We now come to Equations of Mathematical Physics by A. V. Bitsazde.

About the book:

The present book consists of an introduction and six chapters. The introduction discusses basic notions and definitions of the traditional course of mathematical physics and also mathematical models of some phenomena in physics and engineering. Chapters 1 and 2
are devoted to elliptic partial differential equations. Here much emphasis is placed on the Cauchy- Riemann system of partial differential equations, that is on fundamentals of the theory of analytic functions, which facilitates the understanding of the role played in mathematical physics by the theory of functions of a complex variable.

In Chapters 3 and 4 the structural properties of the solutions of hyperbolic and parabolic partial differential equations are studied and much attention is paid to basic problems of the theory of wave equation and heat conduction equation.

In Chapter 5 some elements of the theory of linear integral equations are given. A separate section of this chapter is devoted to singular
integral equations which are frequently used in applications. Chapter 6 is devoted to basic practical methods for the solution of partial differential equations. This chapter contains a number of typical examples demonstrating the essence of the Fourier method of separation of variables, the method of integral transformations, the finite difference method, the melthod of asymptotic expansions and also the variational methods.

To study the book it is sufficient for the reader to be familiar with an ordinary classical course on mathematical analysis studied in colleges. Since such a course usually does not involve functional analysis, the embedding theorems for function’ spaces are not included in the present book.

The book was translated from the Russian by V. M. Volosov and I. G. Volosova and was first published by Mir Publishers in 1980.

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Contents
Preface 5
Introduction 13

1. Basic Notions and Definitions 13

1. The Notion of a Partial Differential Equation and Its Solution 13
2. Characteristic Form of a Linear Partial Differential Equation. Classification of Linear Partial Differential 15
Equations of the Second Order by Type 15
3. Classification of Higher-Order Partial Differential Equations 17
4. Systems of Partial Differential Equations 18

2. Normal Form of Linear Partial Differential Equations of the Second
Order in Two Independent Variables 20

1. Characteristic Curves and Characteristic Directions 20
2. Transformation of Partial Differential Equations of the Second
Order in Two Independent Variables into the Normal Form 22

3. Simplest Examples of the Three Basic Types of Second-Order Partial Differential Equations 26

1. The Laplace Equation 26
2. Wave Equation 29
3. Heat Conduction Equation 32
4. Statement of Some Problems for Partial Differential Equations 33

4. The Notion of an Integral Equation 35

1. Notation and Basic Definitions 35
2. Classification of Linear Integral Equations 36

5. Simplified Mathematical Models for Some Phenomena in Physics and Engineering 37

1. Electrostatic Field 37
2. Oscillation of a Membrane 40
3. Propagation of Heat 43
4. The Motion of a Material Point under theAction of the Force of Gravity 44

Chapter 1.
Elliptic Partial Differential Equations 47

1. Basic Properties of Harmonic Functions 47

1. Definition of a Harmonic Function and Some of Its Basic Properties 47
2. Integral Representation of Harmonic Functions 50
3. Mean-Value Formulas 51
4. The Extremum Principle for the Dirichlet Problem. Uniqueness of the Solution 53

2. The Notion of Green’s Function. Solution of the Dirichlet Problem for a Ball and for a Half-Space 55

1. Green’s Function of the Dirichlet Problem for the Laplace Equation 55
2. Solution of the Dirichlet Problem for a Ball. Poisson’s Formula 57
3. Verification of Boundary Conditions 60
4. Solution of the Dirichlet Problem for a Half-Space 61
5. Some Important Consequences of Poisson’s Formula. Theorems of Liouville and Harnack 63

3. Potential Function for a Volume Distribution of Mass 65

1. Continuity of Volume Potential and Its Derivatives of the First Order 65
2. Existence of the Derivatives of the Second Order of Volume Potential 67
3. The Poisson Equation 69
4. Gauss Formula 72

