#
M 383D/CSE 386D Methods of Applied Mathematics II.

M 383D, Unique #53240 and CSE 386D, Unique #61045

*Spring 2020*

**Revised in response to the COVID-19 outbreak, and subject to further modification**
### Instructor:

Prof. Todd Arbogast

E-Mail: arbogast@oden.utexas.edu

Office hours: W 9:30-10:30 a.m. via Zoom

Please use Piazza (available in Canvas) to discuss questions regarding the material and homework, so all students may benefit from the discussion. You can also use e-mail, but the rest of the class will not see the message.
### Teaching Assistant:

Mr. Isaac Smith

E-Mail: ismith@math.utexas.edu

Please use e-mail to discuss questions regarding the grading of specific assignments.
### Textbook:

A bound set of lecturer-prepared notes (2019 version) will be made available
for purchase from the UT Copy Center, Union location, UNB
2.214.
### Meeting:

Online via Zoom (available in Canvas), TTH 11:00 a.m. - 12:15 p.m. The
lectures will be recorded and accessible later (either in Canvas or possibly on U-Tube).
### Class Web Sites:

We will use the University's Canvas web site. Please check
that your scores are recorded correctly in Canvas. You can access Canvas from
my.utexas.edu.
### Homework, Exams, and Grades:

Homework will be assigned regularly. Students are encouraged to work in virtual groups; however,
each student must write up his or her own work. Two mid-term exams will be given in approximately
weeks six and eleven (the weeks of February 24 and April 20). These will be scheduled in the late
afternoon or evening (to allow more time for the exam), after a vote of the class. The final exam
will be comprehensive and given Wednesday, May 13, 2:00-5:00 p.m. The later two exams may be
proctored remotely using one of the proctoring tools (such as Proctorio).

Grades will be recorded in the University Canvas system, so students can check their scores
directly. The final grade (not recorded in canvas) will use the plus/minus system and be based on
the homework and the three exams, with somewhat greater emphasis on the final exam.

### 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 II Preliminary Exam in mathematics and the second half of the
Area A-CAM Option Preliminary Exam in CSEM. It is assumed that the first semester was an
introduction to functional analysis, covering the topics listed below. The second semester builds
upon the first semester.
**Semester I.**

- Preliminaries
- Normed Linear Spaces and Banach Spaces
- Hilbert Spaces
- Spectral Theory and Compact Operators
- Distributions

**Semester II.** The following topics will be covered, as time permits.
- The Fourier Transform (3 weeks)
- The L
^{1}(R^{d}) Theory
- The Schwartz Space Theory
- The L
^{2}(R^{d}) Theory
- The S′ Theory
- Some Applications

- Sobolev spaces (4 weeks)
- Definitions and Basic Properties
- Extensions from Ω to R
^{d}
- The Sobolev Embedding Theorem
- Compactness
- The H
^{s} Sobolev Spaces
- A Trace Theorem
- The W
^{s,p}(Ω) Sobolev Spaces

- Boundary Value Problems (3 weeks)
- Second Order Elliptic Partial Differential Equations
- A Variational Problem and Minimization of Energy
- The Closed Range Theorem and Operators Bounded Below
- The Lax-Milgram Theorem
- Application to Second Order Elliptic Equations
- Galerkin Methods
- Green's Functions

- Differential Calculus in Banach Spaces (3 weeks)
- Differentiation
- Fixed Points and Contractive Maps
- Nonlinear Equations
- Higher Derivatives
- Extrema

- The Calculus of Variations (1 week, if time permits)
- The Euler-Lagrange Equations
- Constrained Extrema and Lagrange Multipliers
- Lower Semi-Continuity and Existence of Minima

### The University of Texas at Austin Student Honor Code:

"As a student of The University of Texas at Austin, I shall abide by the core values of the
University and uphold academic integrity."
### The University of Texas at Austin Code of Conduct:

The core values of The University of Texas at Austin are learning, discovery, freedom, leadership,
individual opportunity, and responsibility. Each member of the university is expected to uphold
these values through integrity, honesty, trust, fairness, and respect toward peers and community.
### Students with Disabilities:

The University provides upon request appropriate academic accommodations for qualified students with
disabilities. Contact the Office of the Dean of Students at 471-6259, 471-4641 TTY, and notify your
instructor early in the semester.
### Religious Holidays:

Appropriate academic accommodation for major religious holidays is provided upon request.
### Emergency Classroom Evacuation:

Occupants of University of Texas buildings are required to evacuate when a fire alarm is activated.
Alarm activation or announcement requires exiting and assembling outside. Familiarize yourself with
all exit doors of each classroom and building you may occupy. Remember that the nearest exit door
may not be the one you used when entering the building. Do not re-enter a building unless given
instructions by the Austin Fire Department, the University Police Department, or the Fire Prevention
Services office.
### Counselling and Mental Health Services:

Available at the Counseling and Mental Health Center, Student Services
Building (SSB), 5th floor, M-F 8:00 a.m. to 5:00 p.m., phone 512-471-3515, web site www.cmhc.utexas.edu. Crisis Line is always available 24/7 at 512-471-2255.