Practical Applied Mathematics: Modelling, Analysis, Approximation

by Sam Howison

• ISBN13: 9780521603690
• ISBN10: 0521603692
• Format: Paperback
• Copyright: 2005-04-11
• Publisher: Cambridge University Press

Summary

Drawing from a wide variety of mathematical subjects, this book aims to show how mathematics is realised in practice in the everyday world. Dozens of applications are used to show that applied mathematics is much more than a series of academic calculations. Mathematical topics covered include distributions, ordinary and partial differential equations, and asymptotic methods as well as basics of modelling. The range of applications is similarly varied, from the modelling of hair to piano tuning, egg incubation and traffic flow. The style is informal but not superficial. In addition, the text is supplemented by a large number of exercises and sideline discussions, assisting the reader's grasp of the material. Used either in the classroom by upper-undergraduate students, or as extra reading for any applied mathematician, this book illustrates how the reader's knowledge can be used to describe the world around them.

Table of Contents

Preface	xi
Part I Modelling techniques
1 The basics of modelling	3	(12)
1.1 Introduction	3	(1)
1.2 What do we mean by a model?	4	(2)
1.3 Principles of modelling: physical laws and constitutive relations	6	(5)
1.4 Conservation laws	11	(1)
1.5 General remarks	12	(1)
1.6 Exercises	12	(3)
2 Units, dimensions and dimensional analysis	15	(13)
2.1 Introduction	15	(1)
2.2 Units and dimensions	16	(2)
2.3 Electric fields and electrostatics	18	(2)
2.4 Sources and further reading	20	(1)
2.5 Exercises	20	(8)
3 Nondimensionalisation	28	(22)
3.1 Nondimensionalisation and dimensional parameters	28	(8)
3.2 The Navier-Stokes equations and Reynolds numbers	36	(4)
3.3 Buckingham's Pi-theorem	40	(2)
3.4 Sources and further reading	42	(1)
3.5 Exercises	42	(8)
4 Case studies: hair modelling and cable laying	50	(13)
4.1 The Euler-Bernoulli model for a beam	50	(2)
4.2 Hair modelling	52	(1)
4.3 Undersea cable laying	53	(1)
4.4 Modelling and analysis	54	(4)
4.5 Sources and further reading	58	(1)
4.6 Exercises	58	(5)
5 Case study: the thermistor (1)	63	(9)
5.1 Heat and current flow in thermistors	63	(3)
5.2 Nondimensionalisation	66	(1)
5.3 A thermistor in a circuit	67	(2)
5.4 Sources and further reading	69	(1)
5.5 Exercises	69	(3)
6 Case study: electrostatic painting	72	(9)
6.1 Electrostatic painting	72	(1)
6.2 Field equations	73	(2)
6.3 Boundary conditions	75	(1)
6.4 Nondimensionalisation	76	(1)
6.5 Sources and further reading	77	(1)
6.6 Exercises	77	(4)
Part II Analytical techniques
7 Partial differential equations	81	(23)
7.1 First-order quasilinear partial differential equations: theory	81	(4)
7.2 Example: Poisson processes	85	(2)
7.3 Shocks	87	(3)
7.4 Fully nonlinear equations: Charpitt's method	90	(4)
7.5 Second-order linear equations in two variables	94	(3)
7.6 Further reading	97	(1)
7.7 Exercises	97	(7)
8 Case study: traffic modelling	104	(10)
8.1 Simple models for traffic flow	104	(3)
8.2 Traffic jams and other discontinuous solutions	107	(3)
8.3 More sophisticated models	110	(1)
8.4 Sources and further reading	111	(1)
8.5 Exercises	111	(3)
9 The delta function and other distributions	114	(26)
9.1 Introduction	114	(1)
9.2 A point force on a stretched string; impulses	115	(2)
9.3 Informal definition of the delta and Heaviside functions	117	(3)
