MATH 8361 Advanced Partial Differential Equations
This course considers waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations. Topics include: first-order equations: characteristic ODEs, existence of smooth solutions, conservation law equations, shocks, rarefaction, integral solutions; second-order partial differential equations and classification; Wave equation: fundamental solutions in one, two and three dimensions, Duhamel's principle, energy methods, finite propagation speed; the Laplace equation: mean-value property, smoothness, maximum principle, uniqueness of solutions, Harnack’s inequalities, Liouville's theorem; fundamental solution to the Poisson Equation, Green's functions, energy methods. the fundamental solution to the heat equation, the maximum principle, uniqueness of solutions on a bounded domain, and energy methods. In addition, the theory of second order linear PDEs will be covered, including the existence of weak solutions, regularity, maximum principles. The courses will include fixed point methods, and method of subsolutions and supersolutions. The PDE models considered in the course appear in physical models and have numerous applications in physics and engineering.
Prerequisite
Consent of instructor.
Offered
As scheduled