Mathematical Modeling

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Elsevier, 2007 - Mathematics - 335 pages
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Mathematical Modeling 3e is a general introduction to an increasingly crucial topic for today's mathematicians. Unlike textbooks focused on one kind of mathematical model, this book covers the broad spectrum of modeling problems, from optimization to dynamical systems to stochastic processes. Mathematical modeling is the link between mathematics and the rest of the world. Meerschaert shows how to refine a question, phrasing it in precise mathematical terms. Then he encourages students to reverse the process, translating the mathematical solution back into a comprehensible, useful answer to the original question. This textbook mirrors the process professionals must follow in solving complex problems.

Each chapter in this book is followed by a set of challenging exercises. These exercises require significant effort on the part of the student, as well as a certain amount of creativity. Meerschaert did not invent the problems in this book--they are real problems, not designed to illustrate the use of any particular mathematical technique. Meerschaert's emphasis on principles and general techniques offers students the mathematical background they need to model problems in a wide range of disciplines.

This new edition will be accompanied by expanded and enhanced on-line support for instructors. MATLAB material will be added to complement existing support for Maple, Mathematica, and other software packages, and the solutions manual will be provided both in hard copy and on the web.

* Increased support for instructors, including MATLAB material as well as other on-line resources
* New sections on time series analysis and diffusion models
* Additional problems with international focus such as whale and dolphin populations, plus updated optimization problems
 

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Contents

ONE VARIABLE OPTIMIZATION
3
12 Sensitivity Analysis
9
13 Sensitivity and Robustness
13
14 Exercises
15
MULTIVARIABLE OPTIMIZATION
19
22 Lagrange Multipliers
29
23 Sensitivity Analysis and Shadow Prices
39
24 Exercises
48
62 ContinuousTime Models
176
63 The Euler Method
179
64 Chaos and Fractals
191
65 Exercises
204
PROBABILITY MODELS
219
INTRODUCTION TO PROBABILITY MODELS
221
72 Continuous Probability Models
226
73 Introduction to Statistics
229

COMPUTATIONAL METHODS FOR OPTIMIZATION
55
32 Multivariable Optimization
64
33 Linear Programming
72
34 Discrete Optimization
89
35 Exercises
100
DYNAMIC MODELS
111
INTRODUCTION TO DYNAMIC MODELS
113
42 Dynamical Systems
118
43 Discrete Time Dynamical Systems
124
44 Exercises
130
ANALYSIS OF DYNAMIC MODELS
137
52 Eigenvalue Methods for Discrete Systems
142
53 Phase Portraits
147
54 Exercises
162
SIMULATION OF DYNAMIC MODELS
169
74 Diffusion
234
75 Exercises
239
STOCHASTIC MODELS
249
82 Markov Processes
259
83 Linear Regression
269
84 Time Series
278
85 Exercises
288
SIMULATION OF PROBABILITY MODELS
299
92 The Markov Property
305
93 Analytic Simulation
315
94 Exercises
321
Afterword
329
Index
333
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About the author (2007)

Mark M. Meerschaert is Chairperson of the Department of Statistics and Probability at Michigan State University and an Adjunct Professor in the Department of Physics at the University of Nevada. Professor Meerschaert has professional experience in the areas of probability, statistics, statistical physics, mathematical modeling, operations research, partial differential equations, ground water and surface water hydrology. He started his professional career in 1979 as a systems analyst at Vector Research, Inc. of Ann Arbor and Washington D.C., where he worked on a wide variety of modeling projects for government and industry. Meerschaert earned his doctorate in Mathematics from the University of Michigan in 1984. He has taught at the University of Michigan, Albion College, Michigan State University, the University of Nevada in Reno, and the University of Otago in Dunedin, New Zealand. His current research interests include limit theorems and parameter estimation for infinite variance probability models, heavy tail models in finance, modeling river flows with heavy tails and periodic covariance structure, anomalous diffusion, continuous time random walks, fractional derivatives and fractional partial differential equations, and ground water flow and transport. For more details, see his personal web page http://www.stt.msu.edu/~mcubed

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