Fourier Series and Boundary Value Problems (MATH 115)

Fourier Series and Boundary Value Problems
Fall 2008, Math 115 (http://www.math.hmc.edu/math115) (Muddshots)
MW 2:45-4:00 pm, Jacobs B132
Prof. Darryl Yong (dyong@hmc.edu), Olin 1265, x72844
Office Hours: W 1:15-2:30pm, Th 4-5:30pm (open door policy)
Grader: Nate Jones

Course description

Objective: By the end of this course, I hope that

when faced with a problem involving a linear partial differential equation, you will be able to select and carry out an appropriate solution strategy;
you will understand why linear differential equations are essentially solved once the eigenvalues and eigenvectors of the appropriate differential operators are known, and to be able to identify those operators;
you will be comfortable using special functions and orthogonal polynomials, such as Bessel functions and Legendre polynomials.

Topics covered (tentative): Complex variables and residue calculus, Laplace transforms, Fourier transforms, Separation of variables, Fourier series, Sturm-Liouville eigenvalue problems, Bessel functions, orthogonal polynomials, the heat equation, wave equation, and Laplace's equation.

Course Materials

Handouts:

Table of Fourier and Laplace transform formulas and pairs

Lecture notes are available to students in this class accessing this web site using a computer on the Claremont Colleges.

Week	Mon	Wed	Fri
1	9/1 (Labor Day)	9/3 Complex Arithmetic, Functions	9/5
2	9/8 Exp, Log, multiple-valued functions; differentiability, analyticity	9/10 Cauchy-Riemann eqs; complex integration cauchyriemann.nb	9/12 Prob Set 1 due (.tex) Solutions
3	9/15 Cauchy's Theorem; Cauchy's Integral Formula; Taylor series	9/17 Isolated singularities; Laurent series	9/19 Prob Set 2 due (.tex) Solutions
4	9/22 Residues; Residue Theorem	9/24 More residue calculus examples integrateassumptions.nb	9/26 Prob Set 3 due (.tex) Solutions
5	9/29 Intro to integral transforms playsound.nb	10/1 Solving DEs with integral transforms de-example1.nb	10/3 Prob Set 4 due (.tex) Solutions
6	10/6 Free space Green's function for 1-D heat equation, convolution.nb	10/8 Free space Green's fcn using Laplace Application to Black-Scholes PDE	10/10 Prob Set 5 due (.tex) Solutions
7	10/13 Half space Dirichlet problem for Laplace's eq	10/15 Half space Neumann problem for heat eq Method of images	10/17 Prob Set 6 due
8	10/20 (Fall Break)	10/22 Derivation of PDE with reaction, diffusion and convection	10/24
9	10/27 Deriv of heat eq 1st separation of variables example	10/29 More separation of variables Self-adjoint eigenvalue problems	10/31 Prob Set 7 due
10	11/3 Fourier series	11/5 Inhomogeneous problems Eigenfunction expansions	11/7 Prob Set 8 due
11	11/10 Inhomog heat eq Wave equation deriv & sep	11/12 Laplace and Poisson eq in a disc	11/14 Prob Set 9 due
12	11/17 Vibrations of a circular membrane	11/19	11/21 (Thanksgiving)
13	11/24 Vibrating modes of a hanging chain	11/26 Sturm-Liouville eigenvalue problems Heat eq w/ Robin BC	11/28 Prob Set 10 due
14	12/1 Laplace eq in spherical coords	12/3 Shape of the earth	12/5 Prob Set 11 due
15	12/8 Energy states of H atom	12/10 Vibrational modes of a beam	12/12 Prob Set 12 due

There is no required textbook for this course, but you may find some of the books in the list below helpful. I personally own these and many more textbooks and am happy to recommend some to you based on your preferences.

Applied Complex Analysis with Partial Differential Equations by Nakhle H. Asmar, and Gregory C. Jones (Prentice Hall, ISBN: 0130892394)
Fundamentals of Complex Analysis with Applications to Engineering and Science, by E. B. Saff and A. D. Snider (Prentice Hall, 3rd ed ISBN: 0139078746)
Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, by Richard Haberman (Pearson Prentice Hall, 4th ed ISBN: 0130652431)
Boundary Value Problems, by David Powers (Academic Press, 4th ed ISBN: 0125637349, 5th ed ISBN: 0125637381)
Partial Differential Equations for Scientists and Engineers, by Stanley J. Farlow (Dover, ISBN: 048667620X)
Integral Transforms and Their Applications, by Brian Davies (Springer, 3rd ed ISBN: 0387953140)

Grading

Exam 1 Take home due on Monday, Oct 27	25%
Exam 2 Take home due on Friday, Dec 19	25%
Homework	50%

Homework

Most of your learning will occur while completing your homework assignments, so take them seriously, and complete them thoroughly. Your homework should show clearly your solution processes. I encourage you to describe your solution process in words. Poor presentation may result in loss of credit.

No late homeworks will be accepted except for family or medical emergencies. Your lowest homework grade will be dropped.

You are encouraged to work cooperatively on your homework assignments with your classmates. However, every student MUST write up his/her own homework separately. In addition, you must cite any sources of help that you use. If you work with one of your classmates on a problem, be sure to acknowledge that person in your homework write-up; if you use any textbooks or websites, acknowledge that too. Harvey Mudd's honor code is in effect for all students in this course.

I encourage you to use a computer algebra system (CAS) such as Mathematica on your homework, when it is appropriate. The Claremont Consortium has a site-wide license for Mathematica. To install it on your personal computer, look in the Dist-Software directory on charlie.hmc.edu. Try this Google search to look for Mathematica tutorials.

<dyong@hmc.edu>

Last modified: Mon Sep 22 16:42:30 Pacific Daylight Time 2008