Applied Mathematical Modeling
A Multidisciplinary Approach
The practice of modeling is best learned by those armed with fundamental methodologies and exposed to a wide variety of modeling experience. Ideally, this experience could be obtained by working on actual modeling problems. But time constraints often make this difficult. Applied Mathematical Modeling provides a collection of models illustrating the power and richness of the mathematical sciences in supplying insight into the operation of important real-world systems. It fills a gap within modeling texts, focusing on applications across a broad range of disciplines.
The first part of the book discusses the general components of the modeling process and highlights the potential of modeling in practice. These chapters discuss the general components of the modeling process, and the evolutionary nature of successful model building. The second part provides a rich compendium of case studies, each one complete with examples, exercises, and projects.
In keeping with the multidimensional nature of the models presented, the chapters in the second part are listed in alphabetical order by the contributor's last name. Unlike most mathematical books, in which you must master the concepts of early chapters to prepare for subsequent material, you may start with any chapter. Begin with cryptology, if that catches your fancy, or go directly to bursty traffic if that is your cup of tea.
Applied Mathematical Modeling serves as a handbook of in-depth case studies that span the mathematical sciences, building upon a modest mathematical background. Readers in other applied disciplines will benefit from seeing how selected mathematical modeling philosophies and techniques can be brought to bear on problems in their disciplines. The models address actual situations studied in chemistry, physics, demography, economics, civil engineering, environmental engineering, industrial engineering, telecommunications, and other areas.
Table of Contents
THE IMPACT AND BENEFITS OF MATHEMATICAL MODELING Introduction Mathematical Aspects, Alternatives, Attitudes Mathematical Modeling Teaching Modeling Benefits of Modeling Educational Benefits Modeling and Group Competition Other Benefits of Modeling The Role of Axioms in Modeling The Challenge References REMARKS ON MATHEMATICAL MODEL BUILDing Introduction An Example of Mathematical Modeling Model Construction and Validation Model Analysis Some Pitfalls Conclusion References UNDERSTANDING THE UNITED STATES AIDS EPIDEMIC Introduction Prelude: The Postwar Polio Epidemic AIDS: A New Epidemic for America Why an AIDS Epidemic in America? A More Detailed Look at the Model Forays into the Public Policy Arena Modeling the Mature Epidemic AIDS as a Facilitator of Other Epidemics Comparisons with First World Countries Conclusion: A Modeler's Portfolio References A MODEL FOR THE SPREAD OF SLEEPING SICKNESS Introduction The Compartmental Model Mathematical Results Discussion Alternative Models Exercises and Projects References MATHEMATICAL MODELS IN CLASSICAL CRYPTOLOGY Introduction Some Terminology of Cryptology Simple Substitution Systems The Vigenere Cipher and One-Time Pads The Basic Hill System and Variations Exercises and Projects References MATHEMATICAL METHODS IN PUBLIC-KEY CRYPTOLOGY Introduction Cryptosystems Based on Integer Factorization Cryptosystems Based on Discrete Logarithms Digital Signatures Exercises and Projects References NONLINEAR TRANSVERSE VIBRATIONS IN AN ELASTIC MEDIUM Introduction A String Embedded in an Elastic Medium An Approximation Technique Base Equation Solution of Ricatti Equation Exercises and Projects References SIMULATING NETWORKS WITH TIME-VARYING ARRIVALS Introduction The Registration Problem Generating Random Numbers Statistical Tools Arrival Processes Queueing Models Exercises and Projects References MATHEMATICAL MODELING OF UNSATURATED POROUS MEDIA FLOW AND TRANSPORT Introduction Governing Equations Constant-Coefficient Convection-Dispersion Coupling the Equations Summary and Suggestions for Further Study Exercises and Projects References INVENTORY REPLENISHMENT POLICIES AND PRODUCTION STRATEGIES Introduction Piston Production and the Multinomial Model Sleeve Inventory Safety Stocks Comparison of Three Reordering Policies Variable Piston Production Quantities The Supplier's Production Problem Target Selection for Multinomial Distributions The Supplier's Cost Function Target Selection Using Normal Distributions Conclusion Exercises and Projects References MODELING NONLINEAR PHENOMENA BY DYNAMICAL SYSTEMS Introduction Simple Pendulum Periodically Forced Pendulum Exercises and Projects References MODULATED POISSON PROCESS MODELS FOR BURSTY TRAFFIC BEHAVIOR Introduction Workstation Utilization Problem Constructing a Modulated Poisson Process Simulation Techniques Analysis Techniques Exercises and Projects References GRAPH-THEORETIC ANALYSIS OF FINITE MARKOV CHAINS Introduction State Classification Periodicity Conclusion Exercises and Projects References SOME ERROR-CORRECTING CODES AND THEIR APPLICATIONS Introduction Background Coding Theory Computer Memories and Hamming Codes Photographs in Space and Reed-Muller Codes Compact Discs and Reed-Solomon Codes Conclusion Exercises and Projects References BROADCASTING AND GOSSIPING IN COMMUNICATION NETWORKS Introduction Standard Gossiping and Broadcasting Examples of Communication Results from Selected Gossiping Problems Conclusion Exercises and Projects References MODELING THE IMPACT OF ENVIRONMENTAL REGULATIONS ON HYDROELECTRIC REVENUES Introduction Preliminaries Model Formulation Model Development Case Study Exercises and Projects References VERTICAL STABILIZATION OF A ROCKET ON A MOVABLE PLATFORM Introduction Mathematical Model State-Space Control Theory The KNvD Algorithm Exercises and Problems References DISTINGUISHED SOLUTIONS OF A FORCED OSCILLATOR Introduction Linear Model with Modified External Forcing Nonlinear Oscillator Periodically Forced by Impulses A Suspension Bridge Model Model Extension to Two Spatial Dimensions Exercises and Projects References MATHEMATICAL MODELING AND COMPUTER SIMULATION OF A POLYMERIZATION PROCESS Introduction Formulating a Mathematical Model Computational Approach Conclusion Exercises and Problems References THE CLEMSON GRADUATE PROGRAM Introduction Historical Background Transformation of a Department The Clemson Program Communication Skills Program Governance Measures of Success Conclusion References
D. R. Shier, K. T. Wallenius