Instructor: Jon Wilkening

Lectures: MWF 10:10-11:00, Room 81 Evans

Office: 1051 Evans

Office Hours: Mon 11:10-11:55, Wed 2:15-3:30

Prerequisites: Undergraduate Analysis (104), Complex Analysis (185), and Linear Algebra (110). (224A is not a prerequisite.)

Required Text: I will post excerpts from the following books and papers on bCourses.

Recommended Reading:
1. G. B. Whitham, Linear and Nonlinear Waves
2. Novikov, Manakov, Pitaevskii, Zakharov, Theory of Solitons, The Inverse Scattering Method
3. G. B. Folland, Partial Differential Equations
4. R. Kress, Linear Integral Equations
5. R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves
6. Dyachenko, Kuznetsov, Spector, Zakharov, Analytic description of the free surface dynamics of an ideal fluid, Phys. Lett. A, 221:73-79, 1996.
7. L. C. Evans Partial Differential Equations
8. D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics
9. Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
10. F. Otto, The Geometry of Dissipative Evolution Equations: the Porous Medium Equation , Comm. Partial Diff. Eq., 26, 101-174, (2001).
11. F. W. J. Olver, Asymptotics and Special Functions
12. C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers
13. Kozlov, Maz'ya, Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities
14. Kevorkian and Cole, Multiple Scale and Singular Perturbation Methods
15. J. Neu, Singular Perturbation Theory (unpublished notes)

Syllabus: The course will survey basic theory and practical methods for solving the fundamental problems of mathematical physics. It is intended for graduate students in applied mathematics, physics, engineering or other mathematical sciences. The overall purpose of the course will be to develop non-numerical tools for understanding and approximating solutions of differential equations. A rough outline for 224B is:

• Linear and Nonlinear Waves (KdV, NLS, the inverse scattering method). (refs 1-2)
• Potential theory, integral equations, compact operators. (refs 3,4)
• Euler equations, potential flow, water waves, conformal mapping methods, stability of traveling waves (refs 5,6)
• Stokes equations, Lam'e equations, weak solutions, Lax-Milgram theorem, Korn's inequality, saddle-point problems, inf-sup conditions. (refs 7,8)
• Semigroup theory, parabolic equations, porous medium equation. (refs 9,10)
• asymptotic behavior of solutions of ordinary differential equations, WKB theory. (refs 11,12)
• asymptotic behavior of integrals, method of steepest descent, method of stationary phase. (ref 12)
• asymptotics behavior of elliptic systems near corners, Mellin transform. (ref 13)
• singular perturbation theory and boundary layer theory. (refs 14-15)

Course Material: I will post handouts, assignments and solutions on bCourses. Please e-mail me if you do not have access to the bCourses page.

Grading: 100% Homework.

Homework: 6-7 assignments, due roughly every two weeks.

Comments: The lowest homework score will be dropped.