# Fourier Analysis and Differential Equations II

## Math 4/522, CRN 23746/23754

### Winter, 2020

Lecture: MWF, 10:00-10:50, 340 Tykeson Hall
Office Hours: MF 11:00-11:50, W 1:00-1:50 or by appointment.

### Textbook

Asmar: Partial Differential Equations with Fourier Series and Boundary Value Problems, 2nd ed. Prentice Hall 2005 = Dover Books on Mathematics, 2016.

### Course description

This is the secnd term of a two term introductory course in Partial Differential Equations (PDEs). Standard processes in nature such as propagation of heat, diffusion through a porous solid, vibrating strings, membranes, and solids, lead to various differential equations involving one to three spatial variables and one time variable. In the above examples, these equations are known as the "heat equation" and the "wave equation" though there are many versions of these equations depending on the setup of the situation. Typically in applications, the solutions of these equations that are of interest also subject to conditions at the boundary of the domain (say, of the spatial variables) or conditions at the initial point (say, in time variable). An example is the equation of a vibrating string (a version of the wave equation), subject to the constraint that the endpoints of the string remain fixed in place. The goal of this course is to learn how to solve such equations subject to constraints at the boundary of some region. Because so many physical phenomena are governed by equations like this, considerable effort has gone into developing techniques of solution. One essential tool for this purpose is the Fourier series, which has its origin in solving the heat equation.

### Learning Outcome

The students are expect to understand basics on (1) Sturm-Lioville Theory: using Bessel functions and Legendre polynomials as examples. (2)PDE in spherical coordinates: PDE, Dirichlet problems, spherical harmonics. (3) Fourier transform and its applications: Fourier transform, Fourier cosine and sine transforms, applications in PDE. (4) Select topics in PDE: Schordinger's equaitons, Green's functions etc.

### Homework

There will be a homework assignment for each week. You can check ASSIGNMENTS here. Homework will be collected each Wedensday in class on the material of the previous week. Late homework will not be accepted. Your lowest homework score will be dropped. It is very important to keep up with your homework. Start it early, do not wait until the night before you have to turn it in.

### Exams

There will be one midterm in class on Monday, Feb. 10. If you must miss a test due to extraordinary circumstances, you must ask for my permission and schedule a make-up exam in advance. The final exam is scheduled on 10:15 Tuesday, March 17 in 340 Tykeson Hall.