This one-year course is written on classical lines with a slant towards
modern methods. It will meet the requirements of senior honours under-
graduates and postgraduates who require a systematic introductory text
outlining the theory of partial differential equations. The amount of theory
presented will meet the needs of most theoretical scientists, and pure mathe-
maticians will find here a sound basis for the study of recent advances in the
subject.
Clear exposition of the elementary theory is a key feature of the book,
and the treatment is rigorous. The author begins by outlining some of the
more common equations of mathematical physics. He then discusses the
principles involved in the solution of various types of partial differential
equation. A close study of the existence and uniqueness theorem is reinforced
by proof. Second-order equations, because of their significance, are considered
in detail. Dr. Smith's approach to the analysis is traditional, but by intro-
ducing the concept of generalized functions and demonstrating their relevance,
he helps the reader to gain a better understanding of Hadamard's theory and
to pass painlessly on to the more difficult works by Courant and Hormander.
The text is reinforced by many worked examples, and problems for
solution have been included where appropriate.
Author(s): M. G. Smith
Publisher: Van Nostrand Reinhold
Year: 1967
Language: English
Pages: 226
Preface
CHAPTER 1 THE PARTIAL DIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS 1
1.1 Introduction 1
1.2 The Equations of Analytical Dynamics 2
1.3 The Equations of Electromagnetism 5
1.4 The Equations of Elasticity 7
1.5 The Equations of Fluid Dynamics 10
CHAPTER 2 THE FUNDAMENTAL EXISTENCE AND UNIQUENESS THEOREMS 15
2.1 Definition of the Cauchy Problem 15
2.2 The Formal Solution 16
2.3 The Dominant Solution 17
2.4 Examination of the Hypotheses 19
2.5 Applicability of the Theorem 20
Exercises 21
CHAPTER 3 FIRST-ORDER EQUATIONS 22
3.1 Definitions and Notation 22
3.2 The Linear Equation in Two Independent Variables 22
3.3 The Semi-linear Equation in Two Independent Variables 24
3.4 The Quasi-linear Equation in Two Independent Variables 25
3.5 The Linear Equation in n Independent Variables 26
3.6 General First-order Equation in Two Independent Variables 27
3.7 Determination of a Complete Integral 31
3.8 A Second Approach to the Cauchy Problem 33
3.9 The Solution of a Cauchy Problem 35
3.10 The One-dimensional Hamilton-Jacobi Equation 36
3.11 The First-order Equation in n Independent Variables 38
3.12 An Example of a Cauchy Problem in Four Variables 40
Exercises 42
Answers 43
CHAPTER 4 SECOND-ORDER LINEAR EQUATIONS 44
4.1 Simultaneous First-order Equations 44
4.2 Classification of Linear Equations of Second Order in Two Independent Variables 45
4.3 Illustrative Examples 47
4.4 The Hyperbolic Equation 48
4.5 Further Discussion of the Examples 51
4.6 Reduction to Canonical Form 52
4.7 The Canonical Form of the Examples 54
4.8 Classification of Quasi-linear Equations 55
Exercises 55
Answers 56
CHAPTER 5 LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER IN MORE TI-IAN Two INDEPENDENT VARIABLES 57
5.1 The Cauchy Problem 57
5.2 The Solution of the Cauchy Problem 58
5.3 Classification of the Equations 59
5.4 Characteristic Surfaces and Bicharacteristics 60
5.5 The Adjoint Equation 61
5.6 The Conormal 62
Exercises 63
Answers 64
CHAPTER 6 THE LAPLACE EQUATION 65
6.1 The Linear Properties 65
6.2 The Averaging Properties 66
6.3 Uniqueness Theorem 67
6.4 The Fundamental Solution and its Use 68
6.5 The Equation in Two-dimensional Cartesian Coordinates 70
6.6 An Example by Hadamard 73
6.7 The Equation in Two-dimensional Polar Coordinates 74
6.8 The Equation in Three-dimensional Cartesian Coordinates 78
6.9 The Equation in Spherical Polar Coordinates 78
6.10 The Equation in Cylindrical Polar Coordinates 82
Exercises 87
Answers 87
CHAPTER 7 THE WAVE EQUATION 89
7.1 The Physical Significance 89
7.2 The Wave Equation in One Dimension 90
7.3 The Finite String 91
7.4 The Infinite String 93
7.5 The Related Helmholtz Equation 96
7.6 Waves in a Finite Rectangular Cavity 97
7.7 Waves in a Finite Spherical Cavity 98
7.8 A General Solution in Three Dimensions 99
7.9 rrhe Corresponding Solution in Two Dimensions 102
Exercises 103
Answers 104
CHAPTER 8 THE DIFFUSION OR HEAT-CONDUCTION EQUATION 105
8.1 1-'he Physical Significance 105
8.2 The Finite Bar 106
8.3 The Semi-infinite Bar 106
8.4 Similarity Solutions 109
8.5 The Related Heln1holtz Equation 113
Exercises 113
CHAPTER 9 GREEN'S FUNCTIONS 115
9.1 The Simple Properties 115
9.2 Some Elementary Green's Functions 116
9.3 The Property of Symmetry 117
9.4 The Neumann Problem 118
9.5 The Expansion of Green's Function 119
9.6 The General Linear Elliptic Equation of Second Order 122
Exercises 123
Answers 124
CHAPTER 10 THE RIEMANN THEORY OF THE HYPERBOLIC EQUATION 125
10.1 The Equation in Two Independent Variables 125
10.2 Some Illustrative Examples 128
10.3 The Equation in n Independent Variables 130
10.4 Volterra's Solution of the Wave Equation in Three Variables 132
10.5 Hadamard Finite Values of Infinite Integrals 136
10.6 rrhe Wave Equation in More than Three Variables 141
10.7 Conclusion 144
Exercises 145
Answers 146
CHAPTER 11 GENERALIZED FUNCTIONS 147
11.1 Introduction 147
11.2 The Distribution as a Functional 148
11.3 Some Simple Properties of Generalized Functions 149
11.4 The Convergence of a Sequence of Generalized Functions 150
11.5 The Distribution Derivatives 155
11.6 The Relation with Hadamard Type Integrals 156
11.7 Generalized Functions of More than One Independent Variable 161
11.8 The Resolution of the Dirac Function into Plane Waves 164
11.9 The Application to Elliptic Equations with Constant Coefficients 166
CHAPTER 12 SYSTEMS OF EQUATIONS OF ORDER GREATER THAN TWO 169
12.1 Introduction 169
12.2 Solution of the Cauchy Problem 169
12.3 Generalization to n Independent Variables 171
12.4 The Adjoint Equation System 172
CHAPTER 13 THE EQUATIONS OF FLUID DYNAMICS 174
13.1 The Linear Cone Field 174
13.2 Inviscid Compressible Steady Flow in Two Dimensions 175
13.3 Inviscid Compressible Steady Flow of an Electrically Conducting Fluid 179
13.4 Inviscid Compressible Steady Flow in Three Dimensions 180
13.5 Unsteady One-dimensional Flow 181
13.6 Unsteady Three-dimensional Flow 183
APPENDIX 1 THE GAUSS THEOREM IN n DIMENSIONS
APPENDIX 2 THE VOLUME AND SURFACE AREA OF THE n-DIMENSIONAL UNIT SPHERE 189
APPENDIX 3 THE EXPANSION THEOREMS OF FOURIER
APPENDIX 4 SINGULARITIES OF LEGENDRE FUNCTIONS
APPENDIX 5 AN IDENTITY INVOLVING DERIVATIVES
References
Index