9.4 Examples	120	(2)
9.5 Balancing singularities	122	(3)
9.6 Green's functions	125	(9)
9.7 Sources and further reading	134	(1)
9.8 Exercises	134	(6)
10 Theory of distributions	140	(17)
10.1 Test functions	140	(1)
10.2 The action of a test function	141	(1)
10.3 Definition of a distribution	142	(1)
10.4 Further properties of distributions	143	(1)
10.5 The derivative of a distribution	143	(2)
10.6 Extensions of the theory of distributions	145	(3)
10.7 Sources and further reading	148	(1)
10.8 Exercises	148	(9)
11 Case study: the pantograph	157	(16)
11.1 What is a pantograph?	157	(1)
11.2 The model	158	(2)
11.3 Impulsive attachment for an undamped pantograph	160	(2)
11.4 Solution near a support	162	(2)
11.5 Solution for a whole span	164	(3)
11.6 Sources and further reading	167	(1)
11.7 Exercises	167	(6)
Part III Asymptotic techniques
12 Asymptotic expansions	173	(10)
12.1 Introduction	173	(2)
12.2 Order notation	175	(3)
12.3 Convergence and divergence	178	(2)
12.4 Sources and further reading	180	(1)
12.5 Exercises	180	(3)
13 Regular perturbation expansions	183	(17)
13.1 Introduction	183	(1)
13.2 Example: stability of a spacecraft in orbit	184	(1)
13.3 Linear stability	185	(3)
13.4 Example: the pendulum	188	(2)
13.5 Small perturbations of a boundary	190	(4)
13.6 Caveat expandator	194	(1)
13.7 Exercises	195	(5)
14 Case study: electrostatic painting (2)	200	(5)
14.1 Small parameters in the electropaint model	200	(2)
14.2 Exercises	202	(3)
15 Case study: piano tuning	205	(11)
15.1 The notes of a piano: the tonal system of Western music	205	(3)
15.2 Tuning an ideal piano	208	(1)
15.3 A real piano	209	(2)
15.4 Sources and further reading	211	(1)
15.5 Exercises	211	(5)
16 Boundary layers	216	(19)
16.1 Introduction	216	(1)
16.2 Functions with boundary layers; matching	216	(5)
16.3 Examples from ordinary differential equations	221	(3)
16.4 Case study: cable laying	224	(1)
16.5 Examples for partial differential equations	225	(5)
16.6 Exercises	230	(5)
17 Case study: the thermistor (2)	235	(5)
17.1 Strongly temperature-dependent conductivity	235	(3)
17.2 Exercises	238	(2)
18 'Lubrication theory' analysis in long thin domains	240	(15)
18.1 'Lubrication theory' approximations: slender geometries	240	(1)
18.2 Heat flow in a bar of variable cross-section	241	(3)
18.3 Heat flow in a long thin domain with cooling	244	(2)
18.4 Advection-diffusion in a long thin domain	246	(3)
18.5 Exercises	249	(6)
19 Case study: continuous casting of steel	255	(8)
19.1 Continuous casting of steel	255	(5)
19.2 Exercises	260	(3)
20 Lubrication theory for fluids	263	(22)
20.1 Thin fluid layers: classical lubrication theory	263	(2)
20.2 Thin viscous fluid sheets on solid substrates	265	(6)
20.3 Thin fluid sheets and fibres	271	(4)
20.4 Further reading	275	(1)
20.5 Exercises	275	(10)
21 Case study: turning of eggs during incubation	285	(7)
21.1 Incubating eggs	285	(1)
21.2 Modelling	286	(4)
21.3 Exercises	290	(2)
22 Multiple scales and other methods for nonlinear oscillators	292	(11)
22.1 The Poincaré-Linstedt method	292	(2)
22.2 The method of multiple scales	294	(3)
22.3 Relaxation oscillations	297	(2)
22.4 Exercises	299	(4)
23 Ray theory and the WKB method	303	(15)
23.1 Introduction	303	(1)
23.2 Classical WKB theory	304	(2)
23.3 Geometric optics and ray theory: why do we say light travels in straight lines?	306	(5)
23.4 Kelvin's ship waves	311	(3)
23.5 Exercises	314	(4)
References	318	(3)
Index	321