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

    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
    Looking Ahead

    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
    Looking Ahead

    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...?
    Looking Ahead

    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
    Circuits with Quadratic Resistors
    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
    Looking Ahead

    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
    Looking Ahead

    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
    Looking Ahead

    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."
    —Antonio Cańada Villar (Granada), writing in Zentralblatt MATH 1320 – 1