Course Number:

Credit Hours:

4

Course Title:

Partial Differential Equations 2

Course Description:

Continues MTH G202. Comprises advanced topics in linear and nonlinear partial differential equations, with applications. Topics include pseudodifferential operators and regularity of solutions for elliptic equations; elements of microlocal analysis; propagation of singularities; elements of spectral theory of elliptic operators; properties of eigenvalues and eigenfunctions; variational principle for eigenvalues and its applications; the Schrödinger equation and its meaning in quantum mechanics; parabolic equations and their role in describing heat and diffusion processes; hyperbolic equations and propagation of waves; the Cauchy problem for hyperbolic equations and hyperbolic systems; elements of scattering theory; nonlinear elliptic equations in Riemannian geometry including the Yamabe problem, prescribed scalar curvature problem, and Einstein-Kähler metrics; the Navier-Stokes equations in hydrodynamics; simplest properties and open problem nonlinear hyperbolic equations and shock waves; the Korteweg-de Vries equation and its relation to inverse scattering problems; and solitons and algebra-geometric solutions.