Topics in Applied Mathematics and Modeling: Concise Theory with Case Studies

228 Pages·2023·7.16 MB·English

by  Oscar Gonzalez

59
Topics in Applied
Mathematics
and Modeling
Concise Theory
with Case Studies
Oscar Gonzalez

Topics in Applied
Mathematics
and Modeling
Concise Theory
with Case Studies

UNDERGRADUATE TEXTS • 59
Topics in Applied
Mathematics
and Modeling
Concise Theory
with Case Studies
Oscar Gonzalez
EDITORIAL COMMITTEE
Giuliana Davidoff Tara S. Holm
Steven J. Miller Maria Cristina Pereyra
Gerald B. Folland (Chair)
2020 Mathematics Subject Classification. Primary 00A69, 34A26, 34E10,
37N99, 41A58, 49K15.
For additional informationand updates on this book, visit
www.ams.org/bookpages/amstext-59
Library of Congress Cataloging-in-Publication Data
Names: Gonzalez,Oscar,1968–author.
Title: Topics in applied mathematics and modeling : concise theory with case studies / Oscar
Gonzalez.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2023]|Series: Pureand
appliedundergraduatetexts,ISSN1943–9334;Volume59|Includesbibliographicalreferences
andindex.
Identifiers: LCCN2022034482|ISBN9781470469917(paperback)|9781470472177(ebook)
Subjects: LCSH: Differential equations. | AMS: General – General and miscellaneous specific
topics – General applied mathematics. | Ordinary differential equations – General theory –
Geometric methods in differential equations. | Ordinary differential equations – Asymptotic
theory– Perturbations,asymptotics. |Dynamicalsystemsandergodictheory–Applications
–Noneoftheabove,butinthissection. |Approximationsandexpansions–Approximations
and expansions – Series expansions (e.g. Taylor, Lidstone series, but not Fourier series). |
Calculus of variations and optimal control; optimization – Optimality conditions – Problems
involvingordinarydifferentialequations.
Classification: LCCQA372.G6172023|DDC515/.352–dc23/eng20221014
LCrecordavailableathttps://lccn.loc.gov/2022034482
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Contents
Preface ix
Notetoinstructors xi
Casestudiesandmini-projects xiii
Chapter1. Dimensionalanalysis 1
1.1. Unitsanddimensions 1
1.2. Axiomsofdimensions 2
1.3. Dimensionlessquantities 3
1.4. Changeofunits 4
1.5. Unit-freeequations 4
1.6. Buckingham𝜋-theorem 7
1.7. Casestudy 11
Referencenotes 14
Exercises 14
Mini-project 17
Chapter2. Scaling 19
2.1. Domainsandscales 19
2.2. Scaletransformations 20
2.3. Derivativerelations 22
2.4. Naturalscales 23
2.5. Scalingtheorem 26
2.6. Casestudy 29
Referencenotes 31
v
vi Contents
Exercises 31
Mini-project 34
Chapter3. One-dimensionaldynamics 37
3.1. Preliminaries 37
3.2. Solvabilitytheorem 38
3.3. Equilibria 39
3.4. Monotonicitytheorem 40
3.5. Stabilityofequilibria 41
3.6. Derivativetestforstability 43
3.7. Bifurcationofequilibria 44
3.8. Casestudy 46
Referencenotes 50
Exercises 50
Mini-project 53
Chapter4. Two-dimensionaldynamics 55
4.1. Preliminaries 55
4.2. Solvabilitytheorem 56
4.3. Directionfield,nullclines 57
4.4. Pathequation,firstintegrals 58
4.5. Equilibria 60
4.6. Periodicorbits 62
4.7. Linearsystems 64
4.8. Equilibriainnonlinearsystems 70
4.9. Periodicorbitsinnonlinearsystems 73
4.10. Bifurcation 77
4.11. Casestudy 77
4.12. Casestudy 81
Referencenotes 86
Exercises 86
Mini-project1 92
Mini-project2 92
Mini-project3 93
Chapter5. Perturbationmethods 95
5.1. Perturbedequations 95
5.2. Regularversussingularbehavior 96
5.3. Assumptions,analyticfunctions 98
5.4. Notation,ordersymbols 99
Contents vii
5.5. Regularalgebraiccase 100
5.6. Regulardifferentialcase 107
5.7. Casestudy 110
5.8. Poincaré–Lindstedtmethod 114
5.9. Singularalgebraiccase 118
5.10. Singulardifferentialcase 121
5.11. Casestudy 127
Referencenotes 132
Exercises 133
Mini-project1 139
Mini-project2 140
Mini-project3 141
Chapter6. Calculusofvariations 143
6.1. Preliminaries 143
6.2. Absoluteextrema 145
6.3. Localextrema 147
6.4. Necessaryconditions 150
6.5. First-orderproblems 154
6.6. Simplifications,essentialresults 158
6.7. Casestudy 161
6.8. Naturalboundaryconditions 166
6.9. Casestudy 170
6.10. Second-orderproblems 174
6.11. Casestudy 177
6.12. Constraints 181
6.13. Casestudy 186
6.14. Asufficientcondition 189
Referencenotes 193
Exercises 193
Mini-project1 200
Mini-project2 201
Mini-project3 202
Bibliography 205
Index 207