Course: STAT 31110=CAAM 31110
Title: Integral Equation Methods for PDEs
Instructor: Tristan Goodwill
Class Schedule: Sec 1: TR 12:30 PM–1:50 PM in TBA
Description: Many important PDE problems can be converted into an equivalent integral equation. These integral equation formulations have a number of computationally useful properties. In particular, they make many unbounded scattering problems tractable, enable the construction of efficient solvers for domains with complex boundaries, and lead to well-conditioned linear systems. In this course, we will demonstrate how to derive integral equation formulations for a variety of standard PDE problems. We will also show how Fredholm theory can be used to prove the existence and uniqueness of solutions to these integral equations and discuss the hallmarks of a well-conditioned formulation. Examples will include the Laplace and Helmholtz equations on domains with compact boundaries, the variable coefficient Helmholtz equation, vector-valued scattering problems, and scattering problems involving unbounded interfaces.