Mathematics Applied to Deterministic Problems in the Natural Sciences

8/10/2019 Mathematics Applied to Deterministic Problems in the Natural Sciences

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Mathematics Applied

to Deterministic Problems

in the Natural Sciences

b d

C. C Lin

Massachusetts Institute of Technology

L A. Segel

Weizmarin Institute of Science

with material on elasticity by

G. H Handelman

Rensselaer Polytechnic Institute

S H J T L

Society forIndustrial and A pplied M athematics

ULBDarmstadt

Philadelphia

6595756


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Contents

PART A

AN OVERVIEW OF THE INTERACTION OF

MATHEMATICS AND NATURAL SCIENCE

C H A P T E R 1

W H A T Is AP PL IE D MATHEMAT ICS? 3

1.1. On the nature of applied mathematics 4

The scope, purpose, and practice of applied mathematics 5 / Applied

mathematics contrasted with pure mathematics 6

/

Applied mathematics

contrasted with theoretical science 7 /Applied mathematics in engineering 8 j

Theplan of the present volume 81 A preview of thefollowing volume 9 /

Concepts that unify applied mathematics 10

1.2 Introduction to the analysis of galactic structure 10

Physical laws governing galactic behavior 10/Building blocks of the

universe 11

/

Classification of galaxies 11

/

Composition of galaxies 13

/

Dynamics of stellar systems 14/ Distribution of stars across a galactic disk 161

Density wave theory of galactic spirals 19

1.3 Aggregation of slime mold amebae 22

Some facts about slime mold amebae 22

/

Formulation of a mathematical

model 24 j An exact solution: The uniform state 28 / Analysis of aggregation

onset as an instability 28

/

Interpretation of the analysis 30

A P P E N D I X

1.1 Some views on applied mathematics 31

On the nature of applied mathematics 31

/

On the relationships among pure

mathematics, applied mathematics, and theoretical science 32

/

On the teaching

and practice of applied mathematics 33

C H A P T E R 2

D E T E R M I N I S T I C SYSTEMS AND ORDINARY

D I F F E R E N T I A L

EQUATIONS 36

2.1 Planetary orbits 36

Kepler s laws 37

/

Law of universal gravitation 39

/

The inverse problem:

Orbits of planets and comets 39 j Planetary orbits according to the general

theory of relativity 41 / Comments on choice of methods 411N particles: A

deterministic system 42

/

Linearity 43

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xvi Contents

2 .2 Elements of perturbation theory, including Poincare s method for

periodic orbits 45

Perturbation theory: Elementary

considerations46

/

The

simple pendulum48/

Successive approximations

to

the

motion of the pendulum

50

/

P erturbation

series applied

to

the pendulum problem 51/Poin care's perturbation theory

53

/

Generalization

of the

Po incare

method 54

2 .3

A

system

of

or di na ry differential eq uati ons

57

Th e initial value problem: Statement of theo rems

57

/Proof of the u niqueness

theorem

60

Proof of the existence theorem

61

/

Continuous dependence

on a parameterorinitial conditions63/ Differentiability64/Exampleof

nonuniqueness66/M ethodoffinite differences 66/Further remarks on

the

relation betwe en pure and

applied

mathematics68

CHAPTER

3

RANDOM PROCESSES

AND

PARTIAL DIFFERENTIAL

EQUATIONS

71

3.1 Random walkinone dimension; Langcvin s equation 73

The one-dimensional random walk model 73 Explicit solution

74

/ Mean,

variance, and the gene rating function 75

/

Vse of a stochastic differential

equation

to

obtain Boltzmanris constant from observations ofBrow nian

motion

78

3.2 Asym ptotic series, Laplace's meth od , gamm a function, Stirling's

formula

80

An example: Asymptotic exp ansion via parts in tegration 82 Definitions

in

the theory of asymp totic

expansions83

/Laplace's method 84/Development

of the asymptotic Stirling series for the gamma function 86/Justification

of term-by-term integration

89

3.3

Adifference equation anditslimit 91

A differenc e equ ation for the probability function

91

/Approximation of the

difference equation

by a

differential equation

92/S olution of the differential

equ ation for the probability distribution function 93/Further examination

of the limiting process

94

/R eflecting and absorbing b arriers

95

/C oagulation:

