MAT 512E - Partial Differantial Equat. I

Course Objectives

To introduce the concept of characteristics and the method of characteristics to solve Cauchy problem for the first order quasi-linear and nonlinear equations.

To define first the classical and weak solutions of first order quasi-linear equations and then the linear equations of higher order equations. Also to define the distributional solutions of linear equations

To investigate certain classical and weak solutions of the Wave, the Laplace and the Heat equations.

Course Description

First order equations; the Cauchy problem, the method of characteristics, general solutions, weak solutions. Higher order equations; the Cauchy problem, the Cauchy-Kovalewski theorem. Second order equations in two variables. First order systems. Linear equations and generalized solutions. The one dimensional wave equation. The wave equation in higher dimensions; spherical means, Kirchhoff`s formula and Huygens`principle. The Laplace equation; Dirichlet and Neumann problems, the maximum principle, Green`s functions. The heat equation in a bounded domain, the initial value problem and the fundamental solution, regularity.