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

The analysis and interpretation of mathematical models is an essential part of the modern scientific process. Topics in Applied Mathematics and Modeling is designed for a one-semester course in this area aimed at a wide undergraduate audience in the mathematical sciences. The prerequisite for access is exposure to the central ideas of linear algebra and ordinary differential equations. The subjects explored in the book are dimensional analysis and scaling, dynamical systems, perturbation methods, and calculus of variations. These are immense subjects of wide applicability and a fertile ground for critical thinking and quantitative reasoning, in which every student of mathematics should have some experience. Students who use this book will enhance their understanding of mathematics, acquire tools to explore meaningful scientific problems, and increase their preparedness for future research and advanced studies. The highlights of the book are case studies and mini-projects, which illustrate the mathematics in action. The book also contains a wealth of examples, figures, and regular exercises to support teaching and learning. The book includes opportunities for computer-aided explorations, and each chapter contains a bibliography with references covering further details of the material. Cover Title page Copyright Contents Preface Note to instructors Case studies and mini-projects Chapter 1. Dimensional analysis 1.1. Units and dimensions 1.2. Axioms of dimensions 1.3. Dimensionless quantities 1.4. Change of units 1.5. Unit-free equations 1.6. Buckingham π-theorem 1.7. Case study Reference notes Exercises Mini-project Chapter 2. Scaling 2.1. Domains and scales 2.2. Scale transformations 2.3. Derivative relations 2.4. Natural scales 2.5. Scaling theorem 2.6. Case study Reference notes Exercises Mini-project Chapter 3. One-dimensional dynamics 3.1. Preliminaries 3.2. Solvability theorem 3.3. Equilibria 3.4. Monotonicity theorem 3.5. Stability of equilibria 3.6. Derivative test for stability 3.7. Bifurcation of equilibria 3.8. Case study Reference notes Exercises Mini-project Chapter 4. Two-dimensional dynamics 4.1. Preliminaries 4.2. Solvability theorem 4.3. Direction field, nullclines 4.4. Path equation, first integrals 4.5. Equilibria 4.6. Periodic orbits 4.7. Linear systems 4.8. Equilibria in nonlinear systems 4.9. Periodic orbits in nonlinear systems 4.10. Bifurcation 4.11. Case study 4.12. Case study Reference notes Exercises Mini-project 1 Mini-project 2 Mini-project 3 Chapter 5. Perturbation methods 5.1. Perturbed equations 5.2. Regular versus singular behavior 5.3. Assumptions, analytic functions 5.4. Notation, order symbols 5.5. Regular algebraic case 5.6. Regular differential case 5.7. Case study 5.8. Poincaré–Lindstedt method 5.9. Singular algebraic case 5.10. Singular differential case 5.11. Case study Reference notes Exercises Mini-project 1 Mini-project 2 Mini-project 3 Chapter 6. Calculus of variations 6.1. Preliminaries 6.2. Absolute extrema 6.3. Local extrema 6.4. Necessary conditions 6.5. First-order problems 6.6. Simplifications, essential results 6.7. Case study 6.8. Natural boundary conditions 6.9. Case study 6.10. Second-order problems 6.11. Case study 6.12. Constraints 6.13. Case study 6.14. A sufficient condition Reference notes Exercises Mini-project 1 Mini-project 2 Mini-project 3 Bibliography Index Back Cover