**ASTR
5540 Mathematical Methods** Fall 2019 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@lcd.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)