Screen LibraryI-Smooth analysis : theory and applications /
The edition introduces a new class of invariant derivatives and shows their relationships with other derivatives, such as the Sobolev generalized derivative and the generalized derivative of the distribution theory. This is a new direction in mathematics. i-Smooth analysis is the branch of functiona...
Saved in:
| Main Authors: | |
|---|---|
| Corporate Authors: | |
| Published: |
John Wiley & Sons ; Scrivener Publishing,
|
| Publisher Address: | Hoboken, New Jersey : Salem, Massachusetts : |
| Publication Dates: | [2015] |
| Literature type: | eBook |
| Language: | English |
| Subjects: | |
| Online Access: |
http://onlinelibrary.wiley.com/book/10.1002/9781118998519 |
| Summary: |
The edition introduces a new class of invariant derivatives and shows their relationships with other derivatives, such as the Sobolev generalized derivative and the generalized derivative of the distribution theory. This is a new direction in mathematics. 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. |
| Carrier Form: | 1 online resource |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: |
9781118998526 1118998529 9781118998519 1118998510 1118998367 9781118998366 |
| Index Number: | QA372 |
| CLC: | O175 |
| Contents: |
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 curve5 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 distributions8.4 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 functionals10.4 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 Equations13.2 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 |