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Syllabus of Math 611, Section 600
Blocker 148, TR 9:35 am - 10:50 am.

Office Blocker 614A, Telephone (979)862-3257
Home Page: /~kuchment

Section: 600, Time: TR 9:350 am - 10:50 am, Room: BLOC 148
Textbook: L. C. Evans, Partial Differential Equations: 2nd edition
(1 st edition is also usable), American Math. Society, 2010
Office hours: TR 2-3pm, Blocker 614A
Additional office hours can be arranged by appointment

The class starts with a brief (about a week long) excursion into basic facts concerning Ordinary Differential Equations. It will not dwell on solving special types of equations, which you all have seen in an undergraduate ODE class. I will rather concentrate on several major theorems, which are usually not treated well in the undergraduate classes: solutions' existence, uniqueness, and dependence on parameters. We'll also see that all these theorems are consequences of a single geometric one, on straightening a vector field. We'll also discuss how to tell whether an applied problem in hand can be modeled by a (system of) ODE. Surprisingly, one can get an answer to this question.

We will shift then to the Partial Differential Equations, probably one of the most active, diverse, and beautiful areas of pure and applied mathematics. Besides numerous applications inside mathematics (e.g., to geometry, several complex variables), the PDEs form the core part of our scientific understanding of the physical world: from physics to chemistry, to biology, to medical diagnostics, to meteorology, you name it.

The class (except the short initial ODE part) will be based on the well respected textbook by L. Evans. It is planned to cover Part I "Representation formulas for solutions" of the book. This includes a study of the four major PDEs: transport, Laplace, heat, and wave equations, as well as analysis of 1st order non-linear PDEs and other methods of representing solutions (Fourier transform, separation of variables, asymptotics, etc.). Knowing these equations well is a prerequisite for deeper understanding of more general PDEs.

This study will be continued in the next class Math 612 that will most probably cover the Part II of the book "Theory of linear PDEs" (including more general initial value and boundary value elliptic, hyperbolic, and parabolic problems, spectral theory, etc.).

Prerequisite: MATH 410 or equivalent or instructor's approval.

Grading will be based on home assignments and a take-home final exam.

Tentative schedule of the course (watch for updates)
Week Topics and sections Home assignments and recommended exercises Exams
8/27 - 9/03 A survey of main theorems on ODEs HW1, TBA (to be announced), due 9/12 n/a
9/05 - 9/24 Chapter 1 and Sections 2.1, 2.2 of Chapter 2 HW2: TBA (to be announced), due 09/30 n/a
9/26 - 10/15 Chapter 2, Sections 2.3, 2.4 TBA n/a
10/15 - 11/05 Chapter 3 TBA n/a
11/07 - 12/03 Chapter 4 TBA Final exam (take home), to be completed by December 6th
Percentage of points Grade
90% and higher A
80% and higher B
70% and higher C
60% and higher D
Less than 60% F

Recommended reading: All of the books below are written by great experts in differential equations, all are written well and provide interesting and rewarding reading. This list is certainly far from being comprehensive, it just contains some of the instructor's favorites. These books approach the subject from different perspectives, and so reading (or at least browsing through) all of them is a good idea for someone who wants to learn the ODEs and PDEs. Do not try to do this in one semester, though :-).

Ordinary Differential Equations:

Partial Differential Equations

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Last revised July 22nd, 2013