Mathematical Modeling:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2011
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| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XVI, 460 S. Ill., graph. Darst. |
| ISBN: | 9783642203107 |
Internformat
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| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2011 | |
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
1 DETERMINISTIC ANALYSIS OF OBSERVATIONS 1
1.1 MOTIVATION 2
1.2 DATA TRANSFORMATIONS: LINEAR MODELS 3
1.2.1 ENERGY CONSUMPTION 4
1.2.2 KEPLER' S THIRD LAW 5
1.2.3 VEHICULAR STOPPING DISTANCE 8
1.3 MODEL DEVELOPMENT: POLYNOMIAL MODELS 10
1.3.1 THE LAGRANGIAN FORM OF POLYNOMIALS 10
1.3.2 PROPERTIES OF POLYNOMIALS 12
1.3.3 POLYNOMIAL MODELS OF OBSERVATIONS 15
1.4 MODEL EVALUATION: POPULATION MODELING 16
1.4.1 LINEARIZED CONCEPTUAL MODELS 17
1.4.2 POLYNOMIAL MODELS 19
1.4.3 MODEL EVALUATION 20
1.5 THE ADVANTAGE OF MODELING: GLOBAL WARMING MODELING 22 1.5.1 THE
GREENHOUSE EFFECT 22
1.5.2 CO 2 CONCENTRATION AND GLOBAL TEMPERATURE MODELING 24 1.5.3 THE
ADVANTAGE OF MODELING 28
1.6 SUMMARY 31
1.7 EXERCISES 33
2 STOCHASTIC ANALYSIS OF OBSERVATIONS 39
2.1 MOTIVATION 40
2.2 MODEL ERRORS 41
2.2.1 MODEL ERRORS 41
2.2.2 THE CHEBYSHEV ERROR 43
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1010478990
DIGITALISIERT DURCH
IMAGE 2
XII CONTENTS
2.2.3 THE LEAST-ABSOLUTE-DEVIATIONS ERROR 2.2.4 THE LEAST-SQUARES ERROR
2.3 OPTIMAL LINEAR MODELS 2.3.1 MEANS, VARIANCES, AND CORRELATIONS
2.3.2 OPTIMAL LINEAR FUNCTIONS 2.3.3 EXAMPLES 2.4 OPTIMAL QUADRATIC
MODELS 2.4.1 OPTIMAL QUADRATIC FUNCTIONS
2.4.2 VEHICULAR STOPPING DISTANCE 2.4.3 CO 2 CONCENTRATIONS 2.5 OPTIMAL
POWER AND EXPONENTIAL MODELS 2.5.1 OPTIMAL POWER FUNCTIONS
2.5.2 OPTIMAL EXPONENTIAL FUNCTIONS 2.5.3 GLOBAL WARMING 2.6 SUMMARY 2.7
EXERCISES
3 DETERMINISTIC STATES
3.1 MOTIVATION 3.2 DIMENSIONAL ANALYSIS AND SIMILARITY 3.2.1
BUCKINGHAM'S THEOREM 3.2.2 DIMENSIONAL ANALYSIS
3.2.3 SIMILARITY 3.3 APPLICATIONS OF LOW-COMPLEXITY 3.3.1 VEHICULAR
STOPPING DISTANCE
3.3.2 KEPLER'S THIRD LAW 3.3.3 STOKES' LAW 3.4 APPLICATIONS OF
MEDIUM-COMPLEXITY: TIME MEASUREMENT 3.4.1 THE PENDULUM PERIOD
3.4.2 DIMENSIONAL ANALYSIS 3.4.3 COMPARISON WITH OTHER RESULTS 3.5
APPLICATIONS OF HIGH-COMPLEXITY: LIFT 3.5.1 THE PROBLEM
3.5.2 THE RANGE OF PARAMETER VARIATIONS 3.5.3 THE LIFT COEFFICIENT
CALCULATION 3.6 SUMMARY 3.7 EXERCISES
4 STOCHASTIC STATES
4.1 MOTIVATION
44 45 47 47
53 57 59 59 61 62
63 64 66 67
69 70
75
76 77 77
80 84 88 88
91 93 95 95 96 98 101 102 103 104 106 109
115
116
IMAGE 3
CONTENTS XIII
4.2 PROBABILITY DENSITY FUNCTIONS 117
4.2.1 PROBABILITY DENSITY FUNCTIONS 118
4.2.2 DELTA FUNCTIONS 121
4.2.3 AN EXAMPLE: THE UNIFORM PROBABILITY DENSITY FUNCTION 124 4.3
MODELS FOR PROBABILITY DENSITY FUNCTIONS 127
4.3.1 STATISTICALLY MOST-LIKELY PROBABILITY DENSITY FUNCTIONS 127 4.3.2
THE NORMAL PROBABILITY DENSITY FUNCTION 129
4.3.3 THE GAMMA PROBABILITY DENSITY FUNCTION 134
