Differential Equations: A Modeling Approach

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Overview

"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications. "

S.L. Sullivan, Catawba College

Differential Equations: A Modeling Approach introduces differential equations and differential equation modeling to students and researchers in the social sciences. The text explains the mathematics and theory of differential equations. Graphical methods of analysis are emphasized over formal proofs, making the text even more accessible for newcomers to the subject matter. This volume introduces the subject of ordinary differential equations — as well as systems of such equations — to the social science audience. Social science examples are used extensively, and readers are guided through the most elementary models to much more advanced specifications. Emphasis is placed on easily applied and broadly applicable numerical methods for solving differential equations, thereby avoiding complicated mathematical “tricks” that often do not even work with more interesting nonlinear models. Also, graphical methods of analysis are introduced that allow social scientists to rapidly access the power of sophisticated model specifications. This volume also describes in clear language how to evaluate the stability of a system of differential equations (linear or nonlinear) by using the system’s eigenvalues. The mixture of nonlinearity with dynamical systems is a virtual trademark for this author’s approach to modeling, and this theme comes through clearly throughout this volume. This volume’s clarity of exposition encourages social science students of mathematical modeling to begin working with differential equation models that address complex and sophisticated social theories.

Key Features:

  • The text is accessibly written, so that students with minimal mathematical training can understand all of the basic concepts and techniques presented.
  • The author uses social sciences examples to illustrate the relevance of differential equation modeling to readers.
  • Readers can use graphical methods to produce penetrating analysis of differential equation systems.
  • Linear and nonlinear model specifications are explained from a social science perspective. Most interesting differential equation models are nonlinear, and readers need to know how to specify and work with such models in the social sciences.

Learn more about “The Little Green Book” - QASS Series! Click Here

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Editorial Reviews

CHOICE - S.L. Sullivan
"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications. "

—S.L. Sullivan, Catawba College

Choice Magazine
"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications. "-S.L. Sullivan, Catawba College— S.L. Sullivan
CHOICE - S.L. Sullivan
"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications."

S.L. Sullivan, Catawba College

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Product Details

Meet the Author

Courtney Brown is an Associate Professor in the Department of Political Science at Emory University. Dr. Brown has taught differential equation modeling to graduate and undergraduate students for over 20 years. His teaching and research interests also include other quantitative methods, political musicology, science fiction and politics, electoral behavior, political parties, democratic development, and politics and the environment. He has authored five books that deal with differential equation models in the social sciences, including three titles for the Quantitative Applications in the Social Sciences series.

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Table of Contents

Series Editor's Introduction
Acknowledgments
1. Dynamic Models and Social Change
Theoretical Reasons for Using Differential Equations in the Social Sciences
An Example
The Use of Differential Equations in the Natural and Physical Sciences
Deterministic Versus Probabilistic Differential Equation Models
What Is a Differential Equation?
What This Book Is and Is Not
2. First-Order Differential Equations
Analytical Solutions to Linear First-Order Differential Equations
Solving First-Order Differential Equations Using Separation of Variables
An Example From Sociology
Numerical Methods Used to Solve Differential Equations
Summary
Chapter 2 Appendix
3. Systems of First-Order Differential Equations
The Predator-Prey Model
The Phase Diagram
Vector Field and Direction Field Diagrams
The Equilibrium Marsh and Flow Diagrams
Summary
Chapter 3 Appendix
4. Some Classic Social Science Examples of First-Order Systems
Richardson's Arms Race Model
Lanchester's Combat Model
Rapoport's Production and Exchange Model
Summary
5. Transforming Second-Order and Nonautonomous Differential Equations Into Systems of First-Order Differential Equations
Second- and Higher-Order Differential Equations
Nonautonomous Differential Equations
Summary
6. Stability Analyses of Linear Differential Equation Systems
A Motivating Example of How Stability Can Dramatically Change in One System
Scalar Methods
Matrix Methods
Equilibrium Categories
Summarizing the Stability Criteria
7. Stability Analyses of Nonlinear Differential Equation Systems
The Jacobian
Summary
8. Frontiers of Exploration
References
Index
About the Author

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