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E-grāmata: Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations

(West Chester University, Pennsylvania, USA)
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Discovering Evolution Equations with Applications: Volume 1-Deterministic EquationsPalielināt
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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.
Preface xiii
1 A Basic Analysis Toolbox
1(68)
1.1 Some Basic Mathematical Shorthand
1(1)
1.2 Set Algebra
2(1)
1.3 Functions
3(2)
1.4 The Space (R)
5(7)
1.4.1 Order Properties
5(1)
1.4.2 Absolute Value
6(1)
1.4.3 Completeness Property of (E)
7(2)
1.4.4 Topology of M
9(3)
1.5 Sequences in (R)
12(12)
1.5.1 Sequences and Subsequences
12(1)
1.5.2 Limit Theorems
12(7)
1.5.3 Cauchy Sequences
19(2)
1.5.4 A Brief Look at Infinite Series
21(3)
1.6 The Spaces (RN, RN) and (MN(R), MN(R))
24(8)
1.6.1 The Space (RN, RN)
25(1)
1.6.1.1 Geometric and Topological Structure
25(3)
1.6.1.2 Sequences in IR
28(1)
1.6.2 The Space (MN(R), MN(R))
29(3)
1.7 Abstract Spaces
32(9)
1.7.1 Banach Spaces
33(4)
1.7.2 Hubert Spaces
37(4)
1.8 Elementary Calculus in Abstract Spaces
41(11)
1.8.1 Limits
42(1)
1.8.2 Continuity
43(3)
1.8.3 The Derivative
46(2)
1.8.4 "The" Integral
48(4)
1.9 Some Elementary ODEs
52(3)
1.9.1 Separation of Variables
52(1)
1.9.2 First-Order Linear ODEs
53(1)
1.9.3 Higher-Order Linear ODEs
54(1)
1.10 Looking Ahead
55(1)
1.11 Guidance for Exercises
55(14)
1.11.1 Level 1: A Nudge in a Right Direction
55(6)
1.11.2 Level 2: An Additional Thrust in a Right Direction
61(8)
2 Homogenous Linear Evolution Equations in RN
69(28)
2.1 Motivation by Models
69(4)
2.2 The Matrix Exponential
73(9)
2.3 The Homogenous Cauchy Problem: Well-Posedness
82(3)
2.4 Perturbation and Convergence Results
85(2)
2.5 A Glimpse at Long-Term Behavior
87(3)
2.6 Looking Ahead
90(1)
2.7 Guidance for Exercises
91(6)
2.7.1 Level 1: A Nudge in a Right Direction
91(2)
2.7.2 Level 2: An Additional Thrust in a Right Direction
93(4)
3 Abstract Homogenous Linear Evolution Equations
97(66)
3.1 Linear Operators
97(8)
3.1.1 Bounded versus Unbounded Operators
97(5)
3.1.2 Invertible Operators
102(1)
3.1.3 Closed Operators
103(1)
3.1.4 Densely-Defined operators
104(1)
3.2 Motivation by Models
105(11)
3.3 Introducing Semigroups
116(10)
3.3.1 Motivation
116(4)
3.3.2 Uniformly Continuous Semigroups
120(3)
3.3.3 Strongly Continuous Semigroups
123(3)
3.4 The Abstract Homogenous Cauchy Problem
126(4)
3.5 Generation Theorems
130(15)
3.5.1 Hille-Yosida and Feller-Miyadera-Phillips Theorems
131(11)
3.5.2 A First Look at Dissipative Operators
142(3)
3.6 A Useful Perturbation Result
145(2)
3.7 Some Approximation Theory
147(2)
3.8 A Brief Glimpse at Long-Term Behavior
149(1)
3.9 An Important Look Back
150(1)
3.10 Looking Ahead
150(3)
3.11 Guidance for Exercises
153(10)
3.11.1 Level 1: A Nudge in a Right Direction
153(4)
3.11.2 Level 2: An Additional Thrust in a Right Direction
157(6)
4 Nonhomogenous Linear Evolution Equations
163(28)
4.1 Finite-Dimensional Setting
163(8)
4.1.1 Motivation by Models
163(2)
4.1.2 One-Dimensional Case
165(4)
4.1.3 Extension of Theory to RN
169(2)
4.2 Infinite-Dimensional Setting
171(6)
4.2.1 Motivation by Models
171(1)
4.2.2 Theory in a General Banach Space X
172(5)
4.3 Introducing Two New Models
177(7)
4.4 Looking Ahead
184(1)
4.5 Guidance for Exercises
185(6)
4.5.1 Level 1: A Nudge in a Right Direction
185(3)
4.5.2 Level 2: An Additional Thrust in a Right Direction
188(3)
5 Semi-Linear Evolution Equations
191(72)
5.1 Motivation by Models
191(4)
5.1.1 Some Models Revisited
191(1)
5.1.2 Introducing Two New Models
192(3)
5.2 More Tools from Functional Analysis
195(11)
5.2.1 Fixed-Point Theory
195(1)
5.2.1.1 The Contraction Mapping Principle
195(2)
5.2.1.2 Schauder's Fixed Point Theorem