An application

of

first passage theory97

3.4 Further consid eration s pertinent

to the

relationship between

probability

and

partial differential equations

99

M ore o n the Affusion equation and

its

con nection w ith random walk

too

/

Sup erposition of fundamen tal solutions: Th e method of images 1 02/F irst

passage time

as

aflux 104

J

Ge neral initial value problem in

diffusion 104

/

How

twistingabeam can give information aboutDNAmolecules 106

/

Recu rrence

property

in

Brow nian m otion

108

APPENDIX

3.1 O and

osymbols

112


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Contents

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C H A P T E R 4

S U P E R P O S I T I O N , H E A T F L O W , A N D F O U R I E R A N A L Y S I S 114

4.1 Conduction of heat 115

Steady state heat conduction 116/ Differential

equ ation

for

one-dimensional

heat

conduction

117/Initial

boundary

value problem for

one-dimensional

heat

conduction

118/

Past,present,

and future 119I Heat

conduction

in

three-dimensional

space 120/Proof of the

un iqueness

theorem 122jT he

maximum p rinciple

123

/Solutionb ythe metho d of separation of variables

123I

Interpretation; dimensionless representation 1271 Estimate of the time required

to

diffusea

given distance

129

4. 2 Fourier's theore m

131

Summ ation of the

Fourier

sine series 131

/

Proof of the lemmas

1341A

formal

transformation 135j

Fourier

seriesinthe full range

135

/ Summation of Fou rier

series 136fH alf-range series 137

4.3 On the nature of Fourier series 137

Fourierseries for the constant func tion 138

/

F ourier series for the linear

function 139f

Fourier

series for the quadratic fun ction 139/Integration and

differentiation o fFourierseries 139

/

Gibbs phenomenon 141\ Approximation

with least squared error 144/BesseVs inequality and ParsevaVs theorem 146/

Riesz-Fischer theorem 147/A pplications of ParsevaVs theorem 147

C H A P T E R 5

F U R T H E R D E V E L O P M E N T S I N F O U R I E R A N A L Y S I S 150

5.1 Other aspects of heat conduction 150

Variation

of temperature

underground

1 50/N umerical integration of the heat

equation 153/ Heat

conduction

in a

nonuniform

medium155

5.2 Sturm -Liouv ille systems

159

Properties of

eigenvalues

and

eigenfunctions

160

/

Orthogonality and

normalization 161/Expansioninterms of eigenfunctions 162/ Asymptotic

approximationsto

eigenfunctions

and

eigenvalues

163/ Other methodsof

calculating

eigenfunctions

and

eigenvalues

165

5.3 Brief introd uction

t o

Fourier transform

167

Fourier

transform formu las and the Fo urier identity 167/ Solution of the heat

equationb yF ourier transform 170

5.4 G eneralized harmon ic analysis

171

Remarks on functions that cann ot

be

analyzed

by

standard

Fourier

methods 172

/

Fourierseries an alysis of a truncated sinusoidal func tion 173

/

Fo urier integral analysis of a trunc ated sinusoidal fun ction 174/

Generalizationtostationary time sequences 176\A utocorrelation function and

the power spectrum 178/Verification of the cosine transform relation

between the pow er spectrum and the autocorrelation 179

/

Application 180


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Contents

PART

B

SO ME F U N D A M E N T A L P R O C E D U R E S

I L L U S T R A T E D

ON

O R D I N A R Y D I F F E R E N T I A L

E Q U A T I O N S

CHAPTER 6

SIMPLIFICATION, DIMENSIONAL ANALYSIS, AND SCALING 185

6.1 The basic simplification procedure

186

Il lustrations of the procedure 186/ Two chastening examples 187/ Conditioning

and se nsitivity

189

/Zeros of a function

190 I

Second order differential

equations

191 j

R ecommendations

194

62 Dimen sional analysis 195

Putting

a

differential e qu ation into dimen sionless form

195

/

N ondimen sionalization of a functional relationsh ip 198/ Useof scale models

200

I

Summ ary

203

6.3 Scaling

209

Definition of scaling 211/ Scaling the projectile problem 211} Orderof

magnitude213/ Scaling know n funct ions 214/ Orthodoxy 218 fScalingand

perturbation theory

221

/ Scalingunknownfunctions

222

C H P T E R

7

R E G U L R P E R T U R B T I O N T H E O R Y

5

7.1

The

series m ethod applied

to the

simp le pendulum

225

Preliminaries 226/ Series method 227/ Discussion o f resu lts so far 230/

Higher order terms

231

7.2 Project ile p rob lem solved by perturbat ion theory 233

Series method 233/Param etric differentiation 236/ Suc cessive approximations

(method of iteration) 238/ General remarks onregular perturbation theory240

C H A P T E R 8

I L L U S T R A T I O N OFT E C H N I Q U E SON A P H Y S I O L O G I C A L F L O W

P R O B L E M 244

8.1 Physical formulation and dimensional analysis of a model

for

standing gradient osmotically driven flow

244

Some ph ysiological facts 244/ Osmosis and theosmol247/ Factors that affect

