1st Edition

# Differential Equations with MATLAB Exploration, Applications, and Theory

By Mark McKibben, Micah D. Webster Copyright 2015
498 Pages 215 B/W Illustrations
by Chapman & Hall

497 Pages
by Chapman & Hall

Also available as eBook on:

A unique textbook for an undergraduate course on mathematical modeling, Differential Equations with MATLAB: Exploration, Applications, and Theory provides students with an understanding of the practical and theoretical aspects of mathematical models involving ordinary and partial differential equations (ODEs and PDEs). The text presents a unifying picture inherent to the study and analysis of more than 20 distinct models spanning disciplines such as physics, engineering, and finance.

The first part of the book presents systems of linear ODEs. The text develops mathematical models from ten disparate fields, including pharmacokinetics, chemistry, classical mechanics, neural networks, physiology, and electrical circuits. Focusing on linear PDEs, the second part covers PDEs that arise in the mathematical modeling of phenomena in ten other areas, including heat conduction, wave propagation, fluid flow through fissured rocks, pattern formation, and financial mathematics.

The authors engage students by posing questions of all types throughout, including verifying details, proving conjectures of actual results, analyzing broad strokes that occur within the development of the theory, and applying the theory to specific models. The authors’ accessible style encourages students to actively work through the material and answer these questions. In addition, the extensive use of MATLAB® GUIs allows students to discover patterns and make conjectures.

ORDINARY DIFFERENTIAL EQUATIONS
Welcome!
Introduction
This Book Is a Field Guide. What Does That Mean for YOU?
Mired in Jargon - A Quick Language Lesson!
Introducing MATLAB
A First Look at Some Elementary Mathematical Models

A Basic Analysis Toolbox
Some Basic Mathematical Shorthand
Set Algebra
Functions
The Space (R; j_j)
A Closer Look at Sequences in (R; j_j)
The Spaces (RN; k_kRN ) and (MN(R); k_kMN(R)
Calculus of RN-valued and MN(R)-valued Functions
Some Elementary ODEs

A First Wave of Mathematical Models
Newton's Law of Heating and Cooling-Revisited
Pharmocokinetics
Uniform Mixing Models
Combat! Nation in Balance
Springs and Electrical Circuits - The Same, But Different
Boom! - Chemical Kinetics
Going, Going, Gone! A Look at Projectile Motion
Shake, Rattle, Roll!
My Brain Hurts! A Look at Neural Networks
Breathe In, Breathe Out-A Respiratory Regulation Model

Finite-Dimensional Theory - Ground Zero: The Homogenous Case
Introducing the Homogenous Cauchy Problem (HCP)
Lessons Learned from a Special Case
Defining the Matrix Exponential
Putzer's Algorithm
Properties of eAt
The Homogenous Cauchy Problem: Well-Posedness
Higher-Order Linear ODEs
A Perturbed (HCP)
What Happens to Solutions of (HCP) as Time Goes On and On and On...?

Finite-Dimensional Theory - Next Step: The Non-Homogenous Case
Introducing...The Non-Homogenous Cauchy Problem (Non-CP)
Carefully Examining the One-Dimensional Version of (Non-CP)
Existence Theory for General (Non-CP)
Dealing with a Perturbed (Non-CP)
What Happens to Solutions of (Non-CP) as Time Goes On and On and On...?

A Second Wave of Mathematical Models-Now, with Nonlinear Interactions
Newton's Law of Heating and Cooling Subjected to Polynomial Effects
Pharmocokinetics with Concentration-Dependent Dosing
Springs with Nonlinear Restoring Forces
Enyzme Catalysts
Projectile Motion-Revisited
Floor Displacement Model with Nonlinear Shock Absorbers

Finite-Dimensional Theory - Last Step: The Semi-Linear Case
Introducing the Even-More General Semi-Linear Cauchy Problem (Semi-CP)
New Challenges
Behind the Scenes: Issues and Resolutions Arising in the Study of (Semi-CP)
Lipschitz to the Rescue!
Gronwall's Lemma
The Existence and Uniqueness of a Mild Solution for (Semi-CP)
Dealing with a Perturbed (Semi-CP)

ABSTRACT ORDINARY DIFFERENTIAL EQUATIONS
Getting the Lay of a New Land
A Hot Example
The Hunt for a New Abstract Paradigm
A Small Dose of Functional Analysis

Three New Mathematical Models
Turning Up the Heat - Variants of the Heat Equation
Clay Consolidation and Seepage of Fluid through Fissured Rocks
The Classical Wave Equation and its Variants
An Informal Recap: A First Step toward Unification

Formulating a Theory for (A-HCP)
Introducing (A-HCP)
Defining eAt
Properties of eAt
The Abstract Homogeneous Cauchy Problem: Well-Posedness
A Brief Glimpse of Long-Term Behavior

The Next Wave of Mathematical Models - With Forcing
Turning Up the Heat - Variants of the Heat Equation
Seepage of Fluid through Fissured Rocks
The Classical Wave Equation and its Variants

Remaining Mathematical Models
Population Growth-Fisher's Equation
Zombie Apocalypse! - Epidemiological Models
How Did That Zebra Gets Its Stripes? - A First Look at Spatial Pattern Formation
Autocatalysis-Combustion!
Money, Money, Money - A Simple Financial Model

Formulating a Theory for (A-NonCP)
Introducing (A-NonCP)
Existence and Uniqueness of Solutions of (A-NonCP)
Dealing with a Perturbed (A-NonCP)
Long-Term Behavior

A Final Wave of Models - Accounting for Semilinear Effects
Turning Up the Heat - Semi-Linear Variants of the Heat Equation
The Classical Wave Equation with Semilinear Forcing
Population Growth-Fisher's Equation
Zombie Apocalypse! - Epidemiological Models
How Did That Zebra Gets Its Stripes? - A First Look at Spatial Pattern Formation
Autocatalysis-Combustion!

Epilogue

Appendix

Bibliography

Index

### Biography

Mark McKibben, Micah D. Webster

"The purpose of this book is to illustrate the use of MATLAB in the study of several models involving ordinary and partial differential equations. It includes different disciplines such as physics, engineering and finance. It may be useful for engineers, physicists and applied mathematicians and also for advanced undergraduate (or beginning graduate) students who are interested in the utilization of MATLAB in differential equations. ... The volume incorporates many figures and MATLAB exercises and many questions are raised throughout the text so that readers can do their own computer experiments."