{"product_id":"ismooth-analysis-9781118998366","title":"iSmooth Analysis","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cb\u003e\u003ci\u003ei\u003c\/i\u003e-SMOOTH ANALYSIS\u003c\/b\u003e \u003cp\u003e\u003cb\u003eA 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.\u003c\/b\u003e \u003c\/p\u003e\u003cp\u003e\u003ci\u003ei\u003c\/i\u003e-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 \u003ci\u003ei\u003c\/i\u003e-smooth analysis is the investigation of the relation of invariant derivatives with the Sobolev generalized derivative and the generalized derivative of distribution theory. \u003c\/p\u003e\u003cp\u003eUntil now, \u003ci\u003ei\u003c\/i\u003e-smooth analysis has been developed mainly to apply to the theory of functional differential equations, and the goal of this book is to present \u003ci\u003ei\u003c\/i\u003e-smooth analysis as a branch of functional analysis. The notion of the invariant derivative (\u003ci\u003ei\u003c\/i\u003e-derivative) of nonlinear\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\"This is a research monograph dedicated to people interested mainly in generalized differentiation methods applied to numerical solutions of functional-differential equations.\" (Zentralblatt MATH 2016)\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003c\/p\u003e\u003cp\u003ePreface xi\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I Invariant derivatives of functionals and numerical methods for functional differential equations 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 The invariant derivative of functionals 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1 Functional derivatives 3\u003c\/p\u003e \u003cp\u003e1.1 The Frechet derivative 4\u003c\/p\u003e \u003cp\u003e1.2 The Gateaux derivative 4\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Classification of functionals on C[a, b] 5\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Regular functionals 5\u003c\/p\u003e \u003cp\u003e2.2 Singular functionals 6\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Calculation of a functional along a line 6\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Shift operators 6\u003c\/p\u003e \u003cp\u003e3.2 Superposition of a functional and a function 7\u003c\/p\u003e \u003cp\u003e3.3 Dini derivatives 8\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Discussion of two examples 8\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Derivative of a function along a curve 8\u003c\/p\u003e \u003cp\u003e4.2 Derivative of a functional along a curve 9\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 The invariant derivative 11\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 The invariant derivative 11\u003c\/p\u003e \u003cp\u003e5.2 The invariant derivative in the class B[a, b] 12\u003c\/p\u003e \u003cp\u003e5.3 Examples 13\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Properties of the invariant derivative 16\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Principles of calculating invariant derivatives 16\u003c\/p\u003e \u003cp\u003e6.2 The invariant differentiability and invariant continuity 19\u003c\/p\u003e \u003cp\u003e6.3 High order invariant derivatives 20\u003c\/p\u003e \u003cp\u003e6.4 Series expansion 21\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Several variables 21\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Notation 21\u003c\/p\u003e \u003cp\u003e7.2 Shift operator 21\u003c\/p\u003e \u003cp\u003e7.3 Partial invariant derivative 22\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Generalized derivatives of nonlinear functionals 22\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Introduction 22\u003c\/p\u003e \u003cp\u003e8.2 Distributions (generalized functions) 24\u003c\/p\u003e \u003cp\u003e8.3 Generalized derivatives of nonlinear distributions 25\u003c\/p\u003e \u003cp\u003e8.4 Properties of generalized derivatives 27\u003c\/p\u003e \u003cp\u003e8.5 Generalized derivative (multidimensional case) 28\u003c\/p\u003e \u003cp\u003e8.6 The space SD of nonlinear distributions 29\u003c\/p\u003e \u003cp\u003e8.7 Basis on shift 30\u003c\/p\u003e \u003cp\u003e8.8 Primitive 31\u003c\/p\u003e \u003cp\u003e8.9 Generalized solutions of nonlinear differential equations 34\u003c\/p\u003e \u003cp\u003e8.10 Linear differential equations with variables coeffecients 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Functionals on Q[−t ; 0] 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Regular functionals 39\u003c\/p\u003e \u003cp\u003e9.2 Singular functionals 40\u003c\/p\u003e \u003cp\u003e9.3 Specific functionals 40\u003c\/p\u003e \u003cp\u003e9.4 Support of a functional 41\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Functionals on R × Rn × Q[−t; 0] 42\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Regular functionals 42\u003c\/p\u003e \u003cp\u003e10.2 Singular functionals 44\u003c\/p\u003e \u003cp\u003e10.3 Volterra functionals 44\u003c\/p\u003e \u003cp\u003e10.4 Support of a functional 45\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 The invariant derivative 45\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Invariant derivative of a functional 46\u003c\/p\u003e \u003cp\u003e11.2 Examples 48\u003c\/p\u003e \u003cp\u003e11.3 Invariant continuity and invariant differentiability 58\u003c\/p\u003e \u003cp\u003e11.4 Invariant derivative in the class B[−t; 0] 59\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Coinvariant derivative 65\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Coinvariant derivative of functionals 65\u003c\/p\u003e \u003cp\u003e12.2 Coinvariant derivative in a class B[−t; 0] 68\u003c\/p\u003e \u003cp\u003e12.3 Properties of the coinvariant derivative 71\u003c\/p\u003e \u003cp\u003e12.4 