Selected Topics in PDE

3 Credits · Level: Graduate, Undergraduate

This is a PhD level course. This course is a continuation of MATH 4541 . Topics include Nonlinear first-order PDE, the method of characteristics, Hamilton-Jacobi equations, conservation laws, Fourier transform, Laplace transform, and the Radon transform. In the context of similarity solutions plane waves, traveling waves, solitons, group velocity, and phase velocity will be introduced. The method of stationary phase will be discussed in the context of asymptotic solutions. Solutions based on power series and the Cauchy-Kovalevskaya Theorem are studied. The course concludes with the study of semigroups of linear operators. The method of semigroup will be applied to various linear and some non-linear equations and produce strong existence, uniqueness, and stability results.