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

Table of Contents:
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