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i-Smooth Analysis - A. V. Kim

i-Smooth Analysis

Theory and Applications

(Autor)

Buch | Hardcover
296 Seiten
2015
Wiley-Scrivener (Verlag)
978-1-118-99836-6 (ISBN)
CHF 285,95 inkl. MwSt
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i-SMOOTH ANALYSIS A totally new direction in mathematics, this revolutionary new study introduces a new class of invariant derivatives of functions and establishes relations with other derivatives, such as the Sobolev generalized derivative and the generalized derivative of the distribution theory.

i-smooth analysis is the branch of functional analysis that considers the theory and applications of the invariant derivatives of functions and functionals. The important direction of i-smooth analysis is the investigation of the relation of invariant derivatives with the Sobolev generalized derivative and the generalized derivative of distribution theory.

Until now, i-smooth analysis has been developed mainly to apply to the theory of functional differential equations, and the goal of this book is to present i-smooth analysis as a branch of functional analysis. The notion of the invariant derivative (i-derivative) of nonlinear functionals has been introduced in mathematics, and this in turn developed the corresponding i-smooth calculus of functionals and showed that for linear continuous functionals the invariant derivative coincides with the generalized derivative of the distribution theory. This book intends to introduce this theory to the general mathematics, engineering, and physicist communities.

i-Smooth Analysis: Theory and Applications



Introduces a new class of derivatives of functions and functionals, a revolutionary new approach
Establishes a relationship with the generalized Sobolev derivative and the generalized derivative of the distribution theory
Presents the complete theory of i-smooth analysis
Contains the theory of FDE numerical method, based on i-smooth analysis
Explores a new direction of i-smooth analysis, the theory of the invariant derivative of functions
Is of interest to all mathematicians, engineers studying processes with delays, and physicists who study hereditary phenomena in nature.

AUDIENCE

Mathematicians, applied mathematicians, engineers , physicists, students in mathematics

Dr. A.V. Kim, PhD, is the head of the research group of the Institute of Mathematics and Mechanics of the Russian Academy of Sciences (Ural Branch). He graduated from the mathematics department at Ural State University in 1980, received his doctorate from Ural State University in 1987, and received his Doctor of Science degree with his research monograph, “Some problems of Functional Differential Equations theory,” at the Institute of Mathematics and Mechanics in 2001. Before doing his research at the Russian Academy of Natural Science, he taught at the Seoul National University in the School of Electrical Engineering.

Preface xi

Part I Invariant derivatives of functionals and numerical methods for functional differential equations 1

1 The invariant derivative of functionals 3

1 Functional derivatives 3

1.1 The Frechet derivative 4

1.2 The Gateaux derivative 4

2 Classification of functionals on C[a, b] 5

2.1 Regular functionals 5

2.2 Singular functionals 6

3 Calculation of a functional along a line 6

3.1 Shift operators 6

3.2 Superposition of a functional and a function 7

3.3 Dini derivatives 8

4 Discussion of two examples 8

4.1 Derivative of a function along a curve 8

4.2 Derivative of a functional along a curve 9

5 The invariant derivative 11

5.1 The invariant derivative 11

5.2 The invariant derivative in the class B[a, b] 12

5.3 Examples 13

6 Properties of the invariant derivative 16

6.1 Principles of calculating invariant derivatives 16

6.2 The invariant differentiability and invariant continuity 19

6.3 High order invariant derivatives 20

6.4 Series expansion 21

7 Several variables 21

7.1 Notation 21

7.2 Shift operator 21

7.3 Partial invariant derivative 22

8 Generalized derivatives of nonlinear functionals 22

8.1 Introduction 22

8.2 Distributions (generalized functions) 24

8.3 Generalized derivatives of nonlinear distributions 25

8.4 Properties of generalized derivatives 27

8.5 Generalized derivative (multidimensional case) 28

8.6 The space SD of nonlinear distributions 29

8.7 Basis on shift 30

8.8 Primitive 31

8.9 Generalized solutions of nonlinear differential equations 34

8.10 Linear differential equations with variables coeffecients 36

9 Functionals on Q[−t ; 0] 37

9.1 Regular functionals 39

9.2 Singular functionals 40

9.3 Specific functionals 40

9.4 Support of a functional 41

10 Functionals on R × Rn × Q[−t; 0] 42

10.1 Regular functionals 42

10.2 Singular functionals 44

10.3 Volterra functionals 44

10.4 Support of a functional 45

11 The invariant derivative 45

11.1 Invariant derivative of a functional 46

11.2 Examples 48

11.3 Invariant continuity and invariant differentiability 58

11.4 Invariant derivative in the class B[−t; 0] 59

12 Coinvariant derivative 65

12.1 Coinvariant derivative of functionals 65

12.2 Coinvariant derivative in a class B[−t; 0] 68

12.3 Properties of the coinvariant derivative 71

12.4 Partial derivatives of high order 73

12.5 Formulas of i–smooth calculus for mappings 75

13 Brief overview of Functional Differential Equation theory 76

13.1 Functional Differential Equations 76

13.2 FDE types 78

13.3 Modeling by FDE 80

13.4 Phase space and FDE conditional representation 81

14 Existence and uniqueness of FDE solutions 84

14.1 The classic solutions 84

14.2 Caratheodory solutions 92

14.3 The step method for systems with discrete delays 94

15 Smoothness of solutions and expansion into the Taylor series 95

15.1 Density of special initial functions 98

15.2 Expansion of FDE solutions into Taylor series 100

16 The sewing procedure 103

16.1 General case 104

16.2 Sewing (modification) by polynomials 105

16.3 The sewing procedure of the second order 107

16.4 Sewing procedure of the second order for linear delay differential equation 109

2 Numerical methods for functional differential equations 113

17 Numerical Euler method 115

18 Numerical Runge-Kutta-like methods 118

18.1 Methods of interpolation and extrapolation 119

18.2 Explicit Runge-Kutta-like methods 127

18.3 Order of the residual of ERK-methods 132

18.4 Implicit Runge-Kutta-like methods 136

19 Multistep numerical methods 142

19.1 Numerical models 143

19.2 Order of convergence 143

19.3 Approximation order. Starting procedure 145

20 Startingless multistep methods 146

20.1 Explicit methods 147

20.2 Implicit methods 148

20.3 Startingless multistep methods 150

21 Nordsik methods 152

21.1 Methods based on calculation of high order derivatives 155

21.2 Various methods based on the separation of finite-dimensional and infinite-dimensional components of the phase state 158

22 General linear methods of numerical solving functional differential equations 162

22.1 Introduction 162

22.2 Methodology of classification numerical FDE models 173

22.3 Necessary and sufficient conditions of convergence with order p 181

22.4 Asymptotic expansion of the global error 186

23 Algorithms with variable step-size and some aspects of computer realization of numerical models 196

23.1 ERK-like methods with variable step 197

23.2 Methods of interpolation and extrapolation of discrete model prehistory 202

23.3 Choice of the step size 207

23.4 Influence of the approximate calculating functionals of the right-hand side of FDEs 212

23.5 Test problems 217

24 Soft ware package Time-delay System Toolbox 230

24.1 Introduction 230

24.2 Algorithms 230

24.3 The structure of the Time-delay System Toolbox 231

24.4 Descriptions of some programs 232

Part II Invariant and generalized derivatives of functions and functionals 251

25 The invariant derivative of functions 253

25.1 The invariant derivative of functions 253

25.2 Examples 256

25.3 Relationship between the invariant derivative and the Sobolev generalized derivative 258

26 Relation of the Sobolev generalized derivative and the generalized derivative of the distribution theory 261

26.1 Affinitivity of the generalized derivative of the distribution theory and the Sobolev generalized derivative 261

26.2 Multiplication of generalized functions at the Hamel basis 262

Bibliography 267

Index 271

Sprache englisch
Maße 163 x 241 mm
Gewicht 540 g
Themenwelt Mathematik / Informatik Mathematik Analysis
ISBN-10 1-118-99836-7 / 1118998367
ISBN-13 978-1-118-99836-6 / 9781118998366
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