**ASTR 5540
Mathematical Methods** Fall 2017 Toomre

(http://zeus.colorado.edu/astr5540-toomre )

MWF
10:00-10:50am Duane E126

__Juri__** Toomre, ** office: JILA A606: Lab Comput
Dynamics (LCD): Duane G-328;

phones:
303-492-7854, or 303-907-9316 (cell); jtoomre@solarz.colorado.edu

office
hours: MW 11am-noon, and readily by appointment.

This course is intended to
help refine your perspectives about a variety of mathematical methods essential
to many areas of research in astrophysical and planetary sciences. Central to
these broad disciplines is understanding the
properties of differential equations, for these are the building blocks for
most models of the underlying physics and dynamics. We turn to combinations of
analytical and numerical methods for seeking solutions to both ordinary and
partial differential equations.

Part of the material involves brief reviews, followed
by discussion of modern methods, including the use of numerical experiments.
Topics to be covered encompass ordinary differential equations, complex
functions, integral transform techniques, partial differential equations,
special functions and asymptotic methods, and the richness of dynamical systems
that admit chaos. The lectures are supplemented by problem sets, some of which
require use of numerical solutions and experimenting, typically using
workstations or laptops and IDL (Interactive Data Language), Mathematica or Matlab
as appropriate.

__Course textbooks__:

RILEY, KF, HOBSON, MP & BENCE SJ, *Mathematical Methods for Physics and
Engineering,* 2006, Third Edition, Cambridge, ISBN 978-0-521-67971-8.

(optional) RILEY, KF & Hobson, MP, *Student Solution Manual for Mathematical
Methods for Physics and Engineering*, 2006, Third Edition, Cambridge, ISBN
978-0-521-67973-2.

(optional)
PRESS, WH, Teukolsky, SA, Vetterling, WT & Flannery,
BP, *Numerical Recipes: The Art of
Scientific Computing*, 2007, Third Edition, Cambridge, ISBN
978-0-521-88068-8.

__Useful
reference books:__

Acton,
*Numerical Methods That (Usually) Work*,
1970

Arfken & Weber, *Mathematical Methods for Physicists, Sixth
Edition, *2005

Bender
& Orszag, *Advanced
Mathematical Methods for Scientists and Engineers*, 1978

Carrier
& Pearson, *Partial Differential
Equations*, 1991

Mathews
& Walker, *Mathematical Methods of
Physics*, 1970

Strang, *Introduction to Applied Mathematics*,
1986

__General topics
to be discussed (ordering may be adjusted/tuned):__

- Ordinary
differential equations:
- Review basic methods for seeking
solutions in closed form
- Greens functions and
superposition
- Harmonic oscillator applications
- Systems of ODEs,
eigenvectors and eigenvalues
- Numerical solutions of ODEs

- Introduction to computational
approaches
- Initial value problems: explicit,
implicit methods; multi-step, compound methods
- Boundary value problems:
shooting, relaxation
- Dynamical systems with chaos
- Asymptotic methods

- Regular perturbation theory
- Singular perturbations
- WKB approximations and turning
points
- Complex functions

- Review of analyticity and
analytic continuation
- Integration in complex plane,
integral and residue theorems
- Conformal transformations
- Integral transform methods

- Fourier series and integrals
- Fourier and Laplace
transforms
- Applications, signal/noise
analysis
- Partial differential equations

- Classification and boundary
conditions; well-posedness
- Characteristics
- Separation of variables and
variety of tractable examples
- Special functions

- Legendre polynomials and spherical
harmonics
- Bessel functions
- Sturm-Liouville theory, self-adjoint differential equations
- Applications to various PDEs
- Numerical solutions of PDEs

- Introduction to issues
- Parabolic equations: FTCS, von Neuman stability, BTCS,
Crank-Nicolson
- Elliptic equations:
finite-difference 5 point formula
- Solvers: direct (Gauss elim), relaxation (Jacobi, Gauss-Seidel, SOR, multigrid,
ADI), rapid solvers (spectral)