**Description:**

A
great deal can be learned through modeling and mathematical analysis about
real-life phenomena, even before numerical simulations are used to accurately
portray the specific configuration of a situation. Scientific computing also
becomes more effective and efficient if it is preceded by some preliminary
analysis. These important advantages of mathematical modeling are demonstrated
by models of historical importance in an easily understandable way.

The
organization of Mathematical Models and Their Analysis groups models by the
issues that need to be addressed about the phenomena. The new approach shows
how mathematics effective for one modeled phenomenon can be used to analyze
another unrelated problem. For instance, the mathematics of differential equations
useful in understanding the classical physics of planetary models, fluid
motion, and heat conduction is also applicable to the seemingly unrelated
phenomena of traffic flow and congestion, offshore sovereignty, and regulation
of overfishing and deforestation. The formulation and in-depth analysis of
these and other models on modern social issues, such as the management of
exhaustible and renewable resources in response to consumption demands and
economic growth, are of increasing concern to students and researchers of our
time.

The
modeling of current social issues typically starts with a simple but meaningful
model that may not capture all the important elements of the phenomenon.
Predictions extracted from such a model may be informative but not compatible
with all known observations; so the model may require improvements. The cycle
of model formulation, analysis, interpretation, and assessment is made explicit
for the modeler to repeat until a model is validated by consistency with all
known facts.

Contents:

**Preface to the Classics Edition **

**Preface **

**Chapter 1: Groping in the Dark: Introduction** •
Mathematical Models and Mathematical Modeling • Teaching Mathematical Modeling
• Process Themes in Applied Mathematics • The Period of a Simple Pendulum •
Dimensional Analysis • The Atomic Explosion of 1945 • Construction of
Dimensionless Combinations • The Principle of Optimality • Exercises

__Part I: Evolution of
Dynamical Systems__

**Chapter 2: Here Comes the Sun: The Three Laws of Kepler **• The
Two-Body Problem • Kepler’s Second Law • Kepler’s First and Third Laws • An
Applied Mathematics Perspective • Exercises

**Chapter 3: Slower Than Light: The Precession of the
Perihelion of the Planet Mercury** • The Relativistic Effect •
The Perturbation Method • A Simple Initial-Value Problem • The Poincare’s
Method • The Modeling Cycle • Exercises

__Part II: Stability of
Equilibrium Configuration__

**Chapter 4: Swing Low: The Stability of Periodic Orbits **•
Planetary Orbits and Their Stability • A Critical Point Analysis for Relativistic
Planetary Motion • Phase Diagram for Relativistic Planetary Motion •
Approximate Solutions by Numerical Methods • Locating Solution Singularize
Numerically • Trajectories on a Phase Sphere • Parametric Excitation •
Exercises

**Chapter 5: Hair: Euler Buckling and Elastic Stability** •
The Legend of Samson and the Euler Column • Determination of the Buckling Load
• The Nonlinear Elastic Stability Problem • The Potential Energy in a Hinged
Column • The Variational Problem for a Hinged Column • The Spring-Mass Analogy
• The Direct Method of Calculus of Variations • The Rayleigh Quotient •
Exercises

**Chapter 6: A Menace on Any Road: Car Following **•
Lagrangian and Eulerian Formulation of Traffic Flow • Instantaneous Velocity
Control • Velocity Control with Lag Time • Approximate Solution for a Short Lag
Time • Local and Asymptotic Stability • Solution of the Initial-Value Problem •
Fluctuation of Car Spacings • Density and Velocity Relation • Exercises

__Part III: Wave Propagation__

**Chapter 7: The Shock of the Crash: Traffic Flow on a Long
and Crowded Road** • Eulerian Formulation of Traffic Problems •
Conservation of Cars • On the Flow-Density Relation • The Linear Initial-Value
Problem • The Green Light Problem • The Traffic Hump and Shock Formation •
Exercises • Appendix: LasGrange’s Method of Characteristics

**Chapter 8: It’s a Bore: Shallow Water Waves** •
Shallow Water Theory • Smooth Solutions and Linearization • D’Alembert’s
Solution for the One-Dimensional Wave Equation • The Fundamental Solution and
inhomogeneous Wave Equation • The Signaling Problem • A Steadily Advancing
Shock Wave • Reflection of a Bore from a Rigid Wall • Characteristics
Coordinates • Exercises

**Chapter 9: The Sound of Music: Vibrating Strings and
Membranes** • Why is a Guitar More Musical than a Drum? •
Small-Amplitude Motion of an Elastic String • Wave Reflection at a Density
Change • Standing Waves in a Finite String • Forced Vibration of a Finite
String • Small-Amplitude Vibration of an Elastic Membrane • Axlsymmetric Modes
of a Circular Membrane • The Initial-Value Problem and the Rayleigh Quotient •
Exercises

__Part IV: Diffusion__

**Chapter 10: A Hot Rod in Traffic: Sensitivity to a Sharp
Traffic Density Gradient **• Burger’s Equation and the Cole-Hopf
Transformation • One-Dimensional Heat Conduction in a Straight Rod • Infinite
Domain Problem by Fourier Transforms • Fundamental Solution and Dimensional
Analysis • Shock Speed • Travelling-Wave Solution and Shock Structure • The
Equal-Area Rule • Exercises

**Chapter 11: Fishing is Strictly prohibited: The 200-Mile
Fishing Limit **• The Depletion of East Coast Fisheries •
Compensatory Growth of Fish Populations • Dispensation of Growth Models • One
Dimensional Reaction-Diffusion Model • Local Stability of the Trivial State •
Nontrivial Equilibrium States • Local Stability near Bifurcation • Finite
Fishing Effort Rate • Exercises

__Part V: Control and
Optimization __

**Chapter 12: Suburbs are for the Affluent: The Structure
of the Residential District **• Residential Area in a Monocentric
Circular City • Household Optimum • Competitive Equilibrium • Two Income
Classes • Other Improvements for the Basic Model • Other Criteria for
Optimality • Land for Roads and the Distance Coast of Transportation •
Congestion Coast of Transportation • Perturbation Solutions • Allocation of
Land for Roads • Exercises

**Chapter 13: Pay or Save: Neoclassical Economic Growth
Theory** • The One-Sector Neoclassical Growth Model • Fixed
Fraction Saving and the Golden Rule • Social Optimum • Legrange Multipllers and
the Hamltonian • The Trumplke Solution • The Max-Min Optimum Growth • Exercises

**Chapter 14: Justice for All: Exhaustible Resources and
Intergenerational Equity** • Exhaustible Resource Essential to
Production • Social Optimum • Constant Return to Scale • The Max-Min Optimum •
A Cobb-Douglas Production Function • The Effect of Extraction Cost • Investment
Rule for Intergenerational Equity • Exercises • Appendix: Nonexistence of a
Max-Min Optimum for Cobb-Douglas Technology with *b > a *

**Chapter 15: Timberrr: Economically Optimal Forest
Harvesting Schedule** • The Fisher Age • Opportunity Cost • The
Faustmann Rotation • Maximum Sustained Yield • Finite Harvesting Rate and
Ordered Site Access • The Once-and-for-All Forest with Ordered Site Access •
Fisher’s Rule and Nonuniform Initial Tree Age Distribution • A Maximum
Feasibility Harvesting Rate • Harvesting Cost Varying with Harvesting Rate •
The Ongoing Forest with Ordered Site Access • A Finite Sequence of
Age-Dependent Net Revenue Functions • An Infinite Harvesting Sequence and Fraustmann’s
Rotation Changing Net Revenue Functions • Exercises • Appendix: The
Environmental Cost

*Appendix to the
Classics Edition: *Controllability
of Linear Systems

**Bibliography **

**Index**

About the Author:

**Frederic Y. M. Wan **has been a Professor
of Mathematics at the University of California, Irvine since 1995, where he was
also Chancellor for Research and Dean of Graduate Studies, 1995–2000. He has
previously held positions on the mathematics faculty at MIT and as the founding
Director of the Institute of Applied Mathematics and Statistics at the
University of British Columbia, President of the American Academy of Mechanics
(AAM), President of the Canadian Applied Mathematics Society (CAMS/SCMA), and
founding Chair of the Department of Applied Mathematics at the University of
Washington. He was Chair of the Committee of Pure and Applied Mathematics of
the Natural Science and Engineering Council of Canada and Director of the
National Science Foundation’s Division of Mathematical Sciences, the only
person to have held both positions. He is a fellow of AAM, ASME, AAAS, and
SIAM.

Target Audience:

This
book is recommended for advanced undergraduates, early graduate students, or
anyone with a strong background in calculus and ordinary differential
equations, for use in mathematical modeling and introduction to applied
mathematics courses.