MATH 383D/CAM 385D Methods of Applied Mathematics II.

MATH 383D, Unique #57205 and CAM 385D, Unique #64305
Spring 2010

Instructor:

Todd Arbogast
Office: RLM 11.162, Phone: 471-0166
Office: ACE 5.334, Phone: 475-8628
E-Mail: arbogast@ices.utexas.edu

Textbook:

A bound set of lecturer-prepared notes (2008 corrected version) is available for purchase from the UT Copy Center in Welch 2.228.

Meeting:

MWF 9:00-10:00, PHR 2.114. A weekly problem discussion session will be arranged.

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 (officially scheduled for Wednesday, May 12, 7:00-10:00 p.m.). 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 CSEM.

Semester I.

Preliminaries (topology and Lebesgue integration)
Banach Spaces
Hilbert Spaces
Spectral Theory

Semester II.

Distributions (2 weeks, unless covered in Semester I)
- Seminorms and locally convex spaces.
- Test functions and distributions.
- Calculus with distributions.
The Fourier Transform (3 weeks)
- The Schwartz space and tempered distributions.
- The Fourier transform.
- The Plancherel Theorem.
- Convolutions.
- Fundamental solutions of PDE's.
Sobolev spaces (3 weeks)
- Basic Definitions.
- Extention Theorems.
- Imbedding Theorems.
- The Trace Theorem.
Variational Boundary Value Problems (BVP) (3 weeks)
- Weak solutions to elliptic BVP's.
- Variational forms.
- Lax-Milgram Theorem.
- Galerkin approximations.
- Green's functions.
Differential Calculus in Banach Spaces and Calculus of Variations (4 weeks)
- The Frechet derivatives.
- The Chain Rule and Mean Value Theorems.
- Higher order derivatives and Taylor's Theorem.
- Banach's Contraction Mapping Theorem and Newton's Method.
- Inverse and Implicit Function Theorems, and applications to nonlinear functional equations.
- Extremum problems, Lagrange multipliers, and problems with constraints.
- The Euler-Lagrange equation.
- Applications to classical mechanics and geometry.
Some Applications (if time permits)

Some references:

R. A. Adams, Sobolev Spaces, Academic Press, 1975.
J.-P. Aubin, Applied Functional Analysis, Wiley, 1979.
C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order, 1982.
E.W. Cheney and H.A. Koch, Notes on Applied Mathematics, Department of Mathematics, University of Texas at Austin.
L. Debnath and P. Mikusinski, Introduction to Hilbert Spaces with Applications, Academic Press, 1990.
G.B. Folland, Introduction to Partial Differential Equations, Princeton, 1976.
I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, 1963; reprinted by Dover Publications.
J. Jost and X. Li-Jost, Calculus of Variations, Cambridge, 1998,
A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover Publications, 1970
E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978.
E.H. Lieb and M. Loss, Analysis, AMS, 1997.
J.T. Oden & L.F. Demkowicz, Applied Functional Analysis, CRC Press, 1996.
F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
M. Reed & B. Simon, Methods of Modern Physics, Vol. 1, Functional analysis.
W. Rudin, Functional Analysis, McGraw Hill, 1991.
W. Rudin, Real and Complex Analysis, 3rd Ed., McGraw Hill, 1987.
H. Sagan, Introduction to the Calculus of Variations, Dover, 1969.
R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, available at World Wide Web address http://ejde.math.txstate.edu//mono-toc.html.
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton, 1971.
K. Yosida, Functional Analysis, Springer-Verlag, 1980.