# AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS ebook 电子书代购

 Contents Preface page xi 1 Introduction 1 1.1 Preliminaries 1 1.2 Classification 3 1.3 Differential operators and the superposition principle 3 1.4 Differential equations as mathematical models 4 1.5 Associated conditions 17 1.6 Simple examples 20 1.7 Exercises 21 2 First-order equations 23 2.1 Introduction 23 2.2 Quasilinear equations 24 2.3 The method of characteristics 25 2.4 Examples of the characteristics method 30 2.5 The existence and uniqueness theorem 36 2.6 The Lagrange method 39 2.7 Conservation laws and shock waves 41 2.8 The eikonal equation 50 2.9 General nonlinear equations 52 2.10 Exercises 58 3 Second-order linear equations in two indenpendent variables 64 3.1 Introduction 64 3.2 Classification 64 3.3 Canonical form of hyperbolic equations 67 3.4 Canonical form of parabolic equations 69 3.5 Canonical form of elliptic equations 70 3.6 Exercises 73 vii viii Contents 4 The one-dimensional wave equation 76 4.1 Introduction 76 4.2 Canonical form and general solution 76 4.3 The Cauchy problem and d’Alembert’s formula 78 4.4 Domain of dependence and region of influence 82 4.5 The Cauchy problem for the nonhomogeneous wave equation 87 4.6 Exercises 93 5 The method of separation of variables 98 5.1 Introduction 98 5.2 Heat equation: homogeneous boundary condition 5.3 Separation of variables for the wave equation 109 5.4 Separation of variables for nonhomogeneous equations 114 5.5 The energy method and uniqueness 116 5.6 Further applications of the heat equation 119 5.7 Exercises 124 6 Sturm–Liouville problems and eigenfunction expansions 130 6.1 Introduction 130 6.2 The Sturm–Liouville problem 133 6.3 Inner product spaces and orthonormal systems 136 6.4 The basic properties of Sturm–Liouville eigenfunctions and eigenvalues 141 6.5 Nonhomogeneous equations 159 6.6 Nonhomogeneous boundary conditions 164 6.7 Exercises 168 7 Elliptic equations 173 7.1 Introduction 173 7.2 Basic properties of elliptic problems 173 7.3 The maximum principle 178 7.4 Applications of the maximum principle 181 7.5 Green’s identities 182 7.6 The maximum principle for the heat equation 184 7.7 Separation of variables for elliptic problems 187 7.8 Poisson’s formula 201 7.9 Exercises 204 8 Green’s functions and integral representations 208 8.1 Introduction 208 8.2 Green’s function for Dirichlet problem in the plane 209 8.3 Neumann’s function in the plane 219 8.4 The heat kernel 221 8.5 Exercises 223 Contents ix 9 Equations in high dimensions 226 9.1 Introduction 226 9.2 First-order equations 226 9.3 Classification of second-order equations 228 9.4 The wave equation in R2 and R3 234 9.5 The eigenvalue problem for the Laplace equation 242 9.6 Separation of variables for the heat equation 258 9.7 Separation of variables for the wave equation 259 9.8 Separation of variables for the Laplace equation 261 9.9 Schr¨odinger equation for the hydrogen atom 263 9.10 Musical instruments 266 9.11 Green’s functions in higher dimensions 269 9.12 Heat kernel in higher dimensions 275 9.13 Exercises 279 10 Variational methods 282 10.1 Calculus of variations 282 10.2 Function spaces and weak formulation 296 10.3 Exercises 306 11 Numerical methods 309 11.1 Introduction 309 11.2 Finite differences 311 11.3 The heat equation: explicit and implicit schemes, stability, consistency and convergence 312 11.4 Laplace equation 318 11.5 The wave equation 322 11.6 Numerical solutions of large linear algebraic systems 324 11.7 The finite elements method 329 11.8 Exercises 334 12 Solutions of odd-numbered problems 337 A.1 Trigonometric formulas 361 A.2 Integration formulas 362 A.3 Elementary ODEs 362 A.4 Differential operators in polar coordinates 363 A.5 Differential operators in spherical coordinates 363 References 364 Index 366 Preface This book presents an introduction to the theory and applications of partial differential equations (PDEs). The book is suitable for all types of basic courses on PDEs, including courses for undergraduate engineering, sciences and mathematics students, and for first-year graduate courses as well. Having taught courses on PDEs for many years to varied groups of students from engineering, science and mathematics departments, we felt the need for a textbook that is concise, clear, motivated by real examples and mathematically rigorous.We therefore wrote a book that covers the foundations of the theory of PDEs. This theory has been developed over the last 250 years to solve the most fundamental problems in engineering, physics and other sciences. Therefore we think that one should not treat PDEs as an abstract mathematical discipline; rather it is a field that is closely related to real-world problems. For this reason we strongly emphasize throughout the book the relevance of every bit of theory and every practical tool to some specific application. At the same time, we think that the modern engineer or scientist should understand the basics of PDE theory when attempting to solve specific problems that arise in applications. Therefore we took great care to create a balanced exposition of the theoretical and applied facets of PDEs. The book is flexible enough to serve as a textbook or a self-study book for a large class of readers. The first seven chapters include the core of a typical one-semester course. In fact, they also include advanced material that can be used in a graduate course. Chapters 9 and 11 include additional material that together with the first seven chapters fits into a typical curriculum of a two-semester course. In addition, Chapters 8 and 10 contain advanced material on Green’s functions and the calculus of variations. The book covers all the classical subjects, such as the separation of variables technique and Fourier’s method (Chapters 5, 6, 7, and 9), the method of characteristics (Chapters 2 and 9), and Green’s function methods (Chapter 8). At the same time we introduce the basic theorems that guarantee that the problem at xii Preface hand is well defined (Chapters 2–10), and we took care to include modern ideas such as variational methods (Chapter 10) and numerical methods (Chapter 11). The first eight chapters mainly discuss PDEs in two independent variables. Chapter 9 shows how the methods of the first eight chapters are extended and enhanced to handle PDEs in higher dimensions. Generalized and weak solutions are presented in many parts of the book. Throughout the book we illustrate the mathematical ideas and techniques by applying them to a large variety of practical problems, including heat conduction, wave propagation, acoustics, optics, solid and fluid mechanics, quantum mechanics, communication, image processing, musical instruments, and traffic flow. 湛艺 联系QQ：526781618 淘宝旺旺：跟朝流走 有需要的欢迎联系！专业代购电子书 ebook 英文电子书代购