Partial Differential Equations in Physics (eBook)

Front Cover 1
Partial Differential Equations in Physics 4
Copyright Page 5
Foreword 6
Editor's Foreword 8
Table of Contents 10
Chapter 1. Fourier Series and Integrals 14
1. Fourier Series 14
2. Example of a Discontinuous Function.Gibbs’ Phenomenon and Non-Uniform Convergence 20
3. On the Convergence of Fourier Series 27
4. Passage to the Fourier Integral 30
5. Development by Spherical Harmonics 34
6. Generalizations: Oscillating and Osculating Approximations.Anhar-monic Fourier Analysis. An Example of Non-Final Determination of Coefficients 39
Chapter 2. Introduction to Partial Differential Equations 45
7. How the Simplest Partial Differential Equations Arise 45
8. Elliptic, Hyperbolic and Parabolic Type. Theory of Characteristics 49
9. Differences Among Hyperbolic, Elliptic, and Parabolic Differential Equations. The Analytic Character of Their Solutions 53
10. Green’s Theorem and Green’s Function for Linear, and, in Particular, for Elliptic Differential Equations 57
11. Riemann’s Integration of the Hyperbolic Differential Equation 65
12. Green’s Theorem in Heat Conduction . The Principal Solution of Heat Conduction 68
Chapter III. Boundray Value Problems in Heat Conduction 76
13. Heat Conductors Bounded on One Side 76
14. The Problem of the Earth’s Temperature 81
15. The Problem of a Ring-Shaped Heat Conductor 84
16. Linear Heat Conductors Bounded on Both Ends 87
17. Reflection in the Plane and in Space 92
18. Uniqueness of Solution for Arbitrarily Shaped Heat Conductors 95
Chapter IV. Cylinder and Sphere Problems 97
19. Bessel and Hankel Functions 97
20. Heat Equalization in a Cylinder 114
21. More About Bessc.1 Functions 120
22. Spherical Harmonics and Potential Theory 136
23. Green’s Function of Potential Theory for the Sphere. Sphere and Circle Problems for Other Differential Equations 147
24. More About Spherical Harmonies 156
Appendix I: Reflection on a Circular-Cylindrical or Spherical Mirror 172
Appendix II: Additions to the Riemann Problem of Sound Waves in Section 11 177
Chapter V. Eigenfunctions and Eigen Values 179
25. Eigen Values and Eigcnfunctions of the Vibrating Membrane 179
26. General Remarks Concerning the Boundary Value Problems of Acous- tics and of Heat Conduction 190
27. Free and Forccd Oscillations . Crecn’s Function for the Wave Equation 195
28. Infinite Domains and Continuous Spectra of Eigen Values. The Con- dition of Radiation 201
29. The Eigen Value Spectrum of Wave Mechanics . Balmer’s Term 213
30. Green’s Function for the Wave Mechanical Scattering Problem. The Rutherford Formula of Nuclear Physics 219
Appendix I: Normalization of the Eigenfunctions in the Infinite Domain 223
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 227
Appendix III: The Wave Mechanical Eigenfunctions of the Scattering Problem in Parabolic Coordinates 238
Appendix IV: Plane and Spherical Waves in Unlimited Space of an Arbitrary Number of Dimensions 240
Chapter VI. Problems of Radio 249
31. The Hertz Dipole in a Homogeneous Medium Over a Completely Con- ductive Earth 250
32. The Vertical Antenna Over an Arbitrary Earth 259
33. The Horizontal Antenna Over an Arbitrary Earth 270
34. Errors in Range Finding for an Electric Horizontal Antenna 278
35. The Magnetic or Frame Antenna 280
36. Radiation Energy and Earth Absorption 283
Appendix: Radio Waves on the Spherical Earth 292
Exercises for Chapter I 303
Exercises for Chapter II 304
Exercises for Chapter III 305
Exercises for Chapter IV 306
Exercises for Chapter V 308
Exercises for Chapter VI 309
Hints for Solving the Exercises 310
Index 344