Math majors, as well as students in the sciences and engineering who feel the need for a deeper understanding of the theory of PDEs but do not want to take the graduate level course (Math 819) in the subject.
Partial differential equations (PDEs) appear everywhere in mathematics, science and engineering. The major theories in physics are written in terms of PDEs. Modern advances in differential geometry are based on the understanding of PDEs. Many mathematical models in the biological sciences rely on PDEs. A large number of the algorithms that are used in image processing are formulated in terms of PDEs. Much of the theory of random or stochastic processes, and its applications such as to mathematical finance, rely on PDEs.
This course is a rigorous introduction to the theoretical underpinnings of the basic methods and techniques in the modern theory of PDEs. The emphasis is on the exposure to a number of different methods of solution of PDEs and their connection to physical phenomena modeled by the equations. The goals include both learning to solve some basic types of PDEs as well as to understand the motivation behind and inner workings of the techniques involved.
The course will start with a discussion of PDEs and their role in the modeling of physical processes. This will be followed by the study of first-order linear and non-linear PDEs via the method of characteristics. The bulk of the course will comprise second-order linear PDEs, concentrating on the three basic ones: the heat, wave, and Laplace's equations. These will be the main vehicles for learning various methods of solving PDEs, such as exact formulas for solutions, energy methods, maximum principles, Green's functions and identities, separation of variables and Fourier series, Bessel functions, general eigenvalue methods. Depending on time and student interest, other topics may be discussed, such as the theory of distributions and weak solutions, basics of numerical solution of PDEs, or features of non-linear second order PDEs.