Partial Differential Equations - Mathematical Techniques for Engineers

Preface

Contents

Part I Background

1 Vector Fields and Ordinary Differential Equations

1.1 Introduction

1.2 Curves and Surfaces in mathbbRn

1.2.1 Cartesian Products, Affine Spaces

1.2.2 Curves in mathbbRn

1.2.3 Surfaces in mathbbR3

1.3 The Divergence Theorem

1.3.1 The Divergence of a Vector Field

1.3.2 The Flux of a Vector Field over an Orientable Surface

1.3.3 Statement of the Theorem

1.3.4 A Particular Case

1.4 Ordinary Differential Equations

1.4.1 Vector Fields as Differential Equations

1.4.2 Geometry Versus Analysis

1.4.3 An Example

1.4.4 Autonomous and Non-autonomous Systems

1.4.5 Higher-Order Equations

1.4.6 First Integrals and Conserved Quantities

1.4.7 Existence and Uniqueness

1.4.8 Food for Thought

References

2 Partial Differential Equations in Engineering

2.1 Introduction

2.2 What is a Partial Differential Equation?

2.3 Balance Laws

2.3.1 The Generic Balance Equation

2.3.2 The Case of Only One Spatial Dimension

2.3.3 The Need for Constitutive Laws

2.4 Examples of PDEs in Engineering

2.4.1 Traffic Flow

2.4.2 Diffusion

2.4.3 Longitudinal Waves in an Elastic Bar

2.4.4 Solitons

2.4.5 Time-Independent Phenomena

2.4.6 Continuum Mechanics

References

Part II The First-Order Equation

3 The Single First-Order Quasi-linear PDE

3.1 Introduction

3.2 Quasi-linear Equation in Two Independent Variables

3.3 Building Solutions from Characteristics

3.3.1 A Fundamental Lemma

3.3.2 Corollaries of the Fundamental Lemma

3.3.3 The Cauchy Problem

3.3.4 What Else Can Go Wrong?

3.4 Particular Cases and Examples

3.4.1 Homogeneous Linear Equation

3.4.2 Non-homogeneous Linear Equation

3.4.3 Quasi-linear Equation

3.5 A Computer Program

References

4 Shock Waves

4.1 The Way Out

4.2 Generalized Solutions

4.3 A Detailed Example

4.4 Discontinuous Initial Conditions

4.4.1 Shock Waves

4.4.2 Rarefaction Waves

References

5 The Genuinely Nonlinear First-Order Equation

5.1 Introduction

5.2 The Monge Cone Field

5.3 The Characteristic Directions

5.4 Recapitulation

5.5 The Cauchy Problem

5.6 An Example

5.7 More Than Two Independent Variables

5.7.1 Quasi-linear Equations

5.7.2 Non-linear Equations

5.8 Application to Hamiltonian Systems

5.8.1 Hamiltonian Systems

5.8.2 Reduced Form of a First-Order PDE

5.8.3 The Hamilton--Jacobi Equation

5.8.4 An Example

References

Part III Classification of Equations and Systems

6 The Second-Order Quasi-linear Equation

6.1 Introduction

6.2 The First-Order PDE Revisited

6.3 The Second-Order Case

6.4 Propagation of Weak Singularities

6.4.1 Hadamard's Lemma and Its Consequences

6.4.2 Weak Singularities

6.4.3 Growth and Decay

6.5 Normal Forms

References

7 Systems of Equations

7.1 Systems of First-Order Equations

7.1.1 Characteristic Directions

7.1.2 Weak Singularities

7.1.3 Strong Singularities in Linear Systems

7.1.4 An Application to the Theory of Beams

7.1.5 Systems with Several Independent Variables

7.2 Systems of Second-Order Equations

7.2.1 Characteristic Manifolds

7.2.2 Variation of the Wave Amplitude

7.2.3 The Timoshenko Beam Revisited

7.2.4 Air Acoustics

7.2.5 Elastic Waves

References

Part IV Paradigmatic Equations

8 The One-Dimensional Wave Equation

8.1 The Vibrating String

8.2 Hyperbolicity and Characteristics

8.3 The d'Alembert Solution

8.4 The Infinite String

8.5 The Semi-infinite String

8.5.1 D'Alembert Solution

8.5.2 Interpretation in Terms of Characteristics

8.5.3 Extension of Initial Data

8.6 The Finite String

8.6.1 Solution

8.6.2 Uniqueness and Stability

8.6.3 Time Periodicity

8.7 Moving Boundaries and Growth

8.8 Controlling the Slinky?

8.9 Source Terms and Duhamel's Principle

References

9 Standing Waves and Separation of Variables

9.1 Introduction

9.2 A Short Review of the Discrete Case

9.3 Shape-Preserving Motions of the Vibrating String

9.4 Solving Initial-Boundary Value Problems by Separation of Variables

9.5 Shape-Preserving Motions of More General Continuous Systems

9.5.1 String with Variable Properties

9.5.2 Beam Vibrations

9.5.3 The Vibrating Membrane

References

10 The Diffusion Equation

10.1 Physical Considerations

10.1.1 Diffusion of a Pollutant

10.1.2 Conduction of Heat

10.2 General Remarks on the Diffusion Equation

10.3 Separating Variables

10.4 The Maximum--Minimum Theorem and Its Consequences

10.5 The Finite Rod

10.6 Non-homogeneous Problems

10.7 The Infinite Rod

10.8 The Fourier Series and the Fourier Integral

10.9 Solution of the Cauchy Problem

10.10 Generalized Functions

10.11 Inhomogeneous Problems and Duhamel's Principle

References

11 The Laplace Equation

11.1 Introduction

11.2 Green's Theorem and the Dirichlet and Neumann Problems

11.3 The Maximum-Minimum Principle

11.4 The Fundamental Solutions

11.5 Green's Functions

11.6 The Mean-Value Theorem

11.7 Green's Function for the Circle and the Sphere

References

Index