197(2)
5.2.1.3 Compact Operators and Schaefer's Fixed-Point Theorem
199(3)
5.2.1.4 The Fixed-Point Approach
202(1)
5.2.2 A Handful of Integral Inequalities
202(3)
5.2.3 Frechet Differentiability
205(1)
5.3 Some Essential Preliminary Considerations
206(2)
5.4 Growth Conditions
208(4)
5.5 Theory for Lipschitz-Type Forcing Terms
212(24)
5.5.1 Existence and Uniqueness Results
212(12)
5.5.2 Continuous Dependence
224(2)
5.5.3 Extendability of Local Solutions
226(3)
5.5.4 Long-Term Behavior
229(1)
5.5.5 Models Revisited
230(6)
5.6 Theory for Non-Lipschitz-Type Forcing Terms
236(7)
5.7 Theory under Compactness Assumptions
243(7)
5.8 A Summarizing Look Back
250(1)
5.9 Looking Ahead
250(2)
5.10 Guidance for Exercises
252(11)
5.10.1 Level 1: A Nudge in a Right Direction
252(5)
5.10.2 Level 2: An Additional Thrust in a Right Direction
257(6)
6 Functional Evolution Equations
263(38)
6.1 Motivation by Models
263(4)
6.2 Functionals
267(5)
6.3 Theory in the Lipschitz Case
272(3)
6.4 Theory under Compactness Assumptions
275(1)
6.5 Models-New and Old
276(13)
6.6 Looking Ahead
289(1)
6.7 Guidance for Exercises
290(11)
6.7.1 Level 1: A Nudge in a Right Direction
290(5)
6.7.2 Level 2: An Additional Thrust in a Right Direction
295(6)
7 Implicit Evolution Equations
301(32)
7.1 Sobolev-Type Equations
301(14)
7.1.1 Motivation by Models
301(3)
7.1.2 The Abstract Framework
304(2)
7.1.3 Main Results
306(9)
7.2 Neutral Evolution Equations
315(11)
7.2.1 Finite-Dimensional Case
315(5)
7.2.2 Infinite-Dimensional Case
320(6)
7.3 Looking Ahead
326(1)
7.4 Guidance for Exercises
326(7)
7.4.1 Level 1: A Nudge in a Right Direction
326(2)
7.4.2 Level 2: An Additional Thrust in a Right Direction
328(5)
8 Delay Evolution Equations
333(38)
8.1 Motivation by Models
333(3)
8.2 Setting and Formulation of the Problem
336(4)
8.3 Theory for Lipschitz-Type Forcing Terms
340(5)
8.4 Theory for Non-Lipschitz-Type Forcing Terms
345(2)
8.5 Implicit Delay Evolution Equations
347(3)
8.6 Other Forms of Delay
350(2)
8.6.1 Unbounded Delay
350(2)
8.6.2 State-Dependent Delay
352(1)
8.7 Models - New and Old
352(8)
8.8 An Important Look Back!
360(1)
8.9 Looking Ahead
361(1)
8.10 Guidance for Exercises
362(9)
8.10.1 Level 1: A Nudge in a Right Direction
362(4)
8.10.2 Level 2: An Additional Thrust in a Right Direction
366(5)
9 Nonlinear Evolution Equations
371(22)
9.1 A Wealth of New Models
371(5)
9.2 Comparison of the Linear and Nonlinear Settings
376(1)
9.3 The Crandall-Liggett Theory
377(10)
9.3.1 m-Dissipativity
377(4)
9.3.2 Nonlinear Semigroups
381(1)
9.3.3 The Associated Homogenous Cauchy Problem
382(3)
9.3.4 The Nonhomogenous Cauchy Problem
385(1)
9.3.5 Nonlinear Functional Evolution Equations
386(1)
9.4 A Quick Look at Nonlinear Evolution Inclusions
387(3)
9.4.1 Some Models
387(2)
9.4.2 Evolution Inclusions
389(1)
9.5 Some Final Comments
390(1)
9.6 Guidance for Exercises
390(3)
9.6.1 Level 1: A Nudge in a Right Direction
390(2)
9.6.2 Level 2: An Additional Thrust in a Right Direction
392(1)
10 Nonlocal Evolution Equations
393(12)
10.1 Introductory Remarks
393(1)
10.2 Motivation by Models
394(4)
10.3 Some Abstract Theory
398(5)
10.3.1 The Semi-Linear Case
398(3)
10.3.2 The General Functional Case
401(1)
10.3.3 The Nonlinear Case
402(1)
10.4 Final Comments
403(2)
11 Beyond Volume 1
405(6)
11.1 Three New Classes of Evolution Equations
405(3)
11.1.1 Time-Dependent Evolution Equations
405(2)
11.1.2 Quasi-Linear Evolution Equations
407(1)
11.1.3 Integro-Differential Evolution Equations
408(1)
11.2 Next Stop... Stochastic Evolution Equations!: Preface to Volume 2
408(3)
Bibliographic Remarks 411(4)
Bibliography 415(24)
Index 439
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.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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