Course Announcement   MATH 655 Partial Differential Equations I   Fall 2016

Instructor: P. Sacks, 436 Carver  Office Hours: MF 2-3,  TR 11-12

Class meeting time/place: TR 9:30-10:45,  031 Ross Hall

Textbook: Partial Differential Equations, 2nd ed. by L. C. Evans

Prerequisites: MATH 515 or MATH 519 or consent of the instructor

Course description: Introduction to the theory of partial differential equations at the advanced graduate level. The principal concerns are existence, uniqueness and regularity of solutions, methods for representation of solutions and interesting qualitative properties.

Course Outline:

I.                Study of model linear problems

a.     Transport equation

b.     Laplace equation

c.     Heat equation

d.     Wave equation

II.              Nonlinear first order equations

a.     Method of characteristics

b.     Scalar conservation laws

III.            Special methods

a.     Separation of variables

b.     ODE methods – plane wave, travelling wave  and similarity solutions

c.     Integral transform methods

d.     Useful changes of variable

IV.            General theory of linear PDEs of arbitrary order

a.     Fundamental solutions

b.     Local solvability – Malgrange-Ehrenpreis and Cauchy-Kowalewski theorems, Lewy example

c.     Hypoellipticity

d.     Wave front sets

Items I-III correspond roughly to Chapters 1-4 in the text. MATH 656, the Spring 2017 continuation of the course, will cover material from Chapters 5-9, specifically Sobolev spaces, general theory of linear elliptic, parabolic and hyperbolic PDEs of second order, and introduction to techniques for the study of nonlinear PDEs.

The course grade will be based on 6-7 homework assignments. Click here for homework assignments.

Other References:

If a student has a disability that qualifies under the Americans with Disabilities Act and Section 504 of the Rehabilitation Act and requires accommodations, he/she should contact the Disability Resources (DR) office for information on appropriate policies and procedures. DR is located on the main floor of the Student Services Building, Room 1076; their phone is 515-294-7220.