Praise for the First Edition:"This book is well conceived and well written. The author has succeeded in producing a text on nonlinear PDEs that is not only quite readable but also accessible to students from diverse backgrounds."—SIAM ReviewA practical introduction to nonlinear PDEs and their real-world applicationsNow in a Second Edition, this popular book on nonlinear partial differential equations (PDEs) contains expanded coverage on the central topics of applied mathematics in an elementary, highly readable format and is accessible to students and researchers in the field of pure and applied mathematics. This book provides a new focus on the increasing use of mathematical applications in the life sciences, while also addressing key topics such as linear PDEs, first-order nonlinear PDEs, classical and weak solutions, shocks, hyperbolic systems, nonlinear diffusion, and elliptic equations. Unlike comparable books that typically only use formal proofs and theory to demonstrate results, An Introduction to Nonlinear Partial Differential Equations, Second Edition takes a more practical approach to nonlinear PDEs by emphasizing how the results are used, why they are important, and how they are applied to real problems.The intertwining relationship between mathematics and physical phenomena is discovered using detailed examples of applications across various areas such as biology, combustion, traffic flow, heat transfer, fluid mechanics, quantum mechanics, and the chemical reactor theory. New features of the Second Edition also include:Additional intermediate-level exercises that facilitate the development of advanced problem-solving skillsNew applications in the biological sciences, including age-structure, pattern formation, and the propagation of diseasesAn expanded bibliography that facilitates further investigation into specialized topicsWith individual, self-contained chapters and a broad scope of coverage that offers instructors the flexibility to design courses to meet specific objectives, An Introduction to Nonlinear Partial Differential Equations, Second Edition is an ideal text for applied mathematics courses at the upper-undergraduate and graduate levels. It also serves as a valuable resource for researchers and professionals in the fields of mathematics, biology, engineering, and physics who would like to further their knowledge of PDEs.
Author(s): Logan J.D.
Series: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
Edition: 2ed
Publisher: Wiley
Year: 2008
Language: English
Pages: 415
Tags: Математика;Дифференциальные уравнения;Дифференциальные уравнения в частных производных;
Cover......Page 1
Series: Pure and Applied Mathematics......Page 3
An Introduction to Nonlinear Partial Differential Equations (Second Edition)......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 12
1. Introduction to Partial Differential Equations......Page 16
1.1.1 Equations and Solutions......Page 17
1.1.2 Classification......Page 20
1.1.3 Linear versus Nonlinear......Page 23
1.1.4 Linear Equations......Page 26
1.2.1 One Dimension......Page 35
1.2.2 Higher Dimensions......Page 38
1.3 Constitutive Relations......Page 40
1.4 Initial and Boundary Value Problems......Page 50
1.5.1 Traveling Waves......Page 60
1.5.2 Plane Waves......Page 65
1.5.3 Plane Waves and Transforms......Page 67
1.5.4 Nonlinear Dispersion......Page 69
2. First-Order Equations and Characteristics......Page 76
2.1.1 Advection Equation......Page 77
2.1.2 Variable Coefficients......Page 79
2.2 Nonlinear Equations......Page 83
2.3 Quasilinear Equations......Page 87
2.3.1 The General Solution......Page 91
2.4 Propagation of Singularities......Page 96
2.5 General First-Order Equation......Page 101
2.5.1 Complete Integral......Page 106
2.6 A Uniqueness Result......Page 109
2.7.1 Age Structure......Page 111
2.7.2 Structured Predator-Prey Model......Page 116
2.7.3 Chemotherapy......Page 118
2.7.4 Mass Structure......Page 120
2.7.5 Size-Dependent Predation......Page 121
3. Weak Solutions to Hyperbolic Equations......Page 128
3.1 Discontinuous Solutions......Page 129
3.2 Jump Conditions......Page 131
3.2.1 Rarefaction Waves......Page 133
3.2.2 Shock Propagation......Page 134
3.3 Shock Formation......Page 140
3.4 Applications......Page 146
3.4.1 Traffic Flow......Page 147
3.4.2 Plug Flow Chemical Reactors......Page 151
3.5 Weak Solutions: A Formal Approach......Page 155
3.6.1 Equal-Area Principle......Page 163
3.6.2 Shock Fitting......Page 167
3.6.3 Asymptotic Behavior......Page 169
4. Hyperbolic Systems......Page 174
4.1.1 Shallow-Water Waves......Page 175
4.1.2 Small-Amplitude Approximation......Page 178
4.1.3 Gas Dynamics......Page 179
4.2 Hyperbolic Systems and Characteristics......Page 184
4.2.1 Classification......Page 185
4.3.1 Jump Conditions for Systems......Page 194
4.3.2 Breaking Dam Problem......Page 196
4.3.3 Receding Wall Problem......Page 198
4.3.4 Formation of a Bore......Page 202
4.3.5 Gas Dynamics......Page 205
4.4.1 Hodograph Transformation......Page 207
4.4.2 Wavefront Expansions......Page 208
4.5 Weakly Nonlinear Approximations......Page 216
4.5.1 Derivation of Burgers' Equation......Page 217
5. Diffusion Processes......Page 224
5.1 Diffusion and Random Motion......Page 225
5.2 Similarity Methods......Page 232
5.3 Nonlinear Diffusion Models......Page 239
5.4 Reaction-Diffusion; Fisher's Equation......Page 249
5.4.1 Traveling Wave Solutions......Page 250
5.4.2 Perturbation Solution......Page 253
5.4.3 Stability of Traveling Waves......Page 255
5.4.4 Nagumo's Equation......Page 257
5.5 Advection-Diffusion; Burgers' Equation......Page 260
5.5.1 Traveling Wave Solution......Page 261
5.5.2 Initial Value Problem......Page 262
5.6 Asymptotic Solution to Burgers' Equation......Page 265
5.6.1 Evolution of a Point Source......Page 267
Appendix: Dynamical Systems......Page 272
6. Reaction-Diffusion Systems......Page 282
6.1 Reaction-Diffusion Models......Page 283
6.1.1 Predator-Prey Model......Page 285
6.1.2 Combustion......Page 286
6.1.3 Chemotaxis......Page 289
6.2 Traveling Wave Solutions......Page 292
6.2.1 Model for the Spread of a Disease......Page 293
6.2.2 Contaminant Transport in Groundwater......Page 299
6.3 Existence of Solutions......Page 307
6.3.1 Fixed-Point Iteration......Page 308
6.3.2 Semilinear Equations......Page 312
6.3.3 Normed Linear Spaces......Page 315
6.3.4 General Existence Theorem......Page 318
6.4.1 Maximum Principles......Page 324
6.4.2 Comparison Theorems......Page 329
6.5 Energy Estimates and Asymptotic Behavior......Page 332
6.5.1 Calculus Inequalities......Page 333
6.5.2 Energy Estimates......Page 335
6.5.3 Invariant Sets......Page 341
6.6 Pattern Formation......Page 348
7. Equilibrium Models......Page 360
7.1 Elliptic Models......Page 361
7.2 Theoretical Results......Page 367
7.2.1 Maximum Principle......Page 368
7.2.2 Existence Theorem......Page 370
7.3.1 Linear Eigenvalue Problems......Page 373
7.3.2 Nonlinear Eigenvalue Problems......Page 376
7.4.1 Ordinary Differential Equations......Page 379
7.4.2 Partial Differential Equations......Page 383
References......Page 402
Index......Page 410
Series: Pure and Applied Mathematics......Page 413
