Syllabus for MATH 611, SECTION 600, Fall 2019

Introduction to Ordinary and Partial Differential Equations

Office hours: TWR 1:30-2:30. Also, by appointment.

Class times and place: MWF 11:30-12:20, Blocker 163.

Section web page: www.math.tamu.edu/~phoward/M611.html

Textbook: Partial Differential Equations, 2nd Edition, by Lawrence C. Evans, Graduate Studies in Mathematics vol. 19 (2010), American Mathematics Society.

Prerequisites: The only listed prerequisite is a year of advanced calculus (often called real analysis or real variables), listed as M409-410 in the Texas A&M course catalog. 

Catalog Description: Basic theory of ordinary differential equations; existence and uniqueness, dependence on parameters, phase portraits, vector fields. Partial differential equations of first order, method of characteristics. Basic linear partial differential equations: Laplace equation, heat (diffusion) equation, wave equation and transport equation. Solution techniques and qualitative properties.

Homework Assignments: A homework assignment will be made both in class and on the course web site each Friday, and will be due the following Friday. Homework assignments will typically consist of four problems, worth ten points each. Work will be accepted up to a week late, though five points will be deducted for each class period by which the assignment is late. 

Exams: There will be two exams, a midterm and a final. The midterm exam will be in the evening, 7:00-9:00 p.m., Wednesday Oct. 23. The final exam for this class will be on Wednesday, Dec. 11, 10:30 a.m.-12:30 p.m. Please make a note of these dates.

Grades: Final grades will be determined in the following manner: Homework assignments: 50%; Exams: 25% each. Grade ranges will be graduate standard: 89.50-100, A; 79.50-89.49, B; 69.50-79.49, C, 59.50-69.49, D; below 59.50, F.

Learning outcomes: Students will be able to: find exact solutions to linear ODE systems with constant coefficients; identify conditions under which nonlinear ODE systems have unique solutions that depend continuously on parameters and initial data; identify and classify PDE as linear, semilinear, quasilinear, or nonlinear; identify linear PDE as elliptic, hyperbolic, or parabolic; find exact solutions to special cases of Laplace's equation and understand qualitative properties of solutions to Laplace's equation as characterized by the Maximum Principle, Liouville's Theorem, and Harnack's inequality; find exact solutions to special cases of the heat equation and understand qualitative properties of solutions to the heat equation as characterized by the parabolic mean value formula and Maximum Principle; find exact solutions to special cases of the wave equation and understand qualitative properties of solutions to the wave equation; solve first order quasilinear and nonlinear equations with the method of characteristics; find exact solutions to special cases of the Hamilton-Jacobi equation via the Hopf-Lax formula.

Make-up policy: Make-ups for exams will only be given if the student can provide a documented University-approved excuse (see University Regulations). According to University Student Rules students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise the student forfeits his or her right to a make-up.

Scholastic Dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. "An Aggie does not lie, cheat, or steal or tolerate those who do." Please refer to the Honor Council Rules and Procedures, available at the Office of the Aggie Honor System.

Copyright policy: All printed materials disseminated in class or on the web are protected by copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Students with Disabilities: The following statement was provided by the Department of Disability Services: The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, in Cain Hall, Room B118, or call 845-1637. For additional information visit http://disability.tamu.edu.

Class Schedule: Roughly speaking, we should cover the following material on the following schedule:

Week of Monday	Material Covered
August  26	Linear ODE systems with constant coefficients.
September 2	Metric spaces; contraction maps; the Picard-Lindelhöf Theorem; continuation; continuous dependence. (Mon. Sept. 2 is last day for drop/add).
September 9	Elements of real analysis; Lebesgue measure and integration; limit theorems; Lp spaces; approximation.
September 16	Mollifiers.
September 23	PDE notation and classification; linear transport equations.
September 30	Laplace's Equation: physicality; fundamental solutions; mean value principle.
October 7	Laplace's equation: maximum principle; uniqueness; regularity; Liouville's Theorem; Harnack's Inequality.
October 14	Laplace's equation: Green's functions.
October 21	The heat equation: physicality; fundamental solutions. (Midterm Wed. Oct. 23.)
October 28	The heat equation: parabolic mean value formula; maximum principle.
November 4	The wave equation: physicality; d'Alembert's formula; spherical means; Hadamard's method of descent.
November 11	Method of characteristics: examples; existence theory. (Friday Nov. 15 is last day for Q-drop.)
November 18	Method of characteristics: special cases.  (Monday, Nov. 18 is Bonfire 1999 Remembrance Day.)
November 25	Hamilton-Jacobi equations: physicality; relation to Hamiltonian mechanics. (Thanksgiving break is Nov. 27-29.)
December 2	Hamilton-Jacobi equations: the Hopf-Lax formula. (Class meets Monday and Wednesday; Wed., Dec. 4 is the last day of Fall 2019 classes.)