Differential Equations with MATLAB: Exploration, Applications, and Theory

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