i-Smooth Analysis E-Book

A. V. Kim

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 256

Veröffentlichungsjahr: 2015

Beschreibung

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

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Part I: Invariant Derivatives of Functionals and Numerical Methods for Functional Differential Equations

Chapter 1: The Invariant Derivative of Functionals

1 Functional Derivatives

2 Classification of Functionals on C[a,b]

3 Calculation of a Functional Along a Line

4 Discussion of Two Examples

5 The Invariant Derivative

6 Properties of the Invariant Derivative

7 Several Variables

8 Generalized Derivatives of Nonlinear Functionals

9 Functionals on Q[−τ, 0)

10 Functionals on R × Rn × Q[−τ, 0)

11 The Invariant Derivative

12 Coinvariant Derivative

13 Brief Overview of Functional Differential Equation Theory

14 Existence and Uniqueness of FDE Solutions

15 Smoothness of Solutions and Expansion into the Taylor Series

16 The Sewing Procedure

Chapter 2: Numerical Methods for Functional Differential Equations

17 Numerical Euler Method

18 Numerical Runge-Kutta-Like Methods

19 Multistep Numerical Methods

20 Startingless Multistep Methods

21 Nordsik Methods

22 General Linear Methods of Numerical Solving Functional Differential Equations

23 Algorithms with Variable Step-Size and Some Aspects of Computer Realization of Numerical Models

24 Software Package Time-Delay System Toolbox

Part II: Invariant and Generalized Derivatives of Functions and Functionals

25 The Invariant Derivative of Functions

26 Relation of the Sobolev Generalized Derivative and the Generalized Derivative of the Distribution Theory

Bibliography

Index

i-Smooth Analysis

Scrivener Publishing100 Cummings Center, Suite 541JBeverly, MA 01915-6106

Publishers at Scrivener Martin Scrivener([email protected]) Phillip Carmical ([email protected])

Copyright © 2015 by Scrivener Publishing LLC. All rights reserved.

Co-published by John Wiley & Sons, Inc. Hoboken, New Jersey and Scrivener Publishing LLC, Salem, Massachusetts. Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at [phone], outside the United States at [phone] or fax [phone].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. For more information about Scrivener products please visit www.scrivenerpublishing.com.

Library of Congress Cataloging-in-Publication Data:

ISBN 978-1-118-99836-6

To sweet memory of my brother Vassilii

Preface

i-Smooth analysis is the branch of functional analysis, that considers the theory and applications of the invariant derivatives of functions and functionals.

The present book includes two parts of the i-smooth analysis theory. The first part presents the theory of the invariant derivatives of functionals.

The second part of the i-smooth analysis is the theory of the invariant derivatives of functions.

Until now, i-smooth analysis has been developed mainly to apply to the theory of functional differential equations. The corresponding results are summarized in the books [17], [18], [19] and [22].

This edition is an attempt to present i-smooth analysis as a branch of the functional analysis.

There are two classic notions of generalized derivatives in mathematics:

a) The Sobolev generalized derivative of functions [34], [35];

b) The generalized derivative of the distribution theory [32].

In works [17], [18], [19] and [25] the notion of the invariant derivative (i-derivative) of nonlinear functionals was introduced, 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.

The theory is based on the notion and constructions of the invariant derivatives of functionals that was introduced around 1980.

Beginning with the first relevant publication in this direction there arose two questions:

Question A1:Is it possible to introduce a notion of the invariant derivative of functions?

Question A2:Is the invariant derivative of functions concerned with the Sobolev generalized derivative?

This book arose as a result of searching for answers to the questionsA1 & A2: there were found positive answers on both these questions and the corresponding theory is presented in the second part.

Another question that initiated the idea for this EDITION was the following

Question B:Does anything besides a terminological and mathematical relation between the Sobolev generalized derivative of functions and the generalized derivative of distributions?

At first glance the question looks incorrect, because the first derivative concerns the finite dimensional functions whereas the second one applies to functional objects (distributions – linear continuous functionals).

Nevertheless as it is shown in the second part the answer to the questionB came out positive: the mathematical relation between both generalized derivatives can be established by means of the invariant derivative.

