Kim A.V. i-Smooth Analysis: Theory and Applications

Scrivener Publishing/Wiley, 2015. — 285 p. — ISBN 978-1-118-99836-6.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.Invariant derivatives of functionals and numerical methods for functional differential equations
The invariant derivative of functionals
Functional derivatives
Classification of functionals on C[a, b]
Calculation of a functional along a line
Discussion of two examples
The invariant derivative
Properties of the invariant derivative
Several variables
Generalized derivatives of nonlinear functionals
Functionals on Q[−t ; 0]
Functionals on R × Rn × Q[−t; 0]
The invariant derivative
Coinvariant derivative
Brief overview of Functional Differential Equation theory
Existence and uniqueness of FDE solutions
Smoothness of solutions and expansion into the Taylor series
The sewing procedure
Numerical Euler method
Numerical Runge-Kutta-like methods
Multistep numerical methods
Startingless multistep methods
Nordsik methods
General linear methods of numerical solving functional differential equations
Algorithms with variable step-size and some aspects of computer realization of numerical models
Soft ware package Time-delay System Toolbox
Invariant and generalized derivatives of functions and functionals
The invariant derivative of functions
Relation of the Sobolev generalized derivative and the generalized derivative of the distribution theory