Math 583A -- Principles and Methods of Applied Mathematics

Announcements

• Homework 8 (last revised Sat 12/03/11)

• Homework 7 (last revised Wed 11/30/11)

• Homework 6 (last revised Mon 11/14/11)

• Homework 5 (last revised Wed 10/26/11)

• Homework 4 (last revised Wed 10/12/11)

• Homework 3 (last revised Thu 09/22/11)

• Homework 2 (last revised Thu 09/15/11)

• Homework 1 (last revised Tue 09/20/11)

Suggested References

General

• Mauch, Introduction to Methods of Applied Mathematics. This free e-book covers most of the topics we will touch on in this course, and then some.

Phase plane analysis

• Strogatz, Nonlinear Dynamics and Chaos. Standard text on the subject. Very readable, with lots of examples and applications.

• Devaney, Hirsch, and Smale, Differential Equations, Dynamical Systems, and an Introduction to Chaos. A more rigorous treatment of some of the same topics, and more.

Complex variables

• Ahlfors, Complex Analysis. This is a classic in the subject. I like it as for the basic theory, though there is relatively little emphasis on integration techniques.

• Ablowitz and Fokas, Complex Variables. This is a very nice book that covers a big range of topics. Also has many good examples and problems.

• There are substantive chapters in the Mauch book (see above) that deal with complex variables.

Fourier series and transforms: theory + applications

• Strauss, Partial Differential Equations: An Introduction. The part most relevant to us is Ch 5, which contains a good exposition of basic Fourier theory and applications to PDEs (complete with exercises). Many of the type of problems we're interested in can also be solved by separation of variables; this is explained in Ch 4, which I encourage you to read.

• Haberman, Applied Partial Differential Equations. I don't know this book well, but I'm told it's quite readable and has a large number of exercises and examples.

• The Mauch book cited above has chapters on Fourier theory. However, it does not seem to say much about convergence issues, which are important in certain applications, e.g., signal processing and numerical solutions of PDEs. For these topics, please see our course notes. (I don't know of a good undergraduate-level text on harmonic analysis; if you know of one, please let me know.)

• Adams and Guillemin, Measure Theory and Probability. A careful exposition of measure theory and L^2 (Hilbert space) Fourier theory, covering both Fourier series and transforms. For the theoretically-inclined, this is a good place to go for a more careful explanation of the Hilbert space view of Fourier series and transforms. No use of distributions, though, nor does it touch on pointwise convergence or L^p convergence for p != 2.

Real analysis

• Course notes for 527. Non-applied math students: you may still be able to get a copy from the bookstore, or try to borrow one from someone in our class.

• Basic topics (continuity, uniform convergence, etc): see any text on mathematical analysis on the undergraduate level. Two examples: Rudin, Principles of Mathematical Analysis or Apostol, Mathematical Analysis.

• For more advanced topics (e.g., L^p spaces): see Folland, Real Analysis: Modern Techniques and Their Applications

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