Fourier Series and Partial Differential Equations Lecture Notes

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Preface These lecture notes are designed to accompany the first year course “Fourier Series and Partial Differential Equations” and are taken largely from notes originally written by Dr Yves Capdeboscq, Dr Alan Day and Dr Janet Dyson. The first part of this course of lectures introduces Fourier series, concentrating on their practical application rather than proofs of convergence. We will then discuss how the heat equation, wave equation and Laplace’s equation arise in physical models. In each case we will explore basic techniques for solving the equations in several independent variables, and elementary uniqueness theorems. Reading material Fourier series. • D. W. Jordan and P. Smith, Mathematical Techniques (Oxford University Press, 3rd Edition, 2003), Chapter 26. • E. Kreyszig, Advanced Engineering Mathematics (Wiley, 8th Edition, 1999), Chapter 10. • W. A. Strauss, Partial Differential Equations: An Introduction (Wiley, 1st Edition, 1992), Chapter 5. Heat equation. • G. F. Carrier and C. E. Pearson, Partial Differential Equations: Theory and Technique (Academic Press, 2nd Edition, 1998), Chapter 1. • W. A. Strauss, Partial Differential Equations: An Introduction (Wiley, 1st Edition, 1992), Chapters 1–4. • E. Kreyszig, Advanced Engineering Mathematics (Wiley, 8th Edition, 1999), Chapter 12. Wave equation. • G. F. Carrier and C. E. Pearson, Partial Differential Equations: Theory and Technique (Academic Press, 2nd Edition, 1998), Chapter 3. • W. A. Strauss, Partial Differential Equations: An Introduction (Wiley, 1st Edition, 1992), Chapters 1–4. 4
CONTENTS 5 • E. Kreyszig, Advanced Engineering Mathematics (Wiley, 8th Edition, 1999), Chapter 12. Laplace’s equation. • G. F. Carrier and C. E. Pearson, Partial Differential Equations: Theory and Technique (Academic Press, 2nd Edition, 1998), Chapter 4. • W. A. Strauss, Partial Differential Equations: An Introduction (Wiley, 1st Edition, 1992), Chapter 6.
Page 1 and 2: Fourier Series and Partial Differen
Page 3: CONTENTS 3 3.7 Other boundary condi
Page 7 and 8: Chapter 1. Introduction 7 IVP: y
Page 9 and 10: Chapter 2 Fourier series In the fol
Page 11 and 12: Chapter 2. Fourier series 11 Defini
Page 13 and 14: Chapter 2. Fourier series 13 we mus
Page 15 and 16: Chapter 2. Fourier series 15 for ev
Page 17 and 18: Chapter 2. Fourier series 17 (a) s0
Page 19 and 20: Chapter 2. Fourier series 19 2.3.1
Page 21 and 22: Chapter 2. Fourier series 21 Exampl
Page 23 and 24: Chapter 2. Fourier series 23 Note.
Page 25 and 26: Chapter 3 The heat equation In this
Page 27 and 28: Chapter 3. The heat equation 27 whe
Page 29 and 30: Chapter 3. The heat equation 29 Fro
Page 31 and 32: Chapter 3. The heat equation 31 3.6
Page 33 and 34: Chapter 4 The wave equation In this
Page 35 and 36: Chapter 4. The wave equation 35 whe
Page 37 and 38: Chapter 4. The wave equation 37 Fig
Page 39 and 40: Chapter 4. The wave equation 39 Thu
Page 41 and 42: Chapter 4. The wave equation 41 Thu
Page 43 and 44: Chapter 4. The wave equation 43 Exa
Page 45 and 46: Chapter 4. The wave equation 45 4.8
Page 47 and 48: Chapter 4. The wave equation 47 In
Page 49 and 50: Chapter 5. Laplace’s equation in
Page 55 and 56:
Chapter 5. Laplace’s equation in
Page 57 and 58:
Chapter 5. Laplace’s equation in
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Fourier Series and Partial Differential Equations Lecture Notes

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