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Syllabus of Math 611, Section 600

Introduction to Ordinary and Partial Differential Equations

Fall 2011, Instructor Peter Kuchment

Office Rm. Blocker 614A, Telephone (979)862-3257

E-mail: kuchment@math.tamu.edu, Home Page: /~kuchment

Section: 600, Time: MWF 9:10 am - 10:00 am, Room: BLOC 164
Textbook: L. C. Evans, Partial Differential Equations: 2nd edition , American Math. Society, 2010.
Office hours: MW 10:10-11:00, F 10:10-10:30, Blocker 614A
Additional office hours can be arranged by appointment.

Introduction

The class starts with a brief excursion into basic facts concerning Ordinary Differential Equations and then shifts to the Partial Differential Equations.

Besides numerous applications inside mathematics (e.g., to geometry), the PDEs form the core part of our scientific understanding of the physical world: from physics to chemistry, to biology, 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.).

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.

Assignments

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
8/29 - 9/02	A survey of main theorems on ODEs	HW1, PDF file, due 9/07	n/a
9/05 - 9/23	Chapter 1 and Sections 2.1, 2.2 of Chapter 2	HW2: PDF file , due 10/03. The problem #8 is moved to the extra credit part	n/a
9/26 - 10/14	Chapter 2, Sections 2.3, 2.4	TBA	n/a
10/17 - 11/04	Chapter 3	TBA	n/a
11/07 - 12/02	Chapter 4	TBA	Final exam (take home)

GRADING POLICY
Percentage of points	Grade
90% and higher	A
80% and higher	B
70% and higher	C
60% and higher	D
Less than 60%	F

Ordinary Differential Equations

V. I. Arnold, Ordinary Differential Equations (any edition).
A wonderful book that provides a contemporary geometric view of all the main issues of ODEs. Requires a lot of work, but the reader gets rewarded for it with joy and much deeper knowledge. Probably not suitable as the first ODE book.
V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations, Springer Verlag.
Can be considered as extension of the previous book to cover important topics usually not in the standard ODE class.
P. Hartman and P. Bates, Ordinary Differential Equations (Classics in Applied Mathematics, 38), SIAM 2002 (older editions in '73 and '82 by other publishers).
A classical rather comprehensive book on ODEs.
L. S. Pontryagin, Ordinary Differential Equations, Reading, Massachusetts: Addison - Wesley Publishing.
A short and easy textbook that covers introduction to ODEs with main theorems proven.

Partial Differential Equations

Richard Courant and David Hilbert, Methods of Mathematical Physics, two volume set (any edition).
This is an absolutely wonderful classics. In spite of being 87 years old (starting with its 1st edition), it is still an extremely important book, in many instances not surpassed by anything else. A must reading for anyone using PDEs extensively.
G. Eskin, Lectures on Linear Partial Differential Equations, AMS 2011.
From a review: This book is a reader-friendly, relatively short introduction to the modern theory of linear partial differential equations. An effort has been made to present complete proofs in an accessible and self-contained form. The first three chapters are on elementary distribution theory and Sobolev spaces with many examples and applications to equations with constant coefficients. The following chapters study the Cauchy problem for parabolic and hyperbolic equations, boundary value problems for elliptic equations, heat trace asymptotics, and scattering theory. The book also covers microlocal analysis, including the theory of pseudodifferential and Fourier integral operators, and the propagation of singularities for operators of real principal type. Among the more advanced topics are the global theory of Fourier integral operators and the geometric optics construction in the large, the Atiyah-Singer index theorem in R^n, and the oblique derivative problem.
F. John, Partial Differential Equations, Springer Verlag.
Although outdated and more limited that Evans' book, this is still a very good introduction to the mathematics of PDEs.
I. G. Petrovsky, Lectures on Partial Differential Equations, Dover 1991.
A good small textbook on basics of PDEs. Much more limited and outdated than Evans, but still valuable.
Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005.
A nice reading, probably at a somewhat lover level than in the 611 class.
From a review: "This is an introductory book on the subject of partial differential equations which is suitable for a large variety of basic courses on this topic. In particular, it can be used as a textbook or self-study book for large classes of readers with interests in mathematics, engineering, and related fields. Its usefulness stems from its clarity, balance and conciseness, achieved without compromising the mathematical rigor. One particularly attractive feature is the way in which the authors managed to emphasize the relevance of the theoretical tools in connection with practical applications."
Michael E. Taylor, Partial Differential Equations: Basic Theory (Texts in Applied Mathematics, 23), Springer Verlag, and its two consecutive volumes.
A comprehensive set of books covering all major topics of PDEs from contemporary points of view. Very geometric, in most cases equations are considered on manifolds. This makes it a very good and important book, but probably not for the first serious study of PDEs.

Make-up policy:

Make-ups for missed quizzes, home assignments and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with University Student Rules , students are required to notify an instructor by the end of the next working day after missing an exam or quiz. Otherwise, they forfeit their rights to a make-up.

Grade complaints:

Sometimes the instructor might make a mistake grading your work. If you feel that this has happened, you have one week since the graded work was handed back to you to talk to the instructor. If a mistake is confirmed, the grade will be changed. No complaints after that deadline will be considered.

Scholastic dishonesty:

Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules .

Students with Disabilities:

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Copyright policy:

All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

GOOD LUCK IN YOUR STUDIES!!!