Partial Differential Equations in Physics

Pure and Applied Mathematics

Foreword

Editors’ Foreword

Erratum

Chapter I: Fourier Series and Integrals

§1 Fourier Series

§2 Example of a Discontinuous Function. Gibbs’ Phenomenon and Non-Uniform Convergence

§3 On the Convergence of Fourier Series

§4 Passage to the Fourier Integral

§5 Development by Spherical Harmonics

§6 Generalizations: Oscillating and Osculating Approximations. Anharmonic Fourier Analysis. An Example of Non-Final Determination of Coefficients

Chapter II: Introduction to Partial Differential Equations

§ 7 How The Simplest Partial Differential Equations Arise

§ 8 Elliptic, Hyperbolic and Parabolic Type. Theory of Characteristics

§ 9 Differences Among Hyperbolic, Elliptic, and Parabolic Differential Equations. The Analytic Character of Their Solutions

§10 Green’s Theorem and Green’s Function for Linear, and, in Particular, for Elliptic Differential Equations

§11 Riemann’s Integration of the Hyperbolic Differential Equation

§12 Green’s Theorem in Heat Conduction. The Principal Solution of Heat Conduction

Chapter III: Boundary Value Problems in Heat Conduction

§13 Heat Conductors Bounded on One Side

§14 The Problem of the Earth’s Temperature

§15 The Problem of a Ring-Shaped Heat Conductor

§16 Linear Heat Conductors Bounded on Both Ends

§ 17 Reflection in the Plane and in Space

§ 18 Uniqueness of Solution for Arbitrarily Shaped Heat Conductors

Chapter IV: Cylinder and Sphere Problems

§ 19 Bessel and Hankel Functions

§ 20 Heat Equalization in a Cylinder

§ 21 More about Bessel Functions

§ 22 Spherical Harmonics and Potential Theory

§ 23 Green’s Function of Potential Theory for the Sphere. Sphere and Circle Problems for Other Differential Equations

§ 24 More about Spherical Harmonics

Appendix I Reflection on a Circular-Cylindrical or Spherical Mirror

Appendix II Additions to the Riemann Problem of Sound Waves in §11

Chapter V: Eigenfunctions and Eigen Values

§ 25 Eigen Values and Eigenfunctions of the Vibrating Membrane

§ 26 General Remarks Concerning the Boundary Value Problems of Acoustics and of Heat Conduction

§ 27 Free and Forced Oscillations. Green’s Function for the Wave Equation

§ 28 Infinite Domains and Continuous Spectra of Eigen Values. The Condition of Radiation

§ 29 The Eigen Value Spectrum of Wave Mechanics. Balmer’s Term

§ 30. Green’s Function for the Wave Mechanical Scattering Problem. The Rutherford Formula of Nuclear Physics

Appendix I Normalization of yhe Eigenfunctions in the Infinite Domain

Appendix II A New Method for the Solution of the Exterior Boundary Value Problem of the Wave Equation Presented for the Special Case of the Sphere

Appendix III The Wave Mechanical Eigenfunctions of the Scattering Problem in Parabolic Coordinates

Appendix IV Plane And Spherical Waves in Unlimited Space of an Arbitrary Number of Dimensions

Chapter VI: Problems of Radio

§31 The Hertz Dipole in a Homogeneous Medium Over a Completely Conductive Earth

§32 The Vertical Antenna Over an Arbitrary Earth

§33 The Horizontal Antenna Over an Arbitrary Earth

§ 34 Errors in Range Finding for an Electric Horizontal Antenna

§ 35 § 35. The Magnetic or Frame Antenna

§ 36 Radiation Energy and Earth Absorption

Appendix Radio Waves On The Spherical Earth

Exercises For Chapter I

Exercises For Chapter II

Exercises For Chapter III

Exercises For Chapter IV

Exercises For Chapter V

Exercises For Chapter VI

Hints for Solving the Exercises

Index