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Partial Differential Equations
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Table of Contents

(The starred sections form the basic part of the book.) Chapter 1/Where PDEs Come From 1.1 What is a Partial Differential Equation? 1 1.2 First-Order Linear Equations 6 1.3 Flows, Vibrations, and Diffusions 10 1.4 Initial and Boundary Conditions 20 1.5 Well-Posed Problems 25 1.6 Types of Second-Order Equations 28 Chapter 2/Waves and Diffusions 2.1 The Wave Equation 33 2.2 Causality and Energy 39 2.3 The Diffusion Equation 42 2.4 Diffusion on the Whole Line 46 2.5 Comparison of Waves and Diffusions 54 Chapter 3/Reflections and Sources 3.1 Diffusion on the Half-Line 57 3.2 Reflections of Waves 61 3.3 Diffusion with a Source 67 3.4 Waves with a Source 71 3.5 Diffusion Revisited 80 Chapter 4/Boundary Problems 4.1 Separation of Variables, The Dirichlet Condition 84 4.2 The Neumann Condition 89 4.3 The Robin Condition 92 Chapter 5/Fourier Series 5.1 The Coefficients 104 5.2 Even, Odd, Periodic, and Complex Functions 113 5.3 Orthogonality and General Fourier Series 118 5.4 Completeness 124 5.5 Completeness and the Gibbs Phenomenon 136 5.6 Inhomogeneous Boundary Conditions 147 Chapter 6/Harmonic Functions 6.1 Laplace's Equation 152 6.2 Rectangles and Cubes 161 6.3 Poisson's Formula 165 6.4 Circles, Wedges, and Annuli 172 (The next four chapters may be studied in any order.) Chapter 7/Green's Identities and Green's Functions 7.1 Green's First Identity 178 7.2 Green's Second Identity 185 7.3 Green's Functions 188 7.4 Half-Space and Sphere 191 Chapter 8/Computation of Solutions 8.1 Opportunities and Dangers 199 8.2 Approximations of Diffusions 203 8.3 Approximations of Waves 211 8.4 Approximations of Laplace's Equation 218 8.5 Finite Element Method 222 Chapter 9/Waves in Space 9.1 Energy and Causality 228 9.2 The Wave Equation in Space-Time 234 9.3 Rays, Singularities, and Sources 242 9.4 The Diffusion and Schro dinger Equations 248 9.5 The Hydrogen Atom 254 Chapter 10/Boundaries in the Plane and in Space 10.1 Fourier's Method, Revisited 258 10.2 Vibrations of a Drumhead 264 10.3 Solid Vibrations in a Ball 270 10.4 Nodes 278 10.5 Bessel Functions 282 10.6 Legendre Functions 289 10.7 Angular Momentum in Quantum Mechanics 294 Chapter 11/General Eigenvalue Problems 11.1 The Eigenvalues Are Minima of the Potential Energy 299 11.2 Computation of Eigenvalues 304 11.3 Completeness 310 11.4 Symmetric Differential Operators 314 11.5 Completeness and Separation of Variables 318 11.6 Asymptotics of the Eigenvalues 322 Chapter 12/Distributions and Transforms 12.1 Distributions 331 12.2 Green's Functions, Revisited 338 12.3 Fourier Transforms 343 12.4 Source Functions 349 12.5 Laplace Transform Techniques 353 Chapter 13/PDE Problems from Physics 13.1 Electromagnetism 358 13.2 Fluids and Acoustics 361 13.3 Scattering 366 13.4 Continuous Spectrum 370 13.5 Equations of Elementary Particles 373 Chapter 14/Nonlinear PDEs 14.1 Shock Waves 380 14.2 Solitons 390 14.3 Calculus of Variations 397 14.4 Bifurcation Theory 401 14.5 Water Waves 406 Appendix A.1 Continuous and Differentiable Functions 414 A.2 Infinite Series of Functions 418 A.3 Differentiation and Integration 420 A.4 Differential Equations 423 A.5 The Gamma Function 425 References 427 Answers and Hints to Selected Exercises 431 Index 446

About the Author

Dr. Walter A. Strauss is a professor of mathematics at Brown University. He has published numerous journal articles and papers. Not only is he is a member of the Division of Applied Mathematics and the Lefschetz Center for Dynamical Systems, but he is currently serving as the Editor in Chief of the SIAM Journal on Mathematical Analysis. Dr. Strauss' research interests include Partial Differential Equations, Mathematical Physics, Stability Theory, Solitary Waves, Kinetic Theory of Plasmas, Scattering Theory, Water Waves, Dispersive Waves.

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