This is the first course in the two-course qualifying sequence for partial differential equations. For graduate students in the mathematics department, this course is intended to be the part of the preparation for the qualifying exam in PDE; see the graduate student handbook for more information concerning the qualifying exams.

This course provides a first introduction to the theoretical study of PDEs. After an opening discussion on general questions (*What are PDEs? Where do they come from? Why do we study them? What are the questions we ask?*),
the course focuses on demonstrating many fundamental concepts and techniques through examination of four classical types of equations.

- Through studying
*first order equations*, we discuss the method of characteristics and the idea of the Cauchy problem. - Through
*Laplace's equation*(and to some extent*Poisson's equation*), we focus on the concept of the fundamental solution and the Green's function, as well as techniques frequently associated with the general study of "elliptic" PDEs such as the maximum principle and Harnack's inequalities. - Through the
*heat equation*we introduce evolutionary PDEs and the Duhamel principle. General properties of "parabolic" PDEs, especially those related to the maximum principle, we also be discussed. - Through the
*wave equation*we talk about the energy method fundamental to the study of "hyperbolic" PDEs, by way of which we discuss the finite-speed-of-propagation property.

Along the way, as time permits, some additional topics may be discussed.

Professor Willie W. Wong

*Office*: D-303 Wells Hall

*E-mail*: wongwwy@math

*Office hours*: Wednesdays 9:30a - 11:10a or by appointment

The course will not specifically follow any textbook per se, but the vast majority of the material covered in the course (and 100% of that which will appear on the qualifying exam) can be found in L.C.Evans' *Partial Differential Equations*, 2nd edition.

- Supplemental notes on energy method for wave equations.
- Supplemental notes on plane waves and radiation fields.

**Six** sets of homeworks will be assigned, collected, and graded. Your grader is Seonghyeon Jeong (jeongs10@math).
You will have at least one week after the assignment to complete the homework problems; the due dates will be announced on the assignments.
*No late homework will be accepted* under any circumstances.

- HW1 is due Monday Sept. 18
- HW2 is due Friday Sept. 29
- HW3 is due Friday Oct. 13
- HW4 is due Monday Oct. 30
- HW5 is due Wednesday Nov. 15
- HW6 is due Wednesday Nov. 29 (Q10 on this set is extra credit)

There will be **two in-class exams** for the course.

- Exam 1 will take place on
*Wednesday Oct 18*and will cover material concerning Laplace's equation, Poisson's equation, and the heat equation (both homogeneous and inhomogeneous). - Exam 2 will take place on
*Friday Dec 8*and will cover material concerning wave equation and first order partial differential equations.

Solutions to exam 1 is now available.

Solutions to exam 2 is now available.

Each exam will account for 20% of your final grade; each homework assignment will make up 10%.

Collaboration is *encouraged* for the homework assignments, provided:

- Each student turns in his or her individual write-up.
*On a per question basis*each student should list all other students with whom he or she has collaborated.

Collaboration is *not allowed* on the in-class exams. The general MSU rules concerning academic integrity applies.