Equations of Mathematical Physics

CONTENTS
Publisher's Note. xiv
Preface to the second edition. xiv
Preface to the first edition. xv

I. CLASSIFICATION OF PARTIAL DIFFERENTIAL EQUATIONS
1. Classification of partial differential equations of the second order.
1.1. Differential equations with two independent variables. 1
1.2. Classification of equations of the second order with several independent variables. 9
1.3. Canonical forms of linear equations with constant coefficients. 11
Problems on Chapter I. 13

II. EQUATIONS OF THE HYPERBOLIC TYPE
1. Simplest problems leading to equations of the hyperbolic type. Formulation of boundary problems. 15
1.1. Equation of small transverse vibrations of a string. 15
1.2. Equation of longitudinal vibrations of rods and strings. 19
1.3. Vibrational energy of a string. 21
1.4. Derivation of the transmission line equation. 23
1.5. Transverse vibrations of a membrane. 24
1.6. Equations of hydrodynamics and acoustics. 27
1.7. Boundary and initial conditions. 32
1.8. Reduction of the general problem. 37
1.9. Formulation of boundary problems for the case of several variables. 39
1.10. Uniqueness theorem. 39
Problems. 43
2. Method of propagating waves. 45
2.1. D' Alembert's formula. 45
2.2. Physical interpretation. 47
2.3. Continuity of solutions. 54
2.4. Semi-infinite region and the method of extending initial data. 57
2.5. Problems for a finite segment. 65
2.6. Dispersion of waves. 68
2.7. Integral wave equation. 73
2.8. Propagation of discontinuities along the characteristic curves. 78
Problems. 79
3. Method of separation of variables. 82
3.1. Equation of free vibrations of a string. 82
3.2. Interpretation of the solution. 88
3.3. Representation of an arbitrary vibration as a superposition of standing waves. 92
3.4. Inhomogeneous equations. 97
3.5. First general boundary problem 104
3.6. Boundary problems with stationary inhomogeneities. 105
3.7. Steady state problems. 108
3.8. Concentrated force. 113
3.9. General scheme of the method of separation of variables. 116
Problems. 123
4. Problem with data on characteristics. 125
4.1. Formulation of the problem. 125
4.2. Method of successive approximations. 127
Problems. 133
5. Solution of general linear equations of the hyperbolic type. 133
5.1. Conjugate differential operators. 133
5.2. Integral form of solution. 134
5.3. Physical interpretation of Riemann's function. 137
5.4. Equations with constant coefficients. 141
Problems on Chapter II. 145
Appendices to Chapter II. 147
I. Vibrations of strings of musical instruments. 147
II. Vibrations of rods. 150
III. Vibrations of a loaded string. 155
IV. Equations of gas dynamics and theory of shock waves. 163
V. Dynamics of the absorption of gases. 175
VI. Physical analogies. 186

III. EQUATIONS OF THE PARABOLIC TYPE
1. Physical problems leading to equations of the parabolic type. Formulation of boundary-value problems. 191
1.1. Linear problem of heat conduction. 191
1.2. Equation of diffusion. 196
1.3. Heat conduction in space. 197
1.4. Formulation of boundary-value problems. 200
1.5. The maximum value principle. 206
1.6. Uniqueness theorem. 209
1.7. Uniqueness theorem for an infinite region. 212
2. Method of the separation of variables. 213
2.1. Homogeneous boundary-value problem. 213
2.2. Source function. 217
2.3. Boundary problems with discontinuous initial conditions. 220
2.4. Inhomogeneous equation of heat conduction. 228
2.5. First general boundary problem. 231
Problems. 233
3. Problems in an infinite region. 235
3.1. Source function for an infinite region. 235
3.2. Heat propagation in an infinite region. 243
3.3. Boundary-value problems for a semi-infinite straight line. 253
3.4. Steady state problems. 262
Problems on Chapter III. 266
Appendices to Chapter III. 268
I. Temperature waves. 268
II. Effect of radioactive decay on the temperature of the earth's crust. 272
III. Method of similarity in the theory of heat conduction. 278
IV. Problem of freezing. 283
VI. Einstein-Kolmogorov equation. 288
VI. $\delta$-function. 292

