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...
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| Main Authors: | |
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| Corporate Authors: | |
| Published: |
John Wiley & Sons ; Scrivener Publishing,
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| 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 dist |
| 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 |