Introduction to the Theory of Partial Differential Equations

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