.utexas.edu
Office hours: by appointment
Teaching
Assistant: Chenglong Zhang
Office
hours: Tuesdays from 5:00 to 6:30pm – RLM 12.166
Office: RLM 13.152
E-Mail: chenglong@ices.utexas.edu
Meeting Hours: M – W 3:00-4:30pm RLM 11.176
A weekly
discussion session will be held at RLM 12.166 from 5:00pm to 6:30pm.
Homework Problems
and due dates:
Textbook: Arbogast-Bona notes (2011
version).
Homework,
Exams, and Grades: Homework will be assigned regularly.
Students are encouraged to work in groups; however, each student must write up
his or her own work. Two mid-term exams will be given. The final exam will be
comprehensive and given during finals week (location and schedule TBA). The
final grade will be based on the homework and the three exams.
Course
Description:
This is the second semester of a course on methods of applied
mathematics. It is open to mathematics, science, engineering, and finance
students. It is suitable to prepare graduate students for the Applied
Mathematics Preliminary Exam in mathematics and the Area A Preliminary Exam in
CAM.
Semester I.
1. Preliminaries
(topology and Lebesgue integration)
2. Banach Spaces
3. Hilbert
Spaces
4. Spectral
Theory
5.
Distributions
Semester II.
6. The
Fourier Transform (3 weeks)
o
The Schwartz space and tempered
distributions.
o
The Fourier transform.
o
The Plancherel
Theorem.
o
Convolutions.
o Fundamental
solutions of PDE's.
7. Sobolev spaces (3 weeks)
o
Basic Definitions.
o
Extention Theorems.
o
Imbedding Theorems.
o The
Trace Theorem.
8. Variational Boundary Value Problems (BVP) (3
weeks)
o
Weak solutions to elliptic BVP's.
o
Variational
forms.
o
Lax-Milgram
Theorem.
o
Galerkin
approximations.
o Green's
functions.
9. Differential
Calculus in Banach Spaces and Calculus of Variations
(4 weeks)
o
The Frechet
derivatives.
o
The Chain Rule and Mean Value Theorems.
o
Higher order derivatives and Taylor's
Theorem.
o
Banach's
Contraction Mapping Theorem and Newton's Method.
o
Inverse and Implicit Function Theorems, and
applications to nonlinear functional equations.
o
Extremum
problems, Lagrange multipliers, and problems with constraints.
o
The Euler-Lagrange equation.
o Applications
to classical mechanics and geometry.
10.
Some Applications (if time permits)
Some
references:
1. R.
A. Adams, Sobolev Spaces, Academic
Press, 1975.
2. J.-P.
Aubin, Applied Functional Analysis, Wiley, 1979.
3. C.
Caratheodory, Calculus of Variations and Partial
Differential Equations of the First Order, 1982.
4. E.W.
Cheney and H.A. Koch, Notes on Applied Mathematics, Department of
Mathematics, University of Texas at Austin.
5. L.
Debnath and P. Mikusinski, Introduction
to Hilbert Spaces with Applications, Academic Press, 1990.
6. G.B.
Folland, Introduction to Partial Differential
Equations, Princeton, 1976.
7. I.M.
Gelfand and S.V. Fomin, Calculus
of Variations, Prentice-Hall, 1963; reprinted by Dover Publications.
8. J.
Jost and X. Li-Jost,
Calculus of Variations, Cambridge, 1998,
9. A.N.
Kolmogorov and S.V. Fomin, Introductory
Real Analysis, Dover Publications, 1970
10. E.
Kreyszig, Introductory Functional Analysis with
Applications, Wiley, 1978.
11. E.H.
Lieb and M. Loss, Analysis, AMS, 1997.
12. J.T.
Oden & L.F. Demkowicz, Applied
Functional Analysis, CRC Press, 1996.
13. F.W.J.
Olver, Asymptotics
and Special Functions, Academic Press, 1974.
14. M.
Reed & B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.
15. W.
Rudin, Functional Analysis, McGraw Hill, 1991.
16. W.
Rudin, Real and Complex Analysis, 3rd
Ed., McGraw Hill, 1987.
17. H.
Sagan, Introduction to the Calculus of Variations, Dover, 1969.
18. R.E.
Showalter, Hilbert Space Methods for Partial Differential Equations,
available at World Wide Web address http://ejde.math.txstate.edu//mono-toc.html.
19. E.
Stein and G. Weiss, Introduction to Fourier Analysis
on Euclidean Spaces, Princeton, 1971.
20.
K. Yosida, Functional
Analysis, Springer-Verlag, 1980.
