1st Edition
Discovering Evolution Equations with Applications Volume 1-Deterministic Equations
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.
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
Biography
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.
