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More About This Title Partial Differential Equations: An Introduction, 2nd edition
English
Partial Differential Equations presents a balanced and comprehensive introduction to the concepts and techniques required to solve problems containing unknown functions of multiple variables. While focusing on the three most classical partial differential equations (PDEs)—the wave, heat, and Laplace equations—this detailed text also presents a broad practical perspective that merges mathematical concepts with real-world application in diverse areas including molecular structure, photon and electron interactions, radiation of electromagnetic waves, vibrations of a solid, and many more.
Rigorous pedagogical tools aid in student comprehension; advanced topics are introduced frequently, with minimal technical jargon, and a wealth of exercises reinforce vital skills and invite additional self-study. Topics are presented in a logical progression, with major concepts such as wave propagation, heat and diffusion, electrostatics, and quantum mechanics placed in contexts familiar to students of various fields in science and engineering. By understanding the properties and applications of PDEs, students will be equipped to better analyze and interpret central processes of the natural world.
English
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.
English
(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