One of the main goals writing this book was to clarify the nature of the invariant derivatives and their status in the present system of known derivatives. By this reason we do not pay much attention to applications of the invariant derivatives and concentrate on developing the theory.

The edition is not a textbook and is assigned for specialists, so statements and constructions regarding standard mathematical courses are used without justification or additional comments. Proofs of some new propositions contain only key moments if rest of the details are obvious.

Though the edition is not a textbook, the material is appropriate for graduate students of mathematical departments and be interesting for engineers and physicists. Throughout the book generally accepted notation of the functional analysis is used and new notation is used only for the latest notions.

Acknowledgements. At the initial stage of developing the invariant derivative theory the support of the professor V. K. Ivanov had been very important for me: during personal discussions and at his department seminars of various aspects of the theory were discussed. Because of his recommendations and submissions my first works on the matter were published.

At the end of the 1970-s and the beginning of the 1980-s, many questions were cleared up in discussions with my friends and colleagues: PhD-students Alexander Babenko, Alexander Zaslavskii and Alexander Ustyuzhanin. Theory of numerical methods for solving functional differential equations (FDE) based on i-smooth analysis was developed in cooperation with Dr. V. Pimenov1.

The attention and support of professor A. D. Myshkis was important to me during a critical stage of i-smooth analysis development.

I am thankful to Dr. U. A. Alekseeva, professor V. V. Arestov, professor A. G. Babenko, professor Neville J. Ford for their familiarization with the preliminary versions of the books and useful comments and recommendations.

The author is thankful to a book editor for the great work on book improvement and to A. V. Ivanov for preparation of the printing version of the book.

Research was supported by the Russian Foundation for Basic Research (projects 08-01-00141, 14-01-00065, 14-01- 00477, 13-01-00110), the program “Fundamental Sciences for Medicine” of the Presidium of the Russian Academy of Sciences, the Ural-Siberia interdisciplinary project.

1 The author developed a general approach to elaborating numerical methods for FDE and Dr. V. Pimenov developed complete theory, presented in the second chapter of this book

Part I

INVARIANT DERIVATIVES OF FUNCTIONALS AND NUMERICAL METHODS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

Chapter 1

The invariant derivative of functionals

In this chapter we consider the basic constructions of nonlinear i-smooth calculus of nonlinear on C[a, b] functionals.

A study of nonlinear mappings can be realized by local approximations of nonlinear operators by linear operators. Corresponding linear approximations are called derivatives of nonlinear mappings. Depending on the form of linear approximations of various types of derivatives can be introduced. Further

Ck[a, b] is the space of k–times continuous differentiable functions ϕ(·) : [a, b] → R;

C∞[a, b] is the space of infinitely differentiable functions ϕ(·): [a, b] → R.

1 Functional derivatives

In the general case a derivative of a mapping

(1.1)

(X & Y are topological vector spaces) at a point x0X is a linear mapping

(1.2)

which approximates in an appropriate sense the difference

(1.3)

by h. Subject to the specific form of difference approximation difference(1.3) one can obtain various notions of derivatives.

Consider the classic derivatives of functionals1

(1.4)

1.1 The Frechet derivative

The Frechet derivative (strong derivative) of the functional (1.4) at a point ϕ(·) C[a, b] is a linear continuous functional

(1.5)

satisfying the condition

(1.6)

where .

If there exists a functional L, satisfying the above conditions, then it is denoted by ψ → V’[ϕ] ψ and is called the Frechet differential.

1.2 The Gateaux derivative

The Gateaux derivative of the functional (1.4) at a point ϕ C[a, b] is a linear mapping V’Γ[ϕ] : C[a, b] → R, satisfying the condition

(1.7)

where .

2 Classification of functionals on C[a,b]

Many specific classes of functionals have integral forms. The investigation of such integral functionals formed the basis of the general functional analysis theory.

Along with integral functionals (which are called regular functionals) beginning with the works of Dirac and Schwartz mathematicians we generally use singular functionals among which the first and most well known is the δ–function.

Further as a rule integrals are understandood in the Riemann sense 2.

2.1 Regular functionals

Analysing the structure of specific functionals one can single out basic (elementary) types of functionals on []: