Hyperbolic Partial Differential Equations (eBook)

Populations, Reactors, Tides and Waves: Theory and Applications

Matthew Witten (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
253 Seiten
Elsevier Science (Verlag)
978-1-4831-5563-0 (ISBN)

Infos

Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution. This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal equation in the Von Foerster model, and nonlinear age-dependent population dynamics. The next chapters deal with various applications of hyperbolic partial differential equations to such areas as age-structured fish populations, density dependent growth in a cell colony, boll-weevil-cotton crop modeling, age dependent predation and cannibalism, parasite populations, growth of microorganisms, and stochastic perturbations in the Von Foerster model. These topics are followed by discussions of bifurcation of time periodic solutions of the McKendrick equation; the periodic solution of nonlinear hyperbolic problems; and semigroup theory as applied to nonlinear age dependent population dynamics. Other chapters explore the stability of biochemical reaction tanks, an ADI model for the Laplace tidal equations, the Carleman equation, the nonequilibrium behavior of solids that transport heat by second sound, and the nonlinear hyperbolic partial differential equations and dynamic programming. The final chapters highlight two explicitly numerical applications: a predictor-convex corrector method and the Galerkin approximation in hyperbolic partial differential equations. This book will prove useful to practicing engineers, population researchers, physicists, and mathematicians.

