## Recently taught courses at ASU

- APM 524 - Spectral Methods (graduate level), S10, S12, S14
- MAT 272 - Calculus with Analytic Geometry III, S12, F12, F13, F14, S15
- MAT 598 - Topics in Reverse Engineering of Complex Dynamical Networks (graduate level), S12
- MAT 275 - Modern Differential Equations, F11
- APM 505 - Applied/Computational Linear Algebra (graduate level), F14

## APM 524: Spectral Methods

### Course description

Spectral methods are known for their fast convergence. When the function being approximated is smooth, approximations converge exponentially fast. These methods will be covered with emphasis on the solution of differential equations. Approximation theory of global methods will be covered, including Fourier, polynomial, and radial basis function methods. Students will learn theoretical results and practical implementation of algorithms. Depending on the research interests of students, applications in the areas of imaging, mathematical biology, fluid mechanics will be considered.

The course will benefit students interested in numerical methods for approximation of functions and solution of differential equations, including students in engineering. Students should have a previous course in differential equations and basic programming skills, as course assessment will require computer projects.

The figure above is from a simulation of the filamentation of an elliptic vortex in an inviscid fluid. The solution was computed with a Fourier spectral method. Approximations were quickly obtained with the aid of fast Fourier transforms.

### Textbook

The main reference for this course is: L.N. Trefethen. Spectral Methods in MATLAB. SIAM, Philadelphia (2000).

Additional material will be taken from papers and other books. Other refernces include:

- Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb. Spectral Methods for Time Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics (No. 21) Cambridge University Press (2006).
- J. Boyd. Chebyshev and Fourier Spectral Methods, 2d. edition, Dover Publishers (2001).
- G.E. Fasshauer. Meshfree Approximation Methods With MATLAB, World Scientific Publishing Co., Inc. River Edge, NJ (2007).
- R. Peyret. Spectral Methods for Incompressible Viscous Flow. Springer (2002).

### Assessment

Course assessment will be based on homework and a final project. These will often require implementation of algorithms and computations.