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


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.


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.


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.

  1. Preliminaries
  2. Normed Linear Spaces and Banach Spaces
  3. Hilbert Spaces
  4. Spectral Theory and Compact Operators
  5. Distributions
Semester II. The following topics will be covered, as time permits.
  1. The Fourier Transform (3 weeks)
    1. The L1(Rd) Theory
    2. The Schwartz Space Theory
    3. The L2(Rd) Theory
    4. The S′ Theory
    5. Some Applications
  2. Sobolev spaces (4 weeks)
    1. Definitions and Basic Properties
    2. Extensions from Ω to Rd
    3. The Sobolev Embedding Theorem
    4. Compactness
    5. The Hs Sobolev Spaces
    6. A Trace Theorem
    7. The Ws,p(Ω) Sobolev Spaces
  3. Boundary Value Problems (3 weeks)
    1. Second Order Elliptic Partial Differential Equations
    2. A Variational Problem and Minimization of Energy
    3. The Closed Range Theorem and Operators Bounded Below
    4. The Lax-Milgram Theorem
    5. Application to Second Order Elliptic Equations
    6. Galerkin Methods
    7. Green's Functions
  4. Differential Calculus in Banach Spaces (3 weeks)
    1. Differentiation
    2. Fixed Points and Contractive Maps
    3. Nonlinear Equations
    4. Higher Derivatives
    5. Extrema
  5. The Calculus of Variations (1 week, if time permits)
    1. The Euler-Lagrange Equations
    2. Constrained Extrema and Lagrange Multipliers
    3. 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.