The mathematics of PDEs and the wave equation （pde和波动方程的数学）.pdf

The mathematics of PDEs and the wave equation Michael P. Lamoureux ∗ University of Calgary Seismic Imaging Summer School August 7–11, 2006, Calgary Abstract Abstract: We look at the mathematical theory of partial differential equations as applied to the wave equation. In particular, we examine questions about existence and uniqueness of solutions, and various solution techniques. ∗ c Supported by NSERC, MITACS and the POTSI and CREWES consortia. 2006. All rights reserved. 1 OUTLINE 1. Lecture One: Introduction to PDEs • Equations from physics • Deriving the 1D wave equation • One way wave equations • Solution via characteristic curves • Solution via separation of variables • Helmholtz’ equation • Classification of second order, linear PDEs • Hyperbolic equations and the wave equation 2. Lecture Two: Solutions to PDEs with boundary conditions and initial conditions • Boundary and initial conditions • Cauchy, Dirichlet, and Neumann conditions • Well-posed problems • Existence and uniqueness theorems • D’Alembert’s solution to the 1D wave equation • Solution to the n-dimensional wave equation • Huygens principle • Energy and uniqueness of solutions 3. Lecture Three: Inhomogeneous solutions - source terms • Particular solutions and boundary, initial conditions • Solution via variation of parameters • Fundamental solutions • Green’s functions, Green’s theorem • Why the convolution with fundamental solutions? • The Fourier transform and solutions • Analyticit