Instructor: Jon Wilkening

Lectures: MWF 11:10-12:00, Room 3109 Etcheverry Hall

Office: 1051 Evans

Office Hours: Wed 2:30-3:30, Thurs 3:00-4:00

Prerequisites: Undergraduate Analysis (104), Complex Analysis (185), and Linear Algebra (110)

Required Texts:
Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
Lloyd Trefethen, Approximation Theory and Approximation Practice

Recommended Reading:
Gabor Szego, Orthogonal Polynomials
Gerald Folland, Real Analysis
Reed and Simon, Functional Analysis, Vol I
Courant and Hilbert, Methods of Mathematical Physics, vol 1
Coddington and Levinson, Theory of Ordinary Differential Equations
Ivar Stackgold, Green's Functions and Boundary Value Problems
Novikov, Manakov, Pitaevskii, Zakharov, Theory of Solitons, The Inverse Scattering Method

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, focusing on spectral theory.

• Hilbert Spaces, Orthogonal Polynomials, Applications (Multipole Expansions, Toda Lattice, Diffusion)
• Approximation Theory, Fast Fourier Transform, Aliasing, Chebychev Polynomials, Fourier Transform
• Linear Operators on a Hilbert Space, Adjoint Operators, Spectrum and Resolvent, Polar Decomposition
• Spectral Theory for Compact Operators, Green's functions, Fredholm Alternative, Sturm-Liouville theory
• Self-Adjoint and Unitary Operators, Continuous Spectrum, Singular Sturm-Liouville Problems
• Dispersion, Group Velocity, Stationary Phase, KdV and the Inverse Scattering Transform

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: 7 assignments (lowest score dropped)