MA523: Introduction to Partial Differential Equations
SPRING 2017, Purdue University
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: 42719
 Email: wang2482@purdue.edu
Lecture Time and Place:
 TR 12:00  1:15pm, MATH 215
Office Hours:
 TR 1:452:45pm, 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 biweekly.
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 handedin homeworks must be your
own work.
Examinations:
 Midterm Exam: Thursday, March 23, 2017, 8:0010:00 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 (GaussGreen) 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] Meanvalue 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] Nonhomogeneous 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.