PRACTICAL MATHEMATICS
CONTENTS PART I MODELLING TECHNIQUES THE BASICS OF MODELLING INTRODUCTION WHAT DO WE MEAN BY A MODEL? PRINCIPLES OF MODELLING: PHYSICAL LAWS AND CONSTITUTIVE RELATIONS CONSERVATION LAWS GENERAL REMARKS EXERCISES UNITS, DIMENSIONS AND DIMENSIONAL ANALYSIS UNITS AND DIMENSIONS ELECTRIC ?ELDS AND ELECTROSTATICS SOURCES AND FURTHER READING NONDIMENSIONALISATION NONDIMENSIONALISATION AND DIMENSIONAL PARAMETERS THE NAVIER–STOKES EQUATIONS AND REYNOLDS NUMBERS BUCKINGHAM’S PI-THEOREM CASE STUDIES: HAIR MODELLING AND CABLE LAYING THE EULER–BERNOULLI MODEL FOR A BEAM HAIR MODELLING UNDERSEA CABLE LAYING MODELLING AND ANALYSIS CASE STUDY: THE THERMISTOR (1) HEAT AND CURRENT ?OW IN THERMISTORS A THERMISTOR IN A CIRCUIT CASE STUDY: ELECTROSTATIC PAINTING ELECTROSTATIC PAINTING FIELD EQUATIONS BOUNDARY CONDITIONS PART II ANALYTICAL TECHNIQUES PARTIAL DIFFERENTIAL EQUATIONS FIRST-ORDER QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS: THEORY EXAMPLE: POISSON PROCESSES SHOCKS FULLY NONLINEAR EQUATIONS: CHARPITT’S METHOD SECOND-ORDER LINEAR EQUATIONS IN TWO VARIABLES FURTHER READING CASE STUDY: TRAF?C MODELLING SIMPLE MODELS FOR TRAF?C ?OW TRAF?C JAMS AND OTHER DISCONTINUOUS SOLUTIONS MORE SOPHISTICATED MODELS THE DELTA FUNCTION AND OTHER DISTRIBUTIONS A POINT FORCE ON A STRETCHED STRING; IMPULSES INFORMAL DE?NITION OF THE DELTA AND HEAVISIDE FUNCTIONS EXAMPLES BALANCING SINGULARITIES GREEN’S FUNCTIONS CONTENTS THEORY OF DISTRIBUTIONS 10 1 TEST FUNCTIONS 10 2 THE ACTION OF A TEST FUNCTION 10 3 DE?NITION 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 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 ASYMPTOTIC EXPANSIONS 12 1 INTRODUCTION 12 2 ORDER NOTATION 12 3 CONVERGENCE AND DIVERGENCE 12 4 SOURCES AND FURTHER READING 12 5 EXERCISES 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 CASE STUDY: ELECTROSTATIC PAINTING (2) 14 1 SMALL PARAMETERS IN THE ELECTROPAINT MODEL 14 2 EXERCISES CASE STUDY: PIANO TUNING 15 1 THE NOTES OF A PIANO: THE TONAL SYSTEM OF WESTERN MUSIC CONTENTS 15 2 TUNING AN IDEAL PIANO 15 3 A REAL PIANO 15 4 SOURCES AND FURTHER READING 15 5 EXERCISES 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 CASE STUDY: THE THERMISTOR (2) 17 1 STRONGLY TEMPERATURE-DEPENDENT CONDUCTIVITY 17 2 EXERCISES ‘LUBRICATION THEORY’ ANALYSIS IN LONG THIN DOMAINS 18 1 ‘LUBRICATION THEORY’ APPROXIMATIONS: SLENDER GEOMETRIES 18 2 HEAT ?OW IN A BAR OF VARIABLE CROSS-SECTION 18 3 HEAT ?OW IN A LONG THIN DOMAIN WITH COOLING 18 4 ADVECTION–DIFFUSION IN A LONG THIN DOMAIN 18 5 EXERCISES CASE STUDY: CONTINUOUS CASTING OF STEEL 19 1 CONTINUOUS CASTING OF STEEL 19 2 EXERCISES LUBRICATION THEORY FOR ?UIDS 20 1 THIN ?UID LAYERS: CLASSICAL LUBRICATION THEORY 20 2 THIN VISCOUS ?UID SHEETS ON SOLID SUBSTRATES 20 3 THIN ?UID SHEETS AND ?BRES 20 4 FURTHER READING 20 5 EXERCISES CASE STUDY: TURNING OF EGGS DURING INCUBATION 21 1 INCUBATING EGGS 21 2 MODELLING 21 3 EXERCISES MULTIPLE SCALES AND OTHER METHODS FOR NONLINEAR OSCILLATORS 22 1 THE POINCAR´E–LINSTEDT METHOD 22 2 THE METHOD OF MULTIPLE SCALES CONTENTS 22 3 RELAXATION OSCILLATIONS 22 4 EXERCISES 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