4.3.4 THE BETA PROBABILITY DENSITY FUNCTION 138
4.4 DATA ANALYSIS 141
4.4.1 CALCULATION OF STATISTICS 141
4.4.2 THE FIRST FUNDAMENTAL THEOREM OF PROBABILITY 145
4.4.3 THE SECOND FUNDAMENTAL THEOREM OF PROBABILITY 147 4.5 REAL
DISTRIBUTIONS 150
4.5.1 VELOCITY AND TEMPERATURE DATA 150
4.5.2 NORMAL PROBABILITY DENSITY FUNCTIONS 154
4.5.3 STATISTICALLY MOST-LIKELY PROBABILITY DENSITY FUNCTIONS 155 4.6
SUMMARY 158
4.7 EXERCISES 160
5 DETERMINISTIC CHANGES 165
5.1 MOTIVATION 166
5.2 LINEAR CHANGES 167
5.2.1 LINEAR FIRST-ORDER DIFFERENCE EQUATIONS 167
5.2.2 INTEREST 169
5.2.3 LOAN REPAYMENTS 173
5.3 LINEAR CHANGES WITH DELAYS 175
5.3.1 LINEAR SECOND-ORDER DIFFERENCE EQUATIONS 175
5.3.2 SOLUTION FEATURES 178
5.3.3 NATIONAL INCOME 181
5.3.4 RED BLOOD CELL PRODUCTION 183
5.4 NONLINEAR CHANGES 186
5.4.1 ANALYSIS CONCEPTS 186
5.4.2 THE DISCRETE LOGISTIC EQUATION: ANALYSIS 188
5.4.3 THE DISCRETE LOGISTIC EQUATION: SOLUTIONS 190
5.5 DIFFERENCE AND DIFFERENTIAL EQUATIONS 191
5.5.1 DIFFERENCE VERSUS DIFFERENTIAL EQUATIONS 192
5.5.2 FIRST-ORDER DIFFERENCE AND DIFFERENTIAL EQUATIONS 194 5.5.3
SECOND-ORDER DIFFERENCE AND DIFFERENTIAL EQUATIONS 196
IMAGE 4
XIV CONTENTS
5.6 SUMMARY 199
5.7 EXERCISES 201
6 STOCHASTIC CHANGES 207
6.1 MOTIVATION 208
6.2 LINEAR STOCHASTIC CHANGES 210
6.2.1 ONE-POINT STATISTICS 211
6.2.2 CORRELATIONS 213
6.2.3 SOLUTION FEATURES 214
6.2.4 MONTE CARLO SIMULATION 216
6.3 DIFFUSION 218
6.3.1 RANDOM WALK MODEL 218
6.3.2 THE WIENER PROCESS 220
6.3.3 DIFFUSION MODELS 221
6.4 BROWNIAN MOTION 225
6.4.1 BROWNIAN MOTION MODEL 225
6.4.2 DISCRETE BROWNIAN MOTION STATISTICS 227
6.4.3 CONTINUOUS BROWNIAN MOTION STATISTICS 229
6.5 POPULATION DYNAMICS 232
6.5.1 A STOCHASTIC LOGISTIC MODEL 233
6.5.2 ONE-POINT STATISTICS AND CORRELATIONS 234
6.5.3 MODEL APPLICATION 238
6.6 SUMMARY 240
6.7 EXERCISES 242
7 DETERMINISTIC EVOLUTION 247
7.1 MOTIVATION 248
7.2 HEAT AND MASS TRANSFER: BALANCE 249
7.2.1 HEAT AND MASS TRANSFER MODELS 249
7.2.2 FIRST-ORDER LINEAR DIFFERENTIAL EQUATIONS 251
7.2.3 TIME OF DEATH 253
7.2.4 CONTAMINATION OF LAKES 254
7.3 NEWTON'S LAWS OF MOTION: OSCILLATIONS 255
7.3.1 NEWTON'S LAWS OF MOTION 256
7.3.2 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 260
7.3.3 THE DAMPED HARMONIC OSCILLATOR 264
7.4 POPULATION ECOLOGY: GROWTH AND SELF-LIMITATION 266
7.4.1 GROWTH AND SELF-LIMITATION 267
7.4.2 POPULATION DENSITY FUNCTION MODELS 268
7.4.3 APPLICATION TO WORLD POPULATION MODELING 273
IMAGE 5
CONTENTS XV
7.5 POPULATION ECOLOGY: OSCILLATIONS AND COLLAPSE 276
7.5.1 BASIC POPULATION DYNAMICS SCENARIOS 277
7.5.2 THE WORLD3 MODEL 278
7.5.3 A DELAY LOGISTIC MODEL 280
7.6 SUMMARY 286
7.7 EXERCISES 287
8 STOCHASTIC EVOLUTION 295
8.1 MOTIVATION 296
8.2 PDF EVOLUTION EQUATIONS 297
8.2.1 THE KRAMERS-MOYAL EQUATION 297
8.2.2 THE PAWULA THEOREM 300
8.2.3 THE FOKKER-PLANCK EQUATION 301
8.3 SOLUTION TO THE FOKKER-PLANCK EQUATION 304
8.3.1 THE SOLUTION APPROACH 304
8.3.2 THE SOLUTION TO THE FOKKER-PLANCK EQUATION 306
