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

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

*Spring 2020*
### Instructor:

Prof. Todd Arbogast

Office: RLM 11.162,
Phone: 512-471-0166

Office: POB 5.334,
Phone: 512-475-8628

E-Mail: arbogast@oden.utexas.edu

Office hours: W 9:10-10:30 a.m.
Also, unless the instructor has pressing business, he will be
available to help students who find him in one of his offices.
### Teaching Assistant:

Mr. Isaac Smith

Office: RLM 13.152

E-Mail: ismith@math.utexas.edu

Office hours: TBD (to discuss homework that has *already* been graded)
### 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:

TTH 11:00-12:30 p.m., RLM 11.176.

### 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 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 6). 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. 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 (2 weeks, 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.
### Counseling 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.