Spring 2018 Methods of Applied
Mathematics II
Unique# M
383D (54285) and CSE 386D (63650) –
Meeting Hours: M - W 3:30-4:45pm, RLM 12.166
Instructor: Prof.
Irene M. Gamba
Office: RLM 10.166, Phone: 471-7150
E-Mail: gamba@math.utexas.edu
Office hours: by appointment
Teaching Assistant: Matt Rosenzweig
E-Mail: rosenzweig.matthew@math.utexas.edu
Discussion Hours: TBA
Class webpage : S18-Methods for
Applied Mathematics II Unique# M 383D (54285) and
CSE 386D (63650)
Guidance textbook: Arbogast-Bona book or notes
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.
Three mid-term exams will be given in class. The first one on Wednesday February 28th, the second one on Wednesday
April 4rd, and the last one
on Wednesday May 2st.
There will not be a final exam. The final grade will
be based on the homework and the three exams.
Homework problems:
Problem Set 1. Due Wednesday January 31, 2018.
Ch. 6 # 1, 2, 3, 4, 5, 7.
Problem Set 2. Due Wednesday Feb. 14th.
Ch. 6 # 8, 9, 11, 12, 17.
Problem Set 3. Due Wednesday Feb 21st.
Ch. 6 # 13, 21, 24, 26, 29, 30.
Problem Set 4. Due Monday March 26
Ch. 7# 1, 2, 3, 4, 5, 7 a) & b);
Problem Set 5. Due Monday April 2 New Due date: Wednesday April 4
Ch. 7 # 8, 9, 11, 13
Problem Set 6. Due Mon April 9 à New date deadline : Monday April 16
Ch. 7 # 14, 15, 16;
Problem Set 7. Due Wednesday April 25
Ch. 8 # 1, 2, 3, 5, 8, 9, 12,
Problem Set 7. Due Wednesday May 2nd
Ch.8 # 16, 23, 24, 25;
_______________________________________________
Problem Set 9.
Not due
Ch.8 # 27, 28.
Ch.9 # 1, 3, 5, 7, 8 9, 13, 14,
16, 17, 18.
____________________________________________________
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 I & II Preliminary Exam in mathematics and the
Area A Preliminary Exam in the SCEM graduate program.
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.