standing gradient flow

2481

Dimensional analysis of a functional relationship

2491 Po ssibility of a scale model for standing gradient flow

253

%2

A

mathematical model

and its

dimensional analysis

253

Conservationoffluid mass254/ Conservation of solute mass254 f Boundary

conditions

255

Introduction of dimen sionless variables

257

Comparison

of

physical andmathematical approaches to dimensional analysis259


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ont nts

x

8.3 Obtainingthefinal scaled dimensionless formofthe mathematical

model 261

Scaling262/ Estimating the size of the dimensionless parameters 264/An

un successful regular perturbation calculation

265

JRelation between

parameters 265/ Final formulation 267

8 . 4 So lu t ion andinterpretat ion 268

A first ap prox imation to thesolution268 j Comparison with numerical

calculations269/ Interpretation: Physical meaning of the dimensionless

parameters 272 \ Final remarks 274

CHAPTER 9

INTRODUCTION TO SINGULAR PERTURBATION THEORY 277

9.1

Roots of polynomial equations

278

A simple problem 2781A more complicated problem 2 81/The useof scaling284

9 . 2 B o u n d a r y v a l u e p r o b l e m s

for

ordinary di f ferent ial equat ions

28 5

Examination of the exact solution

to a

model problem

285 j

F inding

an

approximate solution

by

singu lar perturbation methods

291

/ Matching

293

/

Further examples

295

C H A P T E R

10

S I N G U L A R

P E R T U R B A T I O N T H E O R Y A P P L I E D

TO A

P R O B L E M

I N B I O C H E M I C A L K I N E T I C S

302

10.1 Form ulation of an initial value problem for a oneenzyme one

substrate chemical reaction 302

The

law

of mass action

302

/ Enzyme catalysis

304

/ Scaling

and

final

formulation

306

10.2 Approximate solutionby singular perturbation methods 308

Michaelis-Menten kineticsas anouter solution308/ Innersolution308/A

uniform approximation 3TO/ Comments onresultsso far 311/ Higher

approximations

311

/

Fu rther analysis for large times

315

/

Fu rther discussion

of the approximate solutions317

C H A P T E R

11

T H R E E T E C H N I Q U E S A P P L I E DTO THE SI M P L E P E N D U L U M 321

11.1 Stabilityofnormal and inverted equilibrium of the pendulum 321

Determining stability of equilibrium 322 Discussion of results323

11 . 2

A

mul t ip l e sca le expans ion

325

Sub stitution of a two -scale series into the pendulu m equ ation 327

/

Solving

lowest order equations

328

/

H igher approximations; removing resonant

terms 328

/

Summaryanddiscussion330

11.3

The

p h a s e p l a n e

334

The

phase p ortrait of an un damped simple pendulum

335

/ Separatrices

337

/

Critical points

338

/ Limit cycles

339

/ Behavior of trajectories near critical

points 339


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xx Contents

PART C

INTRODUCTION TO THEORIES OF CONTINUOUS

FIELDS

C H A P T E R 12

L O N G I T U D I N A L MOTION

OF A BAR 349

12.1 Derivation of the governing equations 349

Geometry349/The material derivative and the Jacobian 352/ Conservation

of mass355/Forceand stress 358/Balance of linear momentum360/ Strain

and stress-strain relations 362/Initial and boundary conditions366/

Linearization

368

12.2 One-dimensional elastic wave propagation

376

The wave equation377/General solution of the wave equation377 j Physical

significance of the solution

378

Solutions

in

complex form 379

/

Analysis

of

sinusoidal waves 380

/

Effects of a discontinuity inproperties 381

12.3 Discontinuous solutions

388

Motion of the discontinuity surface390/Behavior of the discontinuity 396

12.4 Work, energy,

and

vibrations

401

Work and energy 401

/

Avibration problem 403fThe Rayleigh quotient 404

/

Properties of eigenvaluesandeigenfunctions 405\ Anexact solution when

properties areconstant406 fCharacterization of the lowest eigenvalueas the

minimum of the Rayleigh quotient406/Estimate of the lowest eigenvalue for

a wedge407

C H A P T E R

13

T H E

CONTINUOUS MEDIUM

412

13.1 The continuum model 413

Molecular averages 414/ Mass distribution functions416 fThe continuumas

an independent model 417

13.2 Kinematics

of

deformable media

418

Points and particles 419

/

Material and spatial descriptions4201 Streamlines

and particle paths 422/A simple kinematic boundary condition425

13.3 The material derivative

426

13.4 The Jacobian and

its

material derivative

429

APPENDIX

13.1

On the chain rule

of

partial differentiation

433

APPENDIX 13.2

The integral mean value theorem

435

A P P E N D I X 13.3 Similar regions 436


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Contents

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C H A P T E R

14

F I E L D EQUA TIONS OF CONTI NU UM MEC HAN ICS 440

14.1 Conservation of mass 440

Integral

method:

