Course Description:

Introduction to basic concepts of partial differential equations through concrete examples such as Laplace, heat, and wave equations, and first order linear and nonlinear equations.
The emphasis is on derivation of "explicit" solution formulas and understanding the basic properties of the solution. This course is different from a standard course of PDEs for upper level undergraduate students,
which uses mainly separation of variables and Fourier series. This course prepares graduate students in the Department of Mathematics for a written qualifying exam.

Instructor:

Changyou Wang

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 714

Phone Number: 4-2719

Email: wang2482@purdue.edu

Lecture Time and Place:

TR 12:00 - 1:15pm, MATH 215

Office Hours:

TR 1:45-2:45pm, or by appointment

Textbook:

(All of the following are on reserve in math library.)

Main Text:
[E] Partial Differential Equations, by Lawrence C. Evans, second edition

Reference:
[J] Partial Differential Equations, by Fritz John.

Prerequisites:

Good "working" knowledge of vector calculus, linear algebra, and mathematical analysis. A prior course of ordinary differential equations is useful.
(In Purdue, these materials are taught in MA 265, 266, 351, 353, 303, 304, 366, 510, 511, 440+442 and 504.)

Homework:

Homeworks will be assigned roughly bi-weekly. They will be gradually assigned as the course progresses. Please refer to the course announcement below.

Steps must be shown to explain your answers. No credit will be given for just writing down the answers, even if it is correct.

Please staple all loose sheets of your homework to prevent 5% penalty.

Please resolve any error in the grading (homework problems and exams) WINTHIN ONE WEEK after the return of each homework and exam.

No late homeworks are accepted (in principle).

You are encouraged to discuss the homework problems with your classmates but all your handed-in homeworks must be your own work.

Examinations:

Midterm Exam: Thursday, March 23, 2017, 8:00-10:00 pm, MATH 175

Final Exam: TBA

Grading Policy:

Homeworks (35%)

Midterm Exam (25%)

Final Exam (40%)

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade,
plus any other disciplinary action, deemed appropriate.

Course Outline:

The course will cover most of [E] Chapter 2 (transport, Laplace, heat and wave equations) and selected sections of Chapter 3 (nonlinear first order equation)
and Chapter 4 ("miscellaneous" concepts and methods of solutions).

Course Progress and Announcement:

(You should consult this section regularly, for homework assignments, additional materials and announcements.)

Jan 10 (Tuesday):
[E, Ch. 1] Introduction to notations;
[E, Appendix C.2] Divergence (Gauss-Green) Theorem, higher dimensional integration by parts.

Jan 12 (Thursday):
Derivation of minimal surface equations.
[E, Sec 2.1] first order linear partial differential equation with constant coefficients.

Jan 17 (Tuesday):
[E, Sec 7.2.5] classification of second order equations.
[E, 2.2.1] Radially symmetric and fundamental solutions of Laplace equation, Poisson's equation

Jan 19 (Thursday): [E, 2.2.2] Mean-value formulas.

Jan 24 (Tuesday): [E, 2.2.3 a, b] Properties of harmonic functions

Homework 1, due in class or before 4pm Tuesday, Jan 24.
(Slide your homework under the door in case I am not there.)
Solution to Homework 1

Jan 26 (Thursday): [E, 2.2.3 c] Local estimates on harmonic functions

Jan 31 (Tuesday): [E, 2.2.3 c] Local estimates on harmonic functions (Analyticity, Harnack's inequality)

Feb 2 (Thursday): [E, 2.2.4] Green's function

Feb 7 (Tuesday): [E, 2.2.4] Green's functions and Poisson's formula for the half plane

Feb 9 (Thursday): E, 2.2.4] Green's functions and Poisson's formula for balls

Homework 2, due in class or before 4pm Tuesday, Feb 14.
(Slide your homework under the door in case I am not there.)
Solution to Homework 2

Feb 14 (Tuesday): [E, 2.2.5] Energy method

Homework 3, due in class or before 4pm Thursday, Feb. 23.
(Slide your homework under the door in case I am not there.)
Solution to Homework 3

Feb 16 (Thursday): [E, 2.3.1] Heat equation/fundamental solutions

Feb 21 (Tuesday): [E, 2.3.1 c] Non-homogeneous equation/Duhamel's formula

Feb 23 (Thursday): Maximum principle and uniqueness

Homework 4 , due in class or before 4pm Thursday, March 9.
(Slide your homework under the door in case I am not there.)

Feb 28 (Tuesday): [E, 2.3.3] Mean value formula

March 2 (Thursday): [E, 2.3.3] Mean value formula (continued)

March 7 (Tuesday): [E, 2.3.3] Properties of solutions

March 9 (Thursday): [E, 2.3.3] Properties of solutions (continued)

March 13 - March 17 (Spring break, no classes)

March 21 (Tuesday): [E, 2.3.4] More on energy methods

March 23 (Thursday) Review for Midterm Exam.