A Course on Partial Differential Equations (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 197)

Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view. In addition to the three principal examples of the wave equation, the heat equation, and Laplace's equation, the book has chapters on dispersion and the Schrödinger equation, nonlinear hyperbolic conservation laws, and shock waves. The book covers material for an introductory course that is aimed at beginning graduate or advanced undergraduate level students. Readers should be conversant with multivariate calculus and linear algebra. They are also expected to have taken an introductory level course in analysis. Each chapter includes a comprehensive set of exercises, and most chapters have additional projects, which are intended to give students opportunities for more in-depth and open-ended study of solutions of partial differential equations and their properties.

Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view. In addition to the three principal examples of the wave equation, the heat equation, and Laplace's equation, the book has chapters on dispersion and the Schrödinger equation, nonlinear hyperbolic conservation laws, and shock waves.The book covers material for an introductory course that is aimed at beginning graduate or advanced undergraduate level students. Readers should be conversant with multivariate calculus and linear algebra. They are also expected to have taken an introductory level course in analysis. Each chapter includes a comprehensive set of exercises, and most chapters have additional projects, which are intended to give students opportunities for more in-depth and open-ended study of solutions of partial differential equations and their properties.

Contents 4 Preface 8 Chapter 1. Introduction 12 Chapter 2. Wave equations 20 Chapter 3. The heat equation 54 Chapter 4. Laplace’s equation 82 Chapter 5. Properties of the Fourier transform 114 Chapter 6. Wave equations on Rn 132 Chapter 7. Dispersion 162 Chapter 8. Conservation laws and shocks 188 Bibliography 212 Index 214 A,Course,on,Partial,Differential,Equations Introduction -- Wave Equations -- The Heat Equation -- Laplace's Equation -- Properties Of The Fourier Transform -- Wave Equations On Rn -- Dispersion -- Conservation Laws And Shocks. Walter Craig. Includes Bibliographical References And Index. IntroductionWave equationsThe heat equationLaplace's equationProperties of the Fourier transformWave equations on $\mathbb{R}^n$DispersionConservation laws and shocksBibliographyIndex