Inhalt

Front Cover 1
Hyperbolic Partial Differential Equations Populations, Reactors, Tides and Waves: Theory and Application 
4 
Copyright Page 
5 
Table of Contents 
6 
FOREWORD 8
Chapter 1. EDITOR'S REMARKS: 
10 
1. INTRODUCTION 10
2. DISCRETE AGE STRUCTURE 10
3. CONTINUOUS AGE-TIME STRUCTURE: HYPERBOLIC SYSTEMS 10
4. CLOSING COMMENTS 11
REFERENCES 12
Chapter 2. ON THE QUALITATIVE BEHAVIOUR OF POPULATIONSWITH AGE-SPECIFIC INTERACTIONS 14
1. INTRODUCTION 14
2. EXISTENCE OF EQUILIBRIUM SOLUTIONS 16
3. PERSISTENT SOLUTIONS AND ASYMPTOTIC BEHAVIOUR 
19 
4. BIFURCATION OF PERIODIC SOLUTIONS 22
REFERENCES 26
Chapter 3. SIMPLE POPULATION MODELS WITH DIFFUSION 28
INTRODUCTION 28
1. RANDOM DIFFUSION 28
REFERENCES 31
Chapter 4. NONLINEAR AGE-DEPENDENT POPULATION GROWTHUNDER HARVESTING 32
REFERENCES 38
Chapter 5. LOCAL AND GLOBAL STABILITY FOR THESOLUTIONS OF A NONLINEAR RENEWAL EQUATION 40
1. INTRODUCTION 40
2. EXISTENCE AND UNIQUENESS OF THE GLOBAL SOLUTION 41
3. STATIONARY SOLUTIONS 41
4. LOCAL STABILITY 42
5. GLOBAL STABILITY 43
REFERENCES 46
Chapter 6. SOME CONSIDERATIONS ON THE MATHEMATICAL APPROACH TO NONLINEAR AGE DEPENDENT POPULATION DYNAMICS 48
INTRODUCTION 48
1. BASIC THEORY ON (E) 49
REFERENCES 56
APPENDIX 57
Chapter 7. POPULATION MODELS WITH GLOBALLYAGE-DEPENDENT DYNAMICS: O N COMPUTINGTHE STEADY STATE 58
COMPUTING THE STEADY STATE 60
REFERENCES 63
Chapter 8. ASYMPTOTIC BEHAVIOR OF AN AGE-STRUCTUREDFISH POPULATION 64
1. INTRODUCTION 64
2. EXISTENCE AND PROPERTIES OF THE SOLUTIONS OF (1) 65
3. ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS 66
4. SOME FISHERY MODELS 67
REFERENCES 67
Chapter 9. DENSITY DEPENDENT CELLULAR GROWTH IN AN AGE STRUCTURED COLONY 70
INTRODUCTION 70
1. BASIC EQUATIONS 70
2. EXISTENCE 72
3. A SIMPLIFICATION 
73 
REFERENCES 76
APPENDIX 77
Chapter 10.A PDE FORMULATION AND NUMERICAL SOLUTION 
80 
1. INTRODUCTION 80
2. THE MODEL 80
3. COMPUTATIONAL ASPECTS 
84 
REFERENCES 88
APPENDIX 88
Chapter 11. MODELS OF AGE-DEPENDENT PREDATION AND CANNIBALISM VIA THE McKENDRICK EQUATION 90
1. INTRODUCTION 90
2. ONESPECIES: CANNIBALISM 91
3. TWO SPECIES : AGE-INDISCRIMINATE PREDATION VERSUS PREDATION ON NEWBORNS 93
4. TWO SPECIES: PREDATION ON ALL AGES WITH EXTREME AGESMOST VULNERABLE 95
5. POSSIBLE ECOLOGICAL IMPLICATIONS 101
REFERENCES 101
Chapter 12. NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS FOR THE DYNAMICS OF PARASITE POPULATIONS 
102 
INTRODUCTION 102
1. DERIVATION OF THE MODEL 103
2. REDUCTION TO AN INTEGRAL EQUATION 105
3. LOCAL EXISTENCE 108
4. STATIONARY SOLUTIONS 111
5. STABILITY ANALYSIS 
114 
6. EPIDEMIOLOGICAL IMPLICATIONS OF THE EQUILIBRIUM ANALYSIS 116
REFERENCES 117
Chapter 13. STABILITY ANALYSIS OF A DISTRIBUTED PARAMETER MODEL FOR THE GROWTH OF 
118 
1. INTRODUCTION 118
2. STATEMENT OF RESULTS 119
3. PROOF OF THEOREM 1 121
4. PROOF OF THEOREM 2 124
5. PROOF OF THEOREM 3 125
6. PROOF OF THEOREM 4 127
REFERENCES 129
Chapter 14. PARTIAL DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS 130
Chapter 15. ON STOCHASTICITY IN THE VON FOERSTER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION SYSTEM 134
1. INTRODUCTION AND PRELIMINARIES 134
2. STOCHASTICALLY VARYING MORTALITY 
135 
3. APPLICATION TO CELLULAR MODELS 
139 
4. CLOSING REMARKS 143
REFERENCES 143
Chapter 16. BIFURCATION OF TIME PERIODIC SOLUTIONS OF THE McKENDRICK EQUATIONS WITH APPLICATIONS TO POPULATION DYNAMICS 146
1. INTRODUCTION 146
2. PRELIMINARIES 147
3. THE LINEAR THEORY 148
4. A BIFURCATION THEOREM 150
5. ONE PARAMETER OF HOPF-TYPE BIFURCATION 151
6. AN APPLICATION 154
7. PROOFS 157
8. SUMMARY 164
REFERENCES 164
Chapter 17. PERIODIC SOLUTIONS OF NONLINEAR HYPERBOLIC PROBLEMS 166
1. INTRODUCTION 166
2. PERIODIC SOLUTIONS OF NONLINEAR HYPERBOLIC PROBLEMS—A BRIEF SURVEY 167
3. PERIODIC SOLUTIONS OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 
169 
4. REMARKS ON EXISTENCE RESULTS FOR HYPERBOLIC PROBLEMS 172
REFERENCES 172
Chapter 18. THE SEMIGROUP ASSOCIATED WITH NONLINEAR AGE DEPENDENT POPULATION DYNAMICS 174
1. INTRODUCTION 174
2. THE NONLINEAR SEMIGROUP 175
3. THE INFINITESIMAL GENERATOR 
177 
4. THE EXPONENTIAL REPRESENTATION 180
REFERENCES 183
Chapter 19. STABILITY OF BIOCHEMICAL REACTION TANKS 186
INTRODUCTION 186
THE MATHEMATICAL THEORY 
187 
SIMULATION OF LIMIT CYCLE SOLUTIONS 189
CONCLUSIONS 192
REFERENCES 193
Chapter 20. A SPECIAL ADI MODEL FOR THE 
194 
1. BACKGROUND 194
2. STRUCTURE OF THE TWO-DIMENSIONAL GRAVITY WAVE SYSTEM 194
3. FINITE DIFFERENCE FORMS OF THE EQUATIONS 196
4. SPECIAL NUMERICAL FEATURES OF THE MODEL 198
4. NUMERICAL STABILITY AND BOUNDEDNESS 201
5. MODELLING PERSPECTIVES 
203 
REFERENCES 204
Chapter 21. INITIAL BOUNDARY VALUE PROBLEMS FOR THE CARLEMAN EQUATION 206
INTRODUCTION 206
2. MAIN RESULTS 
207 
3. FURTHER CONSIDERATIONS 211
REFERENCES 211
Chapter 22. ON THE NONEQUILIBRIUM BEHAVIOR OF SOLIDS THAT TRANSPORT HEAT BY SECOND SOUND 214
1. CONSTITUTIVE RELATIONS AND FIELD EQUATIONS 214
2. SURFACES OF DISCONTINUITY AND CHARACTERISTIC SPEEDS 216
3. AN ESTIMATE OF THE DEPENDENCE OF ENERGY ON FLUX FOR NaF 220
4. TRAVELING WAVE SOLUTIONS 224
SUMMARY AND DISCUSSION 232
REFERENCES 232
Chapter 23. NONLINEAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS AND DYNAMIC PROGRAMMING 234
1. INTRODUCTION 234
2. THE VARIATIONAL PROBLEM 234
3. DYNAMIC PROGRAMMING APPROACH 234
4. UPPER BOUNDS 235
5. DUALITY AND LOWER BOUNDS 235
6. APPROXIMATE SOLUTION 235
7. SELECTED COMPUTATIONS 235
REFERENCES 235
Chapter 24. EXPLICIT FINITE DIFFERENCE PREDICTOR AND CONVEX CORRECTOR WITH APPLICATIONS TO HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS 236
1. INTRODUCTION 236
3. SOME SIMPLE APPLICATIONS 240
4. NONLINEAR SECOND-ORDER HYPERBOLIC PDE 240
5. A CONCLUDING REMARK 244
REFERENCES 244
Chapter 25. STABLE AND UNSTABLE NUMERICAL BOUNDARY CONDITIONS FOR GALERKIN APPROXIMATIONS TO HYPERBOLIC 
246 
1. INTRODUCTION 246
2. THE MODIFIED METHOD 247
3. A PROBLEM IN 2 SPACE DIM 251
REFERENCES 253
INDEX 254

Produkt-Details

• Erscheint lt. Verlag: 17.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 1-4831-5563-3 / 1483155633
• ISBN-13: 978-1-4831-5563-0 / 9781483155630