IV. EQUATIONS OF ELLIPTIC TYPE
1. Problems reducible to Laplace's equation. 301
1.1. Steady heat flow. Formulation of boundary-value problems. 301
1.2. Potential flow in a fluid. Potential of a stationary current and an electrostatic field. 302
1.3. Laplace's equation in a curvilinear system of coordinates. 304
1.4. Some particular solutions of Laplace's equation. 308
1.5. Harmonic functions and analytic functions of a complex variable. 310
1.6. The inversion transformation. 312
2. General properties of harmonic functions. 314
2.1. Green's formula. Integral representation of a solution. 314
2.2. Some fundamental properties of harmonic functions. 320
2.3. Uniqueness and stability of the first boundary-value problem 324
2.4. Problems with discontinuous boundary conditions. 325
2.5. Isolated singular points. 327
2.6. Regularity of a harmonic function at infinity. 329
2.7. External boundary-value problem. Uniqueness of solution for two- and three-dimensional problems. 330
2.8. Second boundary-value problem. Condition of regularity at infinity. Uniqueness theorem. 334
3. Solution of boundary-value problems for the simplest regions by the method of separation of variables. 338
3.1. First boundary-value problem for a circle. 338
3.2. Poisson's integral. 343
3.3. Case of discontinuous boundary values. 347
4. Source function. 349
4.1. Source function for the equation $\Delta v=0$ and its fundamental properties. 349
4.2. Method of electrostatic images and the source function for a sphere. 354
4.3. Source function for a circle. 357
4.4. Source function for a half-space. 359
5. Potential theory. 360
5.1. Volume potential. 360
5.2. Plane problem. Logarithmic potential. 362
5.3. Improper integrals. 365
5.4. First derivatives of the volume potential. 372
5.5. Second derivatives of the volume potential. 376
5.6. Surface potentials. 380
5.7. Surfaces and Lyapunov's curves. 384
5.8. Discontinuity of the potential of a double layer. 388
5.9. Properties of the potential of a single layer. 392
5.10. Application of surface potentials to the solution of boundary problems. 396
5.11. Integral equations. 401
6. Method of finite differences. 406
6.1. Concept of the method of finite differences for Laplace's equation. 406
6.2. Method of successive approximations for the solution of difference equations. 409
6.3. Modelling methods. 412
Problems on Chapter IV. 413
Appendices to Chapter IV. 416
I. Asymptotic expression for the volume potential. 416
II. Problems of electrostatics. 419
III. Fundamental problem of electrical prospecting. 426
IV. Determination of vector fields. 433
V. Application of the method of conformal mapping to electrostatics. 436
VI. Application of the method of conformal mapping to hydrodynamics. 440
VII. Biharmonic equations. 447

V. WAVE PROPAGATION IN SPACE
1. Problem with initial conditions. Method of averaging. 452
1.1. Method of averaging. 452
1.2. Method of descent. 455
1.3. Physical interpretation. 457
1.4. Method of images. 459
2. Integral relation. 461
2.1. Derivation of the integral relation. 461
2.2. Consequences of the integral relation. 465
3. Vibrations of finite volumes. 468
3.1. General scheme of the method of separation of variables. Standing waves. 468
3.2. Vibrations of a rectangular membrane. 474
3.3. Vibrations of a circular membrane. 478
Problems on Chapter V. 485
Appendices to Chapter V. 486
I. Reduction of the equations of the theory of elasticity to the wave equation. 486
II. Equations of the electromagnetic field. 489

VI. HEAT CONDUCTION IN SPACE
1. Heat conduction in infinite space. 503
1.1. Temperature propagation function. 503
1.2. CO:1duction of heat in infinite space. 508
2. Heat conduction in finite bodies. 512
2.1. Outline of the method of separation of variables. 512
2.2. Cooling of a circular cylinder. 516
2.3. Determination of critical sizes. 519
3. Boundary problems for regions with moving boundaries. 521
3.1. Green's theorem for the equation of heat conduction and the source function. 521
3.2. Solution of the boundary-value problem. 525
3.3. Source function for the segment. 528
4. Thermal potentials. 530
4.1. Properties of thermal potentials of a single and of a double layer. 530
4.2. Solution of boundary-value problems. 533
Problems on Chapter VI. 535
Appendices to Chapter VI. 537
I. Diffusion of a cloud. 537
II. Demagnetization of a cylinder by a coil. 540
III. Method of finite differences for the equation of heat conduction. 545

VII. EQUATIONS OF ELLIPTIC TYPE (CONTINUATION)
1. Fundamental problems leading to the equation $\Delta v+cv=0$. 556
1.1. Steady vibrations. 556
1.2. Diffusion of gas in the presence of decay and with chain reactions. 557
1.3. Diffusion in a moving medium. 558
1.4. Formulation of internal boundary problems for the equation $\Delta v+cv=0$. 559
2. The source function. 560
2.1. The source function. 560
2.2. Integral representation of the solution. 563
2.3. Potentials. 567
3. Problems for an infinite region. The radiation principle. 570
3.1. The equation $\Delta v+cv=-f$ in infinite space. 570
3.2. Principle of vanishing absorption. 571
3.3. Principle of limiting amplitude. 573
3.4. Radiation condition. 575
4. Problems of the mathematical theory of diffraction. 580
4.1. Statement of the problem. 580
4.2. Uniqueness of the solution of the diffraction problem. 581
4.3. Diffraction by a sphere. 584
Problems on Chapter VII. 592
Appendices to Chapter VII. 595
I. Waves in cylindrical guides. 595
II. Electromagnetic vibrations in hollow resonators. 606
III. Skin effect. 616
IV. Propagation of radiowaves over the surface of the earth. 621

SUPPLEMENT. SPECIAL FUNCTIONS
Introduction. 627
1. Equations for special functions. 627
2. Formulation of boundary problems in the case $k(a)=0$. 628

I. CYLINDRICAL FUNCTIONS
1. Cylindrical functions. 637
2. Boundary-value problems for Bessel's equation. 648
3. Various types of cylindrical functions. 652
4. Integral representations. Asymptotic formulae. 661
5. Fourier-Bessel's integral and some integrals containing Bessel functions. 670
6. Representation of cylindrical functions by means of contour integrals. 675

II. SPHERICAL FUNCTIONS
1. Legendre polynomials. 686
2. Harmonic polynomials and spherical functions. 703
3. Some examples of application of spherical functions. 716

III. CHEBYSHEV-HERMITE AND CHEBYSHEV-LAGUERRE POLYNOMIALS
1. Chebyshev-Hermite polynomials. 725
2. Chebyshev-Laguerre polynomials. 728
3. Simplest problems for Schrodinger's equation. 736

TABLES OF THE ERROR INTEGRAL AND SOME CYLINDRICAL FUNCTIONS. 745

Index. 754