{"product_id":"i-smooth-analysis-theory-and-applications-hardback-9781118998366","title":"i-Smooth Analysis; Theory and Applications (Hardback) 9781118998366","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003ei-Smooth Analysis\u003c\/font\u003e\u003cbr\u003e\r\n\u003cfont size=\"5\"\u003eTheory and Applications\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eA. V. Kim (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9781118998366, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 14 July 2015\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e296 pages\u003cbr\u003e24.1 x 16.3 x 2 cm, 0.54 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cem\u003e\u003cfont size=\"3\"\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)\u003c\/font\u003e\u003c\/em\u003e\u003c\/p\u003e\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\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\n\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\n\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 functionals has been introduced in mathematics, and this in turn developed the corresponding \u003ci\u003ei\u003c\/i\u003e-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. \u003c\/p\u003e\n\u003cp\u003e\u003cb\u003e\u003ci\u003ei-Smooth Analysis: Theory and Applications\u003c\/i\u003e\u003c\/b\u003e \u003c\/p\u003e\n\u003cul\u003e\n\u003cli\u003eIntroduces a new class of derivatives of functions and functionals, a revolutionary new approach\u003c\/li\u003e \u003cli\u003eEstablishes a relationship with the generalized Sobolev derivative and the generalized derivative of the distribution theory\u003c\/li\u003e \u003cli\u003ePresents the complete theory of i-smooth analysis\u003c\/li\u003e \u003cli\u003eContains the theory of FDE numerical method, based on i-smooth analysis\u003c\/li\u003e \u003cli\u003eExplores a new direction of i-smooth analysis, the theory of the invariant derivative of functions\u003c\/li\u003e \u003cli\u003eIs of interest to all mathematicians, engineers studying processes with delays, and physicists who study hereditary phenomena in nature.\u003c\/li\u003e\n\u003c\/ul\u003e \u003cