i-Smooth Analysis

Name: i-Smooth Analysis | Theory and Applications
Brand: Wiley-Scrivener
Price: 171.99 EUR
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Theory and Applications

A. V. Kim(Autor*in)

Wiley-Scrivener (Verlag)

Erschienen am 29. Mai 2015

296 Seiten

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978-1-118-99854-0 (ISBN)

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Cover
Title Page
Copyright Page
Contents
Preface
Part I Invariant derivatives of functionals and numerical methods for functional differential equations
1 The invariant derivative of functionals
1 Functional derivatives
1.1 The Frechet derivative
1.2 The Gateaux derivative
2 Classifi cation of functionals on C[a, b]
2.1 Regular functionals
2.2 Singular functionals
3 Calculation of a functional along a line
3.1 Shift operators
3.2 Superposition of a functional and a function
3.3 Dini derivatives
4 Discussion of two examples
4.1 Derivative of a function along a curve
4.2 Derivative of a functional along a curve
5 The invariant derivative
5.1 The invariant derivative
5.2 The invariant derivative in the class B[a, b]
5.3 Examples
6 Properties of the invariant derivative
6.1 Principles of calculating invariant derivatives
6.2 The invariant differentiability and invariant continuity
6.3 High order invariant derivatives
6.4 Series expansion
7 Several variables
7.1 Notation
7.2 Shift operator
7.3 Partial invariant derivative
8 Generalized derivatives of nonlinear functionals
8.1 Introduction
8.2 Distributions (generalized functions)
8.3 Generalized derivatives of nonlinear distributions
8.4 Properties of generalized derivatives
8.5 Generalized derivative (multidimensional case)
8.6 The space SD of nonlinear distributions
8.7 Basis on shift
8.8 Primitive
8.9 Generalized solutions of nonlinear differential equations
8.10 Linear differential equations with variables coeffecients
9 Functionals on Q[-?, 0)
9.1 Regular functionals
9.2 Singular functionals
9.3 Specific functionals
9.4 Support of a functional
10 Functionals on R × Rn × Q[-?, 0)
10.1 Regular functionals
10.2 Singular functionals
10.3 Volterra functionals
10.4 Support of a functional
11 The invariant derivative
11.1 Invariant derivative of a functional
11.2 Examples
11.3 Invariant continuity and invariant differentiability
11.4 Invariant derivative in the class B[-?, 0)
12 Coinvariant derivative
12.1 Coinvariant derivative of functionals
12.2 Coinvariant derivative in a class B[-?, 0)
12.3 Properties of the coinvariant derivative
12.4 Partial derivatives of high order
12.5 Formulas of i-smooth calculus for mappings
13 Brief overview of Functional Differential Equation theory
13.1 Functional Differential Equations
13.2 FDE types
13.3 Modeling by FDE
13.4 Phase space and FDE conditional representation
14 Existence and uniqueness of FDE solutions
14.1 The classic solutions
14.2 Caratheodory solutions
14.3 The step method for systems with discrete delays
15 Smoothness of solutions and expansion into the Taylor series
15.1 Density of special initial functions
15.2 Expansion of FDE solutions into Taylor series
16 The sewing procedure
16.1 General case
16.2 Sewing (modification) by polynomials
16.3 The sewing procedure of the second order
16.4 Sewing procedure of the second order for linear delay differential equation
2 Numerical methods for functional differential equations
17 Numerical Euler method
18 Numerical Runge-Kutta-like methods
18.1 Methods of interpolation and extrapolation
18.2 Explicit Runge-Kutta-like methods
18.3 Order of the residual of ERK-methods
18.4 Implicit Runge-Kutta-like methods
19 Multistep numerical methods
19.1 Numerical models
19.2 Order of convergence
19.3 Approximation order. Starting procedure
20 Startingless multistep methods
20.1 Explicit methods
20.2 Implicit methods
20.3 Startingless multistep methods
21 Nordsik methods
21.1 Methods based on calculation of high order derivatives
21.2 Various methods based on the separation of finite-dimensional and infinite-dimensional components of the phase state
22 General linear methods of numerical solving functional differential equations
22.1 Introduction
22.2 Methodology of classification numerical FDE models
22.3 Necessary and sufficient conditions of convergence with order p
22.4 Asymptotic expansion of the global error
23 Algorithms with variable step-size and some aspects of computer realization of numerical models
23.1 ERK-like methods with variable step
23.2 Methods of interpolation and extrapolation of discrete model prehistory
23.3 Choice of the step size
23.4 Influence of the approximate calculating functionals of the righ-hand side of FDEs
23.5 Test problems
24 Software package Time-delay System Toolbox
24.1 Introduction
24.2 Algorithms
24.3 The structure of the Time-delay System Toolbox
24.4 Descriptions of some programs
Part II Invariant and generalized derivatives of functions and functionals
25 The invariant derivative of functions
25.1 The invariant derivative of functions
25.2 Examples
25.3 Relationship between the invariant derivative and the Sobolev generalized derivative
26 Relation of the Sobolev generalized derivative and the generalized derivative of the distribution theory
26.1 Affinitivity of the generalized derivative of the distribution theory and the Sobolev generalized derivative
26.2 Multiplication of generalized functions at the Hamel basis
Bibliography
Index
EULA

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