Fourier Series and Partial Differential Equations Lecture Notes

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Contents Preface 4 1 Introduction 6 1.1 Initial and boundary value problems . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Initial-value problem (IVP) . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.2 Boundary-value problem (BVP) . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 The equations we shall study . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 The heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Fourier series 9 2.1 Periodic, even and odd functions . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.1 Properties of periodic functions . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 Odd and even functions . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Fourier series for functions of period 2π . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Question 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Sine and cosine series . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.3 Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Convergence of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.2 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Functions of any period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.1 Sine and cosine series . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3 The heat equation 25 3.1 Derivation in one space dimension . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Fourier’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Units and nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Heat conduction in a finite rod . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Initial and boundary value problem (IBVP) . . . . . . . . . . . . . . . . . . 28 3.4.1 Application of Fourier series . . . . . . . . . . . . . . . . . . . . . . . 29 3.5 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.6 Non-zero steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2
CONTENTS 3 3.7 Other boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 The wave equation 33 4.1 Derivation in one space dimension . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Units and nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Normal modes of vibration for a finite string . . . . . . . . . . . . . . . . . 35 4.4 Initial-and-boundary value problems for finite strings . . . . . . . . . . . . . 37 4.4.1 Application of Fourier series . . . . . . . . . . . . . . . . . . . . . . . 38 4.5 Normal modes for a weighted string . . . . . . . . . . . . . . . . . . . . . . 39 4.6 The general solution of the wave equation . . . . . . . . . . . . . . . . . . . 41 4.7 Uniqueness of an IBVP for a finite string . . . . . . . . . . . . . . . . . . . 43 4.8 Waves on infinite strings: D’Alembert’s formula . . . . . . . . . . . . . . . . 45 4.8.1 Characteristic diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Laplace’s equation in the plane 48 5.1 BVP in cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 BVP in polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 Application of Fourier series . . . . . . . . . . . . . . . . . . . . . . . 50 5.2.2 Poisson’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.3.1 Uniqueness for the Dirichlet problem . . . . . . . . . . . . . . . . . . 53 5.3.2 Uniqueness for the Neumann problem . . . . . . . . . . . . . . . . . 54 5.4 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Page 1: Fourier Series and Partial Differen
Page 5 and 6: CONTENTS 5 • E. Kreyszig, Advance
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 51 and 52: Chapter 5. Laplace’s equation in
Page 53 and 54:
Chapter 5. Laplace’s equation in
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