There will be two exams, one midterm (M) approximately half way through the semester and one final (F) in May. 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 is % = 0.5 (F+M) + 0.5 HW.

The following files are LaTeX files for the Homework Assignments and Exams (after they have been given).

• Homework 1 Linear algebra material covering coordinate transformations, diagonalization, orthogonal projections, Fredholm alternative, least squares problems, pseudo-inverses and singular value decomposition.
• Homework 2 Function approximation with different norms, Sturm-Liouville problems, eigenvalue problems in systems, orthogonal polynomials (Chebychev), solution of Volterra and Fredholm integral equations, Volterra systems.
• List of review topics for the midterm and the midterm : Linear algebra topics(SVD, Fredholm Alternative), approximation in Hilbert spaces, solution of separable Fredholm integral equations and integral equation eigenvalue topics.
• Homework 3 Multidimensional Fredholm integral equation, resolvent operators and eigenvalues of compact integral operators, non-self adjoint compact operator, and solvability of Ku=f.
• Take home Final Exam Delta distributions, distributional differentiation, Green's functions for 2nd order boundary value problems
• Homework 4 Adjoints, domains of adjoints for scalar differential operators and system operators. Effect of changing inner product, Sturm-Liouville operators with separated boundary conditions and inverse operators for single BVP and system's of BVP. (Green's function for Lg=d(x-t) done by Maple: postscript file)
• Homework 5 Conversion of BVP to integral equations, Fredholm alternative, necessary conditions for solvability, series solutions, eigenfunctions and solvability in systems.
• Homework 6 Calculus of variations: Euler-Lagrange equations for ODE's and PDE's, geodesic problem, Lagrange multipliers, existence of minimizers.
• Midterm 2 Differential operators, variational calculus, operator systems, Fredholm alternative.
• Homework 7 PDE variational calculus and variationally based estimates for eigenvalue problems. Lagrangian and Hamiltonian mechanics and canonical transformations.
• Takehome Final An extremely easy "gimme" takehome final on Fourier transforms.

• Chapter 1: pg 1-20,24-30,38-42.
• Chapter 2: pg 59-68 (read only), pg 68-70,73 (orthogonal polynomials), pg 74-76, Sturm-Liouville Supplementary material
• Chapter 3: 100-122, 126-127
• Chapter 4:135-161, 166-172
• Chapter 5:183-198,202-203
• Chapter 6:250-261 assumed
• Chapter 7:285-301,304-305,307

• 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.

EXAM Latex file outlining material from M560-61 to be included in Comprehensive PhD exam in August 1999.