# MATH 314 Introduction to Fourier Methods and Partial Differential Equations

### Spring 2017

 INSTRUCTOR: John Stockie Office: K 10518 Phone: 778-782-3553 E-mail: stockie  [at]  math.sfu.ca Course page: http://www.math.sfu.ca/~stockie/teaching/math314/ CLASS TIMES: MWF 11:30-12:20, AQ 2104 TUTORIAL: Wednesday 14:30-15:30, AQ 5016 or AQ 3148.1 (D101) TA: Hans Oeri (email: hoeri  [at]  sfu.ca) MY OFFICE HOURS: Wednesdays 12:30-14:30 PREREQUISITES: MATH 310; and one of MATH 251 (with a grade of B+) or MATH 252 or MATH 254. CALENDAR DESCRIPTION: Fourier series, ODE boundary and eigenvalue problems. Separation of variables for the diffusion wave and Laplace/Poisson equations. Polar and spherical co-ordinate systems. Symbolic and numerical computing, and graphics for PDEs. Quantitative. DETAILED DESCRIPTION: Much of what we perceive in the world around us consists of variations in physical effects like heat, sound & light, where the variations occur over both space and time. Partial differential equations (PDEs) are the mathematical language for describing this sensory landscape in terms of continuous functions. This course contains the core of the traditional boundary value problems curriculum, and will also introduce computational and graphical tools that can be used to analyse PDEs and their solutions. Two central ideas in the theory of linear PDEs are the Fourier series and Fourier transform. These ideas will be developed through the Fourier solutions of a trio of fundamental PDEs: the potential, heat and wave equations. The generalization of these problems to higher dimensions will naturally lead to the study of several "special functions", such as the Bessel function and spherical harmonics which play a central in mathematical physics. The numerical implementation of the Fourier series -- known as the fast Fourier transform (FFT) -- will also be studied since it is one of the most important numerical algorithms in scientific computing. Computational examples will illustrate PDE solutions and underlying concepts, and the numerical algorithms are largely based upon the linear algebra of matrices and vectors. Students are expected to perform the numerical computing and graphics themselves by modifying Matlab scripts that are provided by the instructor. TEXTBOOK: Partial Differential Equations: Analytical and Numerical Methods, 2nd edition, by M. S. Gockenbach (SIAM, 2011). HOMEWORK: Homework assignments will be due approximately every two weeks. Most homework problems will be assigned from the textbook, although additional problems will come from other sources. Some homework problems will involve computing with Matlab and to assist you with these computational problems, every second tutorial session will be held in an SFU computing lab. I encourage you to work together with your classmates on your assignment solutions; however, your written homework submission must be your own work. Plagiarism on homework assignments is a serious academic offense, and I will treat it as such.
 ROUGH OUTLINE:   (numbers in parentheses refer to chapters/sections from the text) Introduction and review -- 1 week Classification of DEs (Ch. 1) Linear algebra, bases and projections (Ch. 3) Existence-uniqueness and the Fredholm alternative Analogy with linear differential operators (5.1) Linear differential operators and Fourier series -- 1.5 weeks Function spaces as vector spaces Symmetry, bases, eigenvalues and eigenfunctions (5.1-5.2) Solving BVPs using Fourier series (5.3) Heat flow and diffusion (Ch. 6) -- 3.5 weeks Convergence theory for Fourier series (12.4-12.6) -- 1 week Wave propagation (Ch. 7) -- 2 weeks Discrete Fourier Transform and FFT (12.2) -- 1 week Multi-dimensional problems (Ch. 11) -- 2 weeks

MARKING SCHEME:
 Assignments (bi-weekly): 25% Midterm Test (Feb. 22, in class): 20% Final examination (Apr. 10): 55%