| Titre : | Partial Differential Equations: An Introduction to Analytical and Numerical Methods (Graduate Texts in Mathematics) |
| Auteurs : | Wolfgang Arendt, Auteur ; Karsten Urban, Auteur |
| Type de document : | Monographie imprimée |
| Editeur : | Springer, 2024 |
| ISBN/ISSN/EAN : | 978-3-031-13381-7 |
| Format : | 1 vol. (476 p.) |
| Langues: | Anglais |
| Résumé : |
This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach. A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses onfinite difference and finite element methods. Computer-aided calculation with Maple™ completes the book. Throughout, three fundamental examples are studied with different tools: Poisson’s equation, the heat equation, and the wave equation on Euclidean domains. The Black–Scholes equation from mathematical finance is one of several opportunities for extension. Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed. |
| Sommaire : |
List of figures xxiii 1 Modeling, or where do differential equations come from 1 1.1 Mathematical modeling . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Transport processes . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 The Black–Scholes equation . . . . . . . . . . . . . . . . . . . . 11 1.6 Let’s get higher dimensional . . . . . . . . . . . . . . . . . . . . 13 1.7* But there’s more . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.8 Classification of partial differential equations . . . . . . . . . . . 25 1.9* Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Classification and characteristics 29 2.1 Characteristics of initial value problems on R . . . . . . . . . . . 30 2.2 Equations of second order . . . . . . . . . . . . . . . . . . . . . . 39 2.3* Nonlinear equations of second order . . . . . . . . . . . . . . . . 43 2.4* Equations of higher order and systems . . . . . . . . . . . . . . . 44 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3 Elementary methods 49 3.1 The one-dimensional wave equation . . . . . . . . . . . . . . . . 50 3.2 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 The Black–Scholes equation . . . . . . . . . . . . . . . . . . . . 90 3.6 Integral transforms . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4 Hilbert spaces 115 4.1 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 xix xx Contents 4.4 Orthogonal projections . . . . . . . . . . . . . . . . . . . . . . . 125 4.5 Linear and bilinear forms . . . . . . . . . . . . . . . . . . . . . . 128 4.6 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.7 Continuous and compact operators . . . . . . . . . . . . . . . . . 138 4.8 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . 139 4.9* Comments on Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 150 4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5 Sobolev spaces and boundary value problems in dimension one 155 5.1 Sobolev spaces in one variable . . . . . . . . . . . . . . . . . . . 156 5.2 Boundary value problems on the interval . . . . . . . . . . . . . . 164 5.3* Comments on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . 176 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 Hilbert space methods for elliptic equations 181 6.1 Mollifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.2 Sobolev spaces on Ω ⊆ Rd . . . . . . . . . . . . . . . . . . . . . 189 6.3 The space H1 0 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.4 Lattice operations on H1(Ω) . . . . . . . . . . . . . . . . . . . . 200 6.5 The Poisson equation with Dirichlet boundary conditions . . . . . 204 6.6 Sobolev spaces and Fourier transforms . . . . . . . . . . . . . . . 207 6.7 Local regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.8 Inhomogeneous Dirichlet boundary conditions . . . . . . . . . . . 219 6.9 The Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . 222 6.10 Elliptic equations with Dirichlet boundary conditions . . . . . . . 231 6.11 H2-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6.12* Comments on Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . 236 6.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7 Neumann and Robin boundary conditions 241 7.1 Gauss’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.2 Proof of Gauss’s theorem . . . . . . . . . . . . . . . . . . . . . . 247 7.3 The extension property . . . . . . . . . . . . . . . . . . . . . . . 254 7.4 The Poisson equation with Neumann boundary conditions . . . . . 258 7.5 The trace theorem and Robin boundary conditions . . . . . . . . . 262 7.6* Comments on Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . 265 7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 8 Spectral decomposition and evolution equations 269 8.1 A vector-valued initial value problem . . . . . . . . . . . . . . . . 270 8.2 The heat equation: Dirichlet boundary conditions . . . . . . . . . 274 8.3 The heat equation: Robin boundary conditions . . . . . . . . . . . 280 8.4 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . 283 8.5 Inhomogeneous parabolic equations . . . . . . . . . . . . . . . . 295 8.6* Space/time variational formulations . . . . . . . . . . . . . . . . 304 8.7* Comments on Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . 308 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Contents xxi 9 Numerical methods 313 9.1 Finite differences for elliptic problems . . . . . . . . . . . . . . . 315 9.2 Finite elements for elliptic problems . . . . . . . . . . . . . . . . 330 9.3* Extensions and generalizations . . . . . . . . . . . . . . . . . . . 355 9.4 Parabolic problems . . . . . . . . . . . . . . . . . . . . . . . . . 360 9.5 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . 379 9.6* Comments on Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . 406 9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 10 Maple®, or why computers can sometimes help 413 10.1 Maple® . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 10.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Appendix 423 A.1 Banach spaces and linear operators . . . . . . . . . . . . . . . . . 423 A.2 The space C(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 A.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 A.4 More details on the Black–Scholes equation . . . . . . . . . . . . 428 References 435 Index of names 439 Index of symbols 443 Inde |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| MAT/1074 | Livre | bibliothèque sciences exactes | Empruntable |