4. Double-Layer and Single-Layer Potentials 74

1. Definition of a Double-Layer Potential 74
2. Formula for the Jump of a Double-Layer Potential Reduction of the Dirichlet Problem to an Integral Equation 77
3. Single-Layer Potential. The Neumann Problem 81
4. The Dirichlet Problem and the Neumann Problem for Unbounded Domains 84

5. Elements of the General Theory of Elliptic Linear Partial
Differential Equations of the Second Order 86

1. Adjoint Operator. Green’s Theorem 86
2. Existence of Solutions of Elliptic Linear Partial Differential Equations of the Second Order 88
3. Boundary-Value Problems 90
4. The Extremum Principle. The Uniqueness of the Solution of the Dirichlet Problem 92
5. Generalized Single-Layer and Double-Layer Potentials 94

Chapter 2.
Cauchy-Riemann System of Partial Differential Equations. Elements of the Theory of Analytic Functions 97

1.The Notion of an Analytic Function of a Complex Variable 97

1. Cauchy-Riemann System of Partial Differential Equations 97
2. The Notion of an Analytic Function 98
3. Examples of Analytic Functions 102
4. Conformal Mapping 104
5. Conformal Mappings Determined by Some Elementary Functions. Inverse Functions. The Notion of a Riemann Surface 109

2. Complex Integrals 116

1. Integration along a Curve in the Complex Plane 116
2. Cauchy’s Theorem 118
3. Cauchy’s Integral Formula 121
4. The Cauchy-Type Integral 124
5. Conjugate Harmonic Functions. Morera’s Theorem 125

3. Some Important Consequences of Cauchy’s Integral Formula 127

1. Maximum Modulus Principle for Analytic Functions 127
2. Weierstrass’ Theorems 129
3. Taylor’s Series 131
4. Uniqueness Theorem for Analytic Functions. Liouville’s Theorem 133
5. Laurent Series 134
6. Singular Points and Residues of an Analytic Function t38
7. Schwarz’s Formula. Solution of Dirichlet Problem 143

4. Analytic Continuation 146

1. The Notion of Analytic Continuation 146
2. The Continuity Principle 146
3. The Riemann-Schwarz Symmetry Principle 148

5. Formulas for Limiting Values of Cauchy-Type Integral and Their Applications 149

1. Cauchy’s Principal Value of a Singular Integral 149
2. Tangential Derivative of a Single-Layer Potential 151
3. Limiting Values of Cauchy-Type Integral 154
4. The Notion of a Piecewise Analytic Function 156
5. Application to Boundary-Value Problems 157

6. Functions of Several Variables 163

1. Notation and Basic Notions 163
2. The Notion of an Analytic Function of Several Variables 164
3. Multiple Power Series 166
4. Cauchy’s Integral Formula and Taylor’s Theorem 168
5. Analytic Functions of Real Variables 170
6. Conformal Mappings in Euclidean Spaces 172

Chapter 3.
Hyperbolic Partial Differential Equations 176

1. Wave Equation 176

1. Wave Equation with Three Spatial Variables. Kirchhoff’s Formula 176
2. Wave Equation with Two Spatial Variables. Poisson’s Formula 178
3. Equation of Oscillation of a String. D’Alemhert’s Formula 179
4. The Notion of the Domains of Dependence, Influence and Propagation 181

2. Non-Homogeneous Wave Equation 183

1. The Case of Three Spatial Variables. Retarded Potential 183
2. The Case of Two, or One, Spatial Variables 184

3. Well-Posed Problems for Hyperbolic Partial Differential Equations 186

1. Uniqueness of the Solution of the Cauchy Problem 186
2. Correctness of the Cauchy Problem for Wave Equation 187
3. General Statement of the Cauchy Problem 188
4. Goursat Problem 191
5. Some Improperly Posed Problems 192