8.3.3 PROCESS AND CONDITIONAL PDF STATISTICS 308
8.4 STOCHASTIC DIFFERENTIAL EQUATIONS 309
8.4.1 NONLINEAR MARKOVIAN STOCHASTIC EQUATIONS 309
8.4.2 RELATIONSHIP TO THE FOKKER-PLANCK EQUATION 311
8.4.3 LINEAR MARKOVIAN STOCHASTIC EQUATIONS 314
8.5 NON-MARKOVIAN STOCHASTIC DIFFERENTIAL EQUATIONS 317
8.5.1 MARKOVIAN AND NON-MARKOVIAN VELOCITY MODELS 317 8.5.2 MARKOVIAN
VELOCITY MODEL ANALYSIS 320
8.5.3 NON-MARKOVIAN VELOCITY MODEL ANALYSIS 322
8.5.4 THE RELEVANCE OF MEMORY EFFECTS 325
8.6 SUMMARY 327
8.7 EXERCISES 330
9 DETERMINISTIC MULTIVARIATE EVOLUTION 335
9.1 MOTIVATION 336
9.2 SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 337
9.2.1 LINEAR SYSTEMS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 337 9.2.2
FEATURES OF SOLUTIONS OF LINEAR FIRST-ORDER SYSTEMS 343 9.2.3 ANALYSIS
OF NONLINEAR EQUATION SYSTEMS 347
9.3 POPULATION ECOLOGY: SPECIES INTERACTIONS 350
9.3.1 MULTIVARIATE POPULATION DYNAMICS EQUATIONS 350
9.3.2 COMPETITION FOR FOOD 352
9.3.3 PREDATOR-PREY INTERACTION 356
IMAGE 6
XVI CONTENTS
9.4 MECHANICAL MOTIONS: THE PENDULUM 360
9.4.1 PENDULUM EQUATIONS 360
9.4.2 LINEAR STABILITY ANALYSIS 363
9.4.3 NONLINEAR STABILITY ANALYSIS 366
9.4.4 PENDULUM MOTIONS 368
9.5 FLUID DYNAMICS: LORENZ'S WEATHER 370
9.5.1 THE LORENZ EQUATIONS 370
9.5.2 LINEAR STABILITY ANALYSIS 372
9.5.3 DETERMINISTIC CHAOS 378
9.6 SUMMARY 383
9.7 EXERCISES 384
10 STOCHASTIC MULTIVARIATE EVOLUTION 391
10.1 MOTIVATION 392
10.2 DATA ANALYSIS CONCEPTS FOR JOINT RANDOM VARIABLES 393 10.2.1 JOINT
PROBABILITY DENSITY FUNCTIONS 394
10.2.2 CONDITIONAL PROBABILITY DENSITY FUNCTIONS 395
10.2.3 APPLICATION TO OPTIMAL MODELING 397
10.3 THE JOINT NORMAL PROBABILITY DENSITY FUNCTION MODEL 399 10.3.1 THE
JOINT NORMAL PROBABILITY DENSITY FUNCTION MODEL 400 10.3.2 DATA ANALYSIS
403
10.3.3 APPLICATION TO RANDOM WALK MODELING 406
10.4 THE FOKKER-PLANCK EQUATION 409
10.4.1 DEFINITION OF MULTIVARIATE PROBABILITY DENSITY FUNCTIONS 410
10.4.2 THE FOKKER-PLANCK EQUATION 411
10.4.3 A SOLUTION TO THE FOKKER-PLANCK EQUATION 415
10.4.4 STOCHASTIC DIFFERENTIAL EQUATIONS 418
10.5 MOLECULAR AND FLUID MOTION 421
10.5.1 MOLECULAR MOTION MODEL 422
10.5.2 FLUID DYNAMICS EQUATIONS 424
10.5.3 APPENDIX: IMPLICATIONS OF THE STOCHASTIC MOLECULAR MODEL 428 10.6
SUMMARY 433
10.7 EXERCISES 435
REFERENCES 441
AUTHOR INDEX 449
SUBJECT INDEX 453
ABOUT THE AUTHOR 461 |
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| indexdate | 2024-07-21T00:13:28Z |
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| isbn | 9783642203107 |
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| title | Mathematical Modeling |
| title_auth | Mathematical Modeling |
| title_exact_search | Mathematical Modeling |
| title_full | Mathematical Modeling Stefan Heinz |
| title_fullStr | Mathematical Modeling Stefan Heinz |
| title_full_unstemmed | Mathematical Modeling Stefan Heinz |
| title_short | Mathematical Modeling |
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