Arbitrary material region 441

/

Integral

method:

Arbitrary

spatial region 444

/

Small box method 444

/

Large box method 446

14.2 Balance of linear momentum 454

An integral form of Newton s second law 454 /Local stress equilibrium 456 j

Action and reaction 457 / The stress tensor 458 /Newton s second law in

differential equation form 461

14.3 Balance of angular moment um 465

Torque and angular momentum 465 I Polar fluids 466 ISymmetry of the stress

tensor 466

/

The

principle of local moment equilibrium 467

/

Local moment

equilibrium for a cube 468

14.4 Energy and entropy 470

Ideal gases 471 /Equilibrium thermodynamics 474 /Effects of inhomogeneity

and motion 475/Energy balance 476/Entropy, temperature, and pressure 477/

Internal energy and deformation rate 479 f Energy and entropy in fluids 480

14.5 On constitutive equations, covariance, and the continuum model 485

Recapitulation of field equations 485 / Introduction to constitutive

equations 486/ Theprinciple ofcovariance487 / Validity of classical continuum

mechanics 490

A P P E N D I X

14.1 Thermodynamics of spatially homogeneous

substances 491

Experiments with a piston 492

/

First law of equilibrium thermodynamics 494

/

Entropy 496

/

Second law of equilibrium thermodynamics 500

A P P E N D I X

14.2 Some historical remarks 501

C H A P T E R 15

I N V I S C I D

FLU ID FL OW 505

15.1 Stress in motionless and inviscid fluids 505

Molecular point of view 506/ Continuum point of view 506/Hydrostatics 508/

Inviscid fluids 510

15.2 Stability of a stratified fluid 515

Governing equations and their exact equilibrium solution 516/ Linearized

equations for the perturbations 518/ Characterization of the growth rate o

as an eigenvalue 521J Qualitative general deductions

525 /

Detailed results for

a particular stratification 525

/

Superposition of normal modes 528 j Nonlinear

effects 5301 Worked example: A model of viscous flow instability 531

15.3 Compression waves in gases 539

Inviscid isentropic flow of a perfect gas 540

/

Waves of small amplitudes 541 \

The speed of sound 542 j Spherical waves 543

/

Nonlinear waves in one

dimension 544

/

Shock waves 547


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Contents

15.4 Uniform flow past a circular cylinder 551

Formulation 551/ Solutionby separation of variables 553/ Interpretation of

solution555

A P P E N D I X

15.1 A proof of D'Alembert's paradox in the

three-dimensional case 560

A P P E N D I X 15.2 Polar and cylindrical coordinates 563

C H A P T E R

16

P O T E N T I A L T H E O R Y

566

16.1 Equations of Laplace and Poisson 566

G ravitational poten tial of discrete m ass distribu tions566/ Gravitational

potential

of

continuousmass distributions 567/Theorems concerning harmonic

funct ions570/ Integral representationfor thesolutiontoPoisson's equation

572

I

Uniqueness

573

16.2 G reen's func t ions 574

Green 's function for theDirichlet problem 574/ Representation of a harmonic

function using Green 's function

575/

Symmetry of the Green 's function

575/

Ex plicit formulas for simple region s 575

/

Widespread uti l ity of source, image,

and reciprocity concepts 577/ Green 's function for theNeumann problem 578/

Green 's func tion for theH elmholtz equation579

16.3 Diffraction ofacous t ic w aves by a h o l e 580

Formulation 580 j Selection of the appropriate Green 's function583/

Derivation of the diffraction integral 584

/

Ap proximate evaluation of th e

diffraction integral586

BlBUOGRAPHY 589

S U P P L E M E N T A R Y B I B L I O G R A P H Y

595

U P D A T E D M A T E R I A L

598

H I N T S A N D A N S W E R S

(for exercises marked with ) 600

A U T H O R S C I T E D

604

S U B JE C T I N D E X

605

Mathematics Applied to Deterministic Problems in the Natural Sciences

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