**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)