本书为“天元基金影印数学丛书”之一。全书内容分为三部分：建模，讲述了建模的一些原则(包括物理定律、本构关系及守恒定律)，量纲分析(包括Bucklngham的Pi-定理)等；分析技巧，讲述了偏微分方程和广义函数的基础知识；渐近分析，讲述了渐近展开的基本概念，正则摄动展开，边界层和多重尺度法等。每部分均介绍了一些生动有趣的例子。

本书是作者总结自己与工程界、科学界相互配合的研究工作中遇到的问题及解决问题的方法而写成的，偏重于宏观范围的物理世界的问题。本书内容分为三部分：建模，讲述了建模的一些原则(包括物理定律、本构关系及守恒定律)，量纲分析(包括Bucklngham的Pi-定理)等；分析技巧，讲述了偏微分方程和广义函数的基础知识；渐近分析，讲述了渐近展开的基本概念，正则摄动展开，边界层和多重尺度法等。每部分均介绍了一些生动有趣的例子。

本书可作为高等学校本科高年级和研究生课程教材。对于具有微积分、向量分析、质点动力学初步、无粘流体力学和偏微分方程初步知识的读者，也是一本很好的建模理论的教材。

Preface

Part Ⅰ Modelling techniques

1 The basics of modelling

1.1 Introduction

1.2 What do we mean by a model?

1.3 Principles of modelling: physical laws and constitutive relations

1.4 Conservation laws

1.5 General remarks

1.6 Exercises

2 Units, dimensions and dimensional analysis

2.1 Introduction

2.2 Units and dimensions

2.3 Electric fields and electrostatics

2.4 Sources and further reading

2.5 Exercises

3 Nondimensionalisation

3.1 Nondimensionalisation and dimensionless parameters

3.2 The Navier-Stokes equations and Reynolds numbers

3.3 Buckingham's Pi-theorem

3.4 Sources and further reading

3.5 Exercises

4 Case studies: hair modelling and cable laying

4.1 The Euler-Bernoulli model for a beam

4.2 Hair modelling

4.3 Undersea cable laying

4.4 Modelling and analysis

4.5 Sources and further reading

4.6 Exercise

5 Case study: the thermistor (1)

5.1 Heat and current flow in thermistors

5.2 Nondimensionalisation

5.3 A thermistor in a circuit

5.4 Sources and further reading

5.5 Exercises

6 Case study: electrostatic painting

6.1 Electrostatic painting

6.2 Field equations

6.3 Boundary conditions

6.4 Nondimensionalisation

6.5 Sources and further reading

6.6 Exercises

Part Ⅱ Analytical techniques

7 Partial differential equations

7.1 First-order quasilinear partial differential equations: theory

7.2 Example: Poisson processes

7.3 Shocks

7.4 Fully nonlinear equations: Charpit's method

7.5 Second-order linear equations in two variables

7.6 Further reading

7.7 Exercises

8 Case study: traffic modelling

8.1 Simple models for traffic flow

8.2 Traffic jams and other discontinuous solutions

8.3 More sophisticated models

8.4 Sources and further reading

8.5 Exercises

9 The delta function and other distributions

9.1 Introduction

9.2 A point force on a stretched string; impulses

9.3 Informal definition of the delta and Heaviside functions

9.4 Examples

9.5 Balancing singularities

9.6 Green's functions

9.7 Sources and further reading

9.8 Exercises

10 Theory of distributions

10.1 Test functions

10.2 The action of a test function

10.3 Definition of a distribution

10.4 Further properties of distributions

10.5 The derivative of a distribution

10.6 Extensions of the theory of distributions

10.7 Sources and further reading

10.8 Exercises

11 Case study: the pantograph

11.1 What is a pantograph?

11.2 The model

11.3 Impulsive attachment for an undamped pantograph

11.4 Solution near a support

11.5 Solution for a whole span

11.6 Sources and further reading

11.7 Exercises

Part Ⅲ Asymptotic techniques

12 Asymptotic expansions

12.1 Introduction

12.2 Order notation

12.3 Convergence and divergence

12.4 Further reading

12.5 Exercises

13 Regular perturbation expansions

13.1 Introduction

13.2 Example: stability of a spacecraft in orbit

13.3 Linear stability

13.4 Example: the pendulum

13.5 Small perturbations of a boundary

13.6 Caveat expandator

13.7 Exercises

14 Case study: electrostatic painting (2)

14.1 Small parameters in the electropaint model

14.2 Exercises

15 Case study: piano tuning

15.1 The notes of a piano: the tonal system of Western music

15.2 Tuning an ideal piano

15.3 A real piano

15.4 Sources and further reading

15.5 Exercises

16 Boundary layers

16.1 Introduction

16.2 Functions with boundary layers; matching

16.3 Examples from ordinary differential equations

16.4 Case study: cable laying

16.5 Examples for partial differential equations

16.6 Exercises

17 Case study: the thermistor (2)

17.1 Strongly temperature-dependent conductivity

17.2 Exercises

18 Lubrication theory' analysis in long thin domains

18.1 Lubrication theory' approximations: slender geometries

18.2 Heat flow in a bar of variable cross-section

18.3 Heat flow in a long thin domain with cooling

18.4 Advection-diffusion in a long thin domain

18.5 Exercises

19 Case study: continuous casting of steel

19.1 Continuous casting of steel

19.2 Exercises

20 Lubrication theory for fluids

20.1 Thin fluid layers: classical lubrication theory

20.2 Thin viscous fluid sheets on solid substrates

20.3 Thin fluid sheets and fibres

20.4 Further reading

20.5 Exercises

21 Case study: turning of eggs during incubation

21.1 Incubating eggs

21.2 Modelling

21.3 Exercises

22 Multiple scales and other methods for nonlinear oscillators

22.1 The Poincare-Linstedt method

22.2 The method of multiple scales

22.3 Relaxation oscillations

22.4 Exercises

23 Ray theory and the WKB method

23.1 Introduction

23.2 Classical WKB theory

23.3 Geometric optics and ray theory: why do we say light travels in straight lines?

23.4 Kelvin's ship waves

23.5 Exercises

References

Index