**Math 540 (Partial Differential Equations). Spring 2022 **

**Instructor:** Steve Krone

**Office:** 421 Brink Hall.

**Office Hours:** by appt.

**Class Time:** MWF 12:30 - 1:20

**Place:** Admin 221

**Prerequisite:** Math 539

**Text:** R. McOwen, *Partial Differential Equations* (1996), Prentice Hall. (Note: 2nd edition is also OK.)

This is a graduate course on partial differential equations. We will treat some of the standard topics, such as existence, uniqueness, and regularity of strong and weak solutions, representations of solutions, and techniques such as scaling, Fourier transforms, energy methods, and Green functions. Although we will treat the classical linear PDE's of mathematical physics, a significant part of the course will involve reaction-diffusion equations. These are nonlinear PDE's that find applications in many areas and are currently one of the hottest topics in PDE. Solutions of reaction-diffusion equations often display interesting spatio-temporal behavior such as traveling waves and spontaneous pattern formation.

Grades will be based on homework, provided everyone remains sufficiently motivated. If I feel you are in need of a "stimulus package," it will come in the form of a final exam.

**Preliminary Course Outline:** (subject to some change in content and order)

- Introduction to the main ideas and types of equations
- Nonlinear Reaction-Diffusion Equations
- traveling wave solutions
- pattern formation, Turing instabilities
- applications

- Wave Equation and 2nd Order Hyperbolic PDE's
- d'Alembert's formula
- Poisson's formula

- Laplace's Equation and 2nd Order Elliptic PDE's
- fundamental solutions
- harmonic functions
- maximum principle
- Green functions
- energy methods

- Heat Equation and 2nd Order Parabolic PDE's
- fundamental solutions
- initial value problem
- Duhamel's principle
- boundary value problems
- maximum principle

- Nonlinear First-Order PDE's
- the method of characteristics
- weak solutions
- scalar conservation laws from gas dynamics, Burgers' equation
- shock waves, entropy condition, Rankine-Hugoniot condition

**Additional References:**

- J.D. Murray, Mathematical Biology, Springer.
- L.C. Evans, Partial Differential Equations, American Math. Soc. (1998).
- F. John, Partial Differential Equations (1982), Springer.
- A.J. Chorin and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics (2000), Springer.