Partial derivatives of high order 73\u003c\/p\u003e \u003cp\u003e12.5 Formulas of i–smooth calculus for mappings 75\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 Brief overview of Functional Differential Equation theory 76\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Functional Differential Equations 76\u003c\/p\u003e \u003cp\u003e13.2 FDE types 78\u003c\/p\u003e \u003cp\u003e13.3 Modeling by FDE 80\u003c\/p\u003e \u003cp\u003e13.4 Phase space and FDE conditional representation 81\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Existence and uniqueness of FDE solutions 84\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 The classic solutions 84\u003c\/p\u003e \u003cp\u003e14.2 Caratheodory solutions 92\u003c\/p\u003e \u003cp\u003e14.3 The step method for systems with discrete delays 94\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Smoothness of solutions and expansion into the Taylor series 95\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Density of special initial functions 98\u003c\/p\u003e \u003cp\u003e15.2 Expansion of FDE solutions into Taylor series 100\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 The sewing procedure 103\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 General case 104\u003c\/p\u003e \u003cp\u003e16.2 Sewing (modification) by polynomials 105\u003c\/p\u003e \u003cp\u003e16.3 The sewing procedure of the second order 107\u003c\/p\u003e \u003cp\u003e16.4 Sewing procedure of the second order for linear delay differential equation 109\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Numerical methods for functional differential equations 113\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Numerical Euler method 115\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Numerical Runge-Kutta-like methods 118\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Methods of interpolation and extrapolation 119\u003c\/p\u003e \u003cp\u003e18.2 Explicit Runge-Kutta-like methods 127\u003c\/p\u003e \u003cp\u003e18.3 Order of the residual of ERK-methods 132\u003c\/p\u003e \u003cp\u003e18.4 Implicit Runge-Kutta-like methods 136\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 Multistep numerical methods 142\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Numerical models 143\u003c\/p\u003e \u003cp\u003e19.2 Order of convergence 143\u003c\/p\u003e \u003cp\u003e19.3 Approximation order. Starting procedure 145\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 Startingless multistep methods 146\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 Explicit methods 147\u003c\/p\u003e \u003cp\u003e20.2 Implicit methods 148\u003c\/p\u003e \u003cp\u003e20.3 Startingless multistep methods 150\u003c\/p\u003e \u003cp\u003e\u003cb\u003e21 Nordsik methods 152\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 Methods based on calculation of high order derivatives 155\u003c\/p\u003e \u003cp\u003e21.2 Various methods based on the separation of finite-dimensional and infinite-dimensional components of the phase state 158\u003c\/p\u003e \u003cp\u003e\u003cb\u003e22 General linear methods of numerical solving functional differential equations 162\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e22.1 Introduction 162\u003c\/p\u003e \u003cp\u003e22.2 Methodology of classification numerical FDE models 173\u003c\/p\u003e \u003cp\u003e22.3 Necessary and sufficient conditions of convergence with order p 181\u003c\/p\u003e \u003cp\u003e22.4 Asymptotic expansion of the global error 186\u003c\/p\u003e \u003cp\u003e\u003cb\u003e23 Algorithms with variable step-size and some aspects of computer realization of numerical models 196\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e23.1 ERK-like methods with variable step 197\u003c\/p\u003e \u003cp\u003e23.2 Methods of interpolation and extrapolation of discrete model prehistory 202\u003c\/p\u003e \u003cp\u003e23.3 Choice of the step size 207\u003c\/p\u003e \u003cp\u003e23.4 Influence of the approximate calculating functionals of the right-hand side of FDEs 212\u003c\/p\u003e \u003cp\u003e23.5 Test problems 217\u003c\/p\u003e \u003cp\u003e\u003cb\u003e24 Soft ware package Time-delay System Toolbox 230\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e24.1 Introduction 230\u003c\/p\u003e \u003cp\u003e24.2 Algorithms 230\u003c\/p\u003e \u003cp\u003e24.3 The structure of the Time-delay System Toolbox 231\u003c\/p\u003e \u003cp\u003e24.4 Descriptions of some programs 232\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II Invariant and generalized derivatives of functions and functionals 251\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e25 The invariant derivative of functions 253\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e25.1 The invariant derivative of functions 253\u003c\/p\u003e \u003cp\u003e25.2 Examples 256\u003c\/p\u003e \u003cp\u003e25.3 Relationship between the invariant derivative and the Sobolev generalized derivative 258\u003c\/p\u003e \u003cp\u003e\u003cb\u003e26 Relation of the Sobolev generalized derivative and the generalized derivative of the distribution theory 261\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e26.1 Affinitivity of the generalized derivative of the distribution theory and the Sobolev generalized derivative 261\u003c\/p\u003e \u003cp\u003e26.2 Multiplication of generalized functions at the Hamel basis 262\u003c\/p\u003e \u003cp\u003eBibliography 267\u003c\/p\u003e \u003cp\u003eIndex 271\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49406964957527,"sku":"9781118998366","price":152.06,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118998366.jpg?v=1730497716","url":"https:\/\/bookcurl.com\/products\/ismooth-analysis-9781118998366","provider":"Book Curl","version":"1.0","type":"link"}