Spring 2022 – Graduate Course ‘Methods of Applied Mathematics II’

 

General Information:

This class will be done online from January 18th to, at least,  January 31st.  Lectures expected to be resumed in the assigned classroom starting on February 2nd.

 

Meeting Hours & Lecture Room: T - TH 12:30-1:45pm,  RLM 10.176

 

Instructor: Prof. Irene M. Gamba     -   Office: RLM 10.166, Phone: 471-7150

E-Mail: gamba@math.utexas.edu      -  Office hours: TBA and  by appointment

 

Discussion Hours: TBA

 

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 on the following tentative dates: The first one on Thursday February 17th, the second one on planned for Thursday March 31st and the last one on Thursday May 6th. There will not be a final exam. The final grade will be based on the homework and the three exams.

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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)
• The Schwartz space and tempered distributions.
• The Fourier transform.
• The Plancherel Theorem.
• Fundamental solutions of PDE's.
7. Sobolev spaces (3 weeks)
• Basic Definitions.
• Extention Theorems.
• Imbedding Theorems.
• The Trace Theorem.
8. Variational Boundary Value Problems (BVP) (3 weeks)
• Weak solutions to elliptic BVP's.
• Variational forms.
• Lax-Milgram Theorem.
• Galerkin approximations.
• Green's functions.
9. 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.
10. Some Applications (if time permits)

Some additional references:

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

 

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities. For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.