Each semester, there will be two exams, one midterm (M) approximately half way through the semester and one final (F). Periodically Homework assignments (HW) will be given. Each Exam will count for 25% of the total grade. The homework will count for 50%. Thus the final grade for each semester is % = 0.5 (F+M) + 0.5 HW.
 

• CHAPTER 1: Finite dimensional Vector spaces (Spectral theory, Fredholm alternative, Least squares and Singular value decomposition)
• CHAPTER   2: Function Spaces (Hilbert spaces, Fourier series, Orthogonal polynomials, Discrete Fourier Transform)
• CHAPTER 3: Integral Equations (Bounded and Compact operators, Spectral theory and resolvents)
• CHAPTER 4: Differential Operators (Distributions, Green's Functions, Adjoints, Fredholm Alternative, Least squares solutions, Eigenfunction expansions, Sturm Louville Theory)
• CHAPTER 5:  Calculus of Variations (Euler Equations, Hamiltonian and Lagrangian Mechanics, Eigenvalue problems)
• CHAPTER 6: Presumed
• CHAPTER 7:  Transform and Spectral Theory (Spectrum of an operator, Fourier and Laplace Transforms, Scattering Theory)
• CHAPTER 10: Asymptotic Expansions (Laplace's method, Steepest descent, Stationary phase)
• CHAPTER 11-12: Perturbation Techniques (Time permitting, selected topics)

• Sturm-Liouville Theory A postscript file summarizing a few issues in Sturm-Liouville theory and associated boundary value problems.
• Compact sets and operators A LaTeX file listing a few definitions and theorems about compact sets, compact operators and operator norms.
• Second Order Boundary Value Problems A LaTeX file (in developement) with proofs concerning the existence of adjoint domains, Green's functions, convertibility to Fredholm integral equations etc.
• Spectral representations of Operators A LaTeX file for two theorems on the spectral representation of bounded and unbounded operators.

GramSchmidt.mws For performing Gram-Schmidt orthogonalization on a finite set of vectors with a user defined inner product. (Maple V Release 4 or higher) Can use for Homework 1.

2nd order Linear to SL code Some code to convert non-self adjoint operators to self adjoint Sturm Liouville operators.

Regular SL eigenvalue problem Some code to solve regular Sturm-Liouville eigenvalue problems with separated boundary conditions.

Operators defined using Fourier Series Some code illustrating how to use Maple to compute T(f) defined on L2(0,1) using a Fourier series. The specific example of T(f) given is a bounded operator whose range is not closed (hence for which the Fredholm alternative does not apply).

Separable Fredholm Integral Equations Separable integral equations are equivalent to matrix equations with appropriate definitions. The code allows for the definition of a separable kernel, creates the equivalent matrix system and then solves the integral equation (when solutions are unique). Eigenvalues of the integral operator are also computed.

Greens Functions for 2nd order BVP To find Greens functions g(x,t) which satisfy Lg=delta(x-t) and prescribed boundary conditions at x=a,x=b, g(x,t) must solve the homogenous equation Lg=0 for x less than t,x greater than t, be continuous at x=t and a jump condition at x=t. This code finds g(x,t) for user defined (2nd order) operators L and boundary conditions.

Eigenvalues of Sturm-Liouville Problems Some code to estimate bounds on eigenvalues of Sturm-Liouville problems with mixed BC.