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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.
Where PDEs Come From | |
What is a Partial Differential Equation? | p. 1 |
First-Order Linear Equations | p. 6 |
Flows, Vibrations, and Diffusions | p. 10 |
Initial and Boundary Conditions | p. 20 |
Well-Posed Problems | p. 25 |
Types of Second-Order Equations | p. 28 |
Waves and Diffusions | |
The Wave Equation | p. 33 |
Causality and Energy | p. 39 |
The Diffusion Equation | p. 42 |
Diffusion on the Whole Line | p. 46 |
Comparison of Waves and Diffusions | p. 54 |
Reflections and Sources | |
Diffusion on the Half-Line | p. 57 |
Reflections of Waves | p. 61 |
Diffusion with a Source | p. 67 |
Waves with a Source | p. 71 |
Diffusion Revisited | p. 80 |
Boundary Problems | |
Separation of Variables, The Dirichlet Condition | p. 84 |
The Neumann Condition | p. 89 |
The Robin Condition | p. 92 |
Fourier Series | |
The Coefficients | p. 104 |
Even, Odd, Periodic, and Complex Functions | p. 113 |
Orthogonality and General Fourier Series | p. 118 |
Completeness | p. 124 |
Completeness and the Gibbs Phenomenon | p. 136 |
Inhomogeneous Boundary Conditions | p. 147 |
Harmonic Functions | |
Laplace's Equation | p. 152 |
Rectangles and Cubes | p. 161 |
Poisson's Formula | p. 165 |
Circles, Wedges, and Annuli | p. 172 |
Green's Identities and Green's Functions | |
Green's First Identity | p. 178 |
Green's Second Identity | p. 185 |
Green's Functions | p. 188 |
Half-Space and Sphere | p. 191 |
Computation of Solutions | |
Opportunities and Dangers | p. 199 |
Approximations of Diffusions | p. 203 |
Approximations of Waves | p. 211 |
Approximations of Laplace's Equation | p. 218 |
Finite Element Method | p. 222 |
Waves in Space | |
Energy and Causality | p. 228 |
The Wave Equation in Space-Time | p. 234 |
Rays, Singularities, and Sources | p. 242 |
The Diffusion and Schrodinger Equations | p. 248 |
The Hydrogen Atom | p. 254 |
Boundaries in the Plane and in Space | |
Fourier's Method, Revisited | p. 258 |
Vibrations of a Drumhead | p. 264 |
Solid Vibrations in a Ball | p. 270 |
Nodes | p. 278 |
Bessel Functions | p. 282 |
Legendre Functions | p. 289 |
Angular Momentum in Quantum Mechanics | p. 294 |
General Eigenvalue Problems | |
The Eigenvalues Are Minima of the Potential Energy | p. 299 |
Computation of Eigenvalues | p. 304 |
Completeness | p. 310 |
Symmetric Differential Operators | p. 314 |
Completeness and Separation of Variables | p. 318 |
Asymptotics of the Eigenvalues | p. 322 |
Distributions and Transforms | |
Distributions | p. 331 |
Green's Functions, Revisited | p. 338 |
Fourier Transforms | p. 343 |
Source Functions | p. 349 |
Laplace Transform Techniques | p. 353 |
PDE Problems from Physics | |
Electromagnetism | p. 358 |
Fluids and Acoustics | p. 361 |
Scattering | p. 366 |
Continuous Spectrum | p. 370 |
Equations of Elementary Particles | p. 373 |
Nonlinear PDEs | |
Shock Waves | p. 380 |
Solitons | p. 390 |
Calculus of Variations | p. 397 |
Bifurcation Theory | p. 401 |
Water Waves | p. 406 |
Appendix | |
Continuous and Differentiable Functions | p. 414 |
Infinite Series of Functions | p. 418 |
Differentiation and Integration | p. 420 |
Differential Equations | p. 423 |
The Gamma Function | p. 425 |
References | p. 427 |
Answers and Hints to Selected Exercises | p. 431 |
Index | p. 446 |
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