4. General Linear Hyperbolic Partial Differential Equation of the Second Order in Two Independent Variables 193

1. Riemann’s Function 193
2. Goursat Problem 196
3. Cauchy Problem 198

Chapter 4.
Parabolic Partial Differential Equations 200

1. Heat Conduction Equation. First Boundary-Value Problem 200
1. Extremum Principle 200
2. First Boundary-Value Problem for Heat Conduction Equation 202

2. Cauchy-Dirichlet Problem 204

1. Statement of Cauchy-Dirichlet Problem and the Proof of the Existence of Its Solution 204
2. Uniqueness and Stability of the Solution of Cauchy-Dirichlet Problem 206
3. Non-Homogeneous Heat Conduction Equation 208

3. On Smoothness of Solutions of Partial Differential Equations 208

1. The Case of Elliptic and Parabolic Partial Differential Equations 208
2. The Case of Hyperbolic Partial Differential Equations 209

Chapter 5.
Integral Equations 210

1. Iterative Method for Solving Integral Equations 210

1. General Remarks 210
2. Solution of Fredholm Integral Equation of the Second Kind for Small Values of the Parameter Using Iterative Method 211
3. Volterra Integral Equation of the Second Kind 213

2. Fredholm Theorems 215

1. Fredholm Integral Equation of the Second Kind with Degenerate Kernel 215
2. The Notions of Iterated and Resolvent Kernels 219
3. Fredholm Integral Equation of the Second Kind with an Arbitrary Continuous Kernel 220
4. The Notion of Spectrum 224
5. Volterra Integral Equation of the Second Kind with Multiple Integral 225
6. Volterra Integral Equation of the First Kind 226

3. Applications of the Theory of Linear Integral Equations of the Second Kind 228

1. Application of Fredholm Alternative to the Theory of Boundary-Value Problems for Harmonic Functions 228
2. Reduction of Cauchy Problem for an Ordinary Linear Differential Equation to a Volterra Integral Equation
of the Second Kind 231
3. Boundary-Value Problem for Ordinary Linear Differential Equations of the Second Order 233

4. Singular Integral Equations 236

1. The Notion of a Singular Integral Equation 236
2. Hilbert’s Integral Equation 237
3. Hilbert Transformation 240
4. Integral Equation of the Theory of the Wing of an Airplane 241
5. Integral Equation with a Kernel Having Logarithmic Singularity 244

Chapter 6.
Basic Practical Methods for the Solution of Partial Differential Equations 246

1. The Method of Separation of Variables 246

1. Solution of Mixed Problem for Equation of Oscillation of a String 246
2. Oscillation of a Membrane 251
3. The Notion of a Complete Orthonormal System of Functions 254
4. Oscillation of Circular Membrane 257
5. Some General Remarks on the Method of Separation of Variables 260
6. Solid and Surface Spherical Harmonics 262
7. Forced Oscillation 264

2. The Method of Integral Transformation 266

1. Integral Representation of Solutions of Ordinary Linear Differential Equations of the Second Order 266
2. Laplace, Fourier and Mellin Transforms 272
3. Application of the Method of Integral Transformations to Partial Differential Equations 275
4. Application of Fourier Transformation to the Solution of Cauchy Problem for the Equation of Oscillation of a String 278
5. Convolution 281
6. Dirac’s Delta Function 284

3. The Method of Finite Differences 286

1. Finite-Difference Approximation of Partial Differential Equations 286
2. Dirichlet Problem for Laplace’s Equation 287
3. First Boundary-Value Problem for Heat Conduction Equation 289
4. Some General Remarks on Finite-Difference Method 290

4. Asymptotic Expansions 291

1. Asymptotic Expansion of a Function of One Variable 291
2. Watson’s Method for Asymptotic Expansion 296
3. Saddle-Point Method 299

5. Variational Methods 303

1. Dirichlet Principle 303
2. Eigenvalue Problem 305
3. Minimizing Sequence 307
4. Ritz Method 308
5. Approximate Solution of Eigenvalue Problem. Bubnov-Galerkin Method 309

Name Index 312
Subject Index 313