Title Mathematical Models and Their Analysis
Subtitle Classics in Applied Mathematics - 79
Author Frederic Y. M. Wan
ISBN 9781611975260
List price USD 74.00
Price outside India Available on Request
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Binding Paperback
No of pages 402
Book size 153 x 229 mm
Publishing year 2018
Original publisher SIAM - Society for Industrial and Applied Mathematics (Eurospan Group)
Published in India by .
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Status New Arrival
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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.


Preface to the Classics Edition


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



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.


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