AN INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS ebook 电子书代购

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 Firstorder 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 Secondorder 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 onedimensional 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 Firstorder equations 226
9.3 Classification of secondorder 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 oddnumbered 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 firstyear 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 realworld 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 selfstudy book for a large
class of readers. The first seven chapters include the core of a typical onesemester
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 twosemester 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.
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