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The mathematics of PDEs and the wave equationMichael P. Lamoureux∗University of CalgarySeismic Imaging Summer SchoolAugust 7–11, 2006, CalgaryAbstractAbstract: We look at the mathematical theory of partial differential equations asapplied to the wave equation. In particular, we examine questions about existence anduniqueness of solutions, and various solution techniques.∗Supported by NSERC, MITACS and the POTSI and CREWES consortia.c©2006. All rights reserved.1
OUTLINE1. 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 equation2. 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 solutions3. 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•Analyticity and avoiding zeros•Spatial Fourier transforms•Radon transform•Things we haven’t covered2
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