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
• Some useful orthogonality formulas
• Solving inhomogeneous ODEs
• Some useful facts about Bessel functions
• Solution of Bessel's equation via power series
• Solution of Legendre' equation via power series

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

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

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.