Almost every year, a new book on mathematical modeling is published, so, why another? The answer springs directly from the fact that it is very rare to find a book that covers modeling with all types of differential equations in one volume. Until now. Mathematical Modeling: Models, Analysis and Applications covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. The book also contains a chapter on discrete modeling, consisting of differential equations, making it a complete textbook on this important skill needed for the study of science, engineering, and social sciences.
More than just a textbook, this how-to guide presents tools for mathematical modeling and analysis. It offers a wide-ranging overview of mathematical ideas and techniques that provide a number of effective approaches to problem solving. Topics covered include spatial, delayed, and stochastic modeling. The text provides real-life examples of discrete and continuous mathematical modeling scenarios. MATLAB® and Mathematica® are incorporated throughout the text. The examples and exercises in each chapter can be used as problems in a project.
Since mathematical modeling involves a diverse range of skills and tools, the author focuses on techniques that will be of particular interest to engineers, scientists, and others who use models of discrete and continuous systems. He gives students a foundation for understanding and using the mathematics that is the basis of computers, and therefore a foundation for success in engineering and science streams.
Table of Contents
About Mathematical Modeling
What Is Mathematical Modeling?
History of Mathematical Modeling
Latest Development in Mathematical Modeling
Various Functional Forms in Mathematical Modeling
Merits and Demerits in Mathematical Modeling
Mathematically Modeling Discrete Process
Introduction to Discrete Models
One Dimensional Density Independent Models
One Dimensional Density Dependent Models
Nonlinear Two-Dimensional Models: Stability and Bifurcation
Continuous Models Using Ordinary Differential Equations
Introduction to Continuous Models
Formation of Various Continuous Models
Steady State Solutions
Stability and Linearization
Phase Plane Diagrams of Linear Systems
Null Cline Approach
Poincare Bendixson Theory
Bifurcation and Limit Cycle
Spatial Models Using Partial Differential Equations
Diffusion Equations: Observations and Consequences
Dispersal Models Using Diffusion
Effect of Density Dependent Dispersal on Population Dynamics
Models for Age Dependent Dispersal
Steady State Solutions and Traveling Waves
Non Linear Stability Analysis for Diffusion Reaction System
Bifurcation Analysis and Limit Cycles
Numerical Solutions of Differential Equations with Diffusion
Pattern Developing Instability in Diffusion Reaction Systems
Effect of Cross Diffusion
Modeling with Delay Differential Equations
An Introduction to Delay Differential Equations
Various Mathematical Models with Time Delay
Delay Induced Model for Tumor Immune Interaction (Single Discrete Delay)
Stage Structure Predator Prey Models with Multiple Delays and Its Global Stability
Models with Distributed Delays
Modeling with Stochastic Differential Equations
Stochastic Process and Stochastic Differential Equations: a Preview
Effect of Additive Noise on a Logistic Equation
Stochastic Stability: Asymptotically Mean Square Stable
Mean Square Stability of a Stochastic Model with Time Delays
Numerical Simulation of Stochastic Differential Equations
Stochastic Metapopulation Models
Examples and Exercises appear at the end of each chapter.
"…the book is rich with examples. There are examples from ecology, physics, chemistry, economy, medicine, sociology, epidemiology, and more, including specific examples that could be of interest in computer simulation … there are many solved problems and exercises … Both features makes the book a good selection for drawing out examples, problems, and exercises to show what differential equations have to offer to the aspiring modeler …"
—Computing Reviews, October 2014
"…the reader may find quite a few interesting examples illustrating several important methods used in applied mathematics. … it may be well used as a valuable source of interesting examples as well as complementary reading in a number of courses."
—Svitlana P. Rogovchenko, Zentralblatt MATH 1298