# Syllabus of Math 611, Section 600 Blocker 624, TR 12:45 - 2:00 pm.

Office Blocker 614A,
E-mail: kuchment@math.tamu.edu

### 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, 2010Office hour: T, W 10 am -- 11 am, Blocker 614AAdditional 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
One week A survey of main theorems on ODEs HW1, due 9/12/2017 n/a
4 weeks Chapter 1 and Sections 2.1, 2.2 of Chapter 2 HW2: TBA (to be announced), due ... n/a
3 weeks Chapter 2, Sections 2.3, 2.4 TBA n/a
1.5 weeks Chapter 3 TBA n/a
4-5 weeks Chapter 4 TBA Final exam (take home), to be completed by December 13th
90% and higher A
80% and higher B
70% and higher C
60% and higher D
Less than 60% F

1. Write solutions neatly and not in microscopic letters. Avoid many crossing outs, etc. This is not only for me to read easier, but for you to be sure that what you write is correct. I will not grade solutions that are messily written and unreadable due to a tiny letter size.
2. Avoid words ``trivial ..., easy to see ..., obvious ...'' and other their synonyms. ``Easy to see'' or ``trivial'' should mean that you can show this in a second, so just do it :-). By the way, errors are usually appearing in ``obvious'' places.
3. Be careful with parentheses and other such standard rules; they can cause mistakes if not followed properly.
4. Make sure you understand your own logic completely. If you don't, I won't. :-)

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:

• 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 about 90 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.
• 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.
• 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.
• Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005.
A very 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. Read it after 611-612!
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