Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations, 1st Edition (Paperback) book cover

Discovering Evolution Equations with Applications

Volume 1-Deterministic Equations, 1st Edition

By Mark McKibben

Chapman and Hall/CRC

466 pages

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Description

Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations provides an engaging, accessible account of core theoretical results of evolution equations in a way that gradually builds intuition and culminates in exploring active research. It gives nonspecialists, even those with minimal prior exposure to analysis, the foundation to understand what evolution equations are and how to work with them in various areas of practice.

After presenting the essentials of analysis, the book discusses homogenous finite-dimensional ordinary differential equations. Subsequent chapters then focus on linear homogenous abstract, nonhomogenous linear, semi-linear, functional, Sobolev-type, neutral, delay, and nonlinear evolution equations. The final two chapters explore research topics, including nonlocal evolution equations. For each class of equations, the author develops a core of theoretical results concerning the existence and uniqueness of solutions under various growth and compactness assumptions, continuous dependence upon initial data and parameters, convergence results regarding the initial data, and elementary stability results.

By taking an applications-oriented approach, this self-contained, conversational-style book motivates readers to fully grasp the mathematical details of studying evolution equations. It prepares newcomers to successfully navigate further research in the field.

Table of Contents

A Basic Analysis Toolbox

Some Basic Mathematical Shorthand

Set Algebra

Functions

The Space (R, |·|)

Sequences in (R, |·|)

The Spaces (RN, ||·||RN) and (MN(R), ||·||MN(R))

Abstract Spaces

Elementary Calculus in Abstract Spaces

Some Elementary ODEs

Looking Ahead

Guidance for Exercises

Homogenous Linear Evolution Equations in RN

Motivation by Models

The Matrix Exponential

The Homogenous Cauchy Problem: Well-Posedness

Perturbation and Convergence Results

A Glimpse at Long-Term Behavior

Looking Ahead

Guidance for Exercises

Abstract Homogenous Linear Evolution Equations

Linear Operators

Motivation by Models

Introducing Semigroups

The Abstract Homogenous Cauchy Problem

Generation Theorems

A Useful Perturbation Result

Some Approximation Theory

A Brief Glimpse at Long-Term Behavior

An Important Look Back

Looking Ahead

Guidance for Exercises

Nonhomogenous Linear Evolution Equations

Finite-Dimensional Setting

Infinite-Dimensional Setting

Introducing Two New Models

Looking Ahead

Guidance for Exercises

Semi-Linear Evolution Equations

Motivation by Models

More Tools from Functional Analysis

Some Essential Preliminary Considerations

Growth Conditions

Theory for Lipschitz-Type Forcing Terms

Theory for Non-Lipschitz-Type Forcing Terms

Theory under Compactness Assumptions

A Summarizing Look Back

Looking Ahead

Guidance for Exercises

Functional Evolution Equations

Motivation by Models

Functionals

Theory in the Lipschitz Case

Theory under Compactness Assumptions

Models—New and Old

Looking Ahead

Guidance for Exercises

Implicit Evolution Equations

Sobolev-Type Equations

Neutral Evolution Equations

Looking Ahead

Guidance for Exercises

Delay Evolution Equations

Motivation by Models

Setting and Formulation of the Problem

Theory for Lipschitz-Type Forcing Terms

Theory for Non-Lipschitz-Type Forcing Terms

Implicit Delay Evolution Equations

Other Forms of Delay

Models—New and Old

An Important Look Back!

Looking Ahead

Guidance for Exercises

Nonlinear Evolution Equations

A Wealth of New Models

Comparison of the Linear and Nonlinear Settings

The Crandall–Liggett Theory

A Quick Look at Nonlinear Evolution Inclusions

Some Final Comments

Guidance for Exercises

Nonlocal Evolution Equations

Introductory Remarks

Motivation by Models

Some Abstract Theory

Final Comments

Beyond Volume 1…

Three New Classes of Evolution Equations

Next Stop… Stochastic Evolution Equations!: The Preface to Volume 2

Bibliography

Index

About the Author

Mark A. McKibben is an associate professor in the mathematics and computer science department at Goucher College in Baltimore, Maryland, USA. Dr. McKibben is the author of more than 25 research articles and a referee for more than 30 journals. His research areas include differential equations, stochastic analysis, and applied functional analysis.

About the Series

Chapman & Hall/CRC Applied Mathematics & Nonlinear Science

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis