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About this book

The objective of this textbook is the construction, analysis, and interpretation of mathematical models to help us understand the world we live in. Rather than follow a case study approach it develops the mathematical and physical ideas that are fundamental in understanding contemporary problems in science and engineering. Science evolves, and this means that the problems of current interest continually change.

What does not change as quickly is the approach used to derive the relevant mathematical models, and the methods used to analyze the models. Consequently, this book is written in such a way as to establish the mathematical ideas underlying model development independently of a specific application. This does not mean applications are not considered, they are, and connections with experiment are a staple of this book.

The book, as well as the individual chapters, is written in such a way that the material becomes more sophisticated as you progress. This provides some flexibility in how the book is used, allowing consideration for the breadth and depth of the material covered.

Moreover, there are a wide spectrum of exercises and detailed illustrations that significantly enrich the material. Students and researchers interested in mathematical modelling in mathematics, physics, engineering and the applied sciences will find this text useful.

The material, and topics, have been updated to include recent developments in mathematical modeling. The exercises have also been expanded to include these changes, as well as enhance those from the first edition.

Review of first edition:

"The goal of this book is to introduce the mathematical tools needed for analyzing and deriving mathematical models. … Holmes is able to integrate the theory with application in a very nice way providing an excellent book on applied mathematics. … One of the best features of the book is the abundant number of exercises found at the end of each chapter. … I think this is a great book, and I recommend it for scholarly purposes by students, teachers, and researchers."

Joe Latulippe, The Mathematical Association of America, December, 2009

Table of Contents

Frontmatter

Chapter 1. Dimensional Analysis

Abstract
Before beginning the material on dimensional analysis, it is worth considering a simple example that demonstrates what we are doing. One that qualifies as simple is the situation of when an object is thrown upwards.
Mark H. Holmes

Chapter 2. Perturbation Methods

Abstract
To introduce the ideas underlying perturbation methods and asymptotic approximations, we will begin with an algebraic equation.
Mark H. Holmes

Chapter 3. Kinetics

Abstract
We now investigate how to model, and analyze, the interactions of multiple species and how these interactions produce changes in their populations. Examples of such problems are below.
Mark H. Holmes

Chapter 4. Diffusion

Abstract
In the last chapter we examined how to use the kinetics of reactions to model the rate of change of populations, or concentrations. We did not consider the consequences of the motion or spatial transport of these populations. There are multiple mechanisms involved with transport, and in this chapter we will examine one of them, and it is the process of diffusion. A simple example of diffusion arises when a perfume bottle is opened. Assuming the air is still, the perfume molecules move through the air because of molecular diffusion.
Mark H. Holmes

Chapter 5. Traffic Flow

Abstract
In this chapter we again investigate the movement of objects along a one-dimensional path, but now the motion is directed rather than random.
Mark H. Holmes

Chapter 6. Continuum Mechanics: One Spatial Dimension

Abstract
In the previous chapter we investigated how to model the spatial motion of objects (cars, molecules, etc.) but omitted the possibility that the objects exert forces on each other. The objective now is to introduce this into the modeling. The situations where this is needed are quite varied and include the deformation of an elastic bar, the stretching of a string, or the flow of air or water.
Mark H. Holmes

Chapter 7. Elastic and Viscoelastic Materials

Abstract
A particularly successful application of continuum mechanics is linear elasticity.
Mark H. Holmes

Chapter 8. Continuum Mechanics: Three Spatial Dimensions

Abstract
The water in the ocean, the air in the room, and a rubber ball have a common characteristic, they appear to completely occupy their respective domains. What this means is that the material occupies every point in the domain.
Mark H. Holmes

Chapter 9. Newtonian Fluids

Abstract
The equations of motion for an incompressible Newtonian fluid are given in Sect. 8.​11.​1.
Mark H. Holmes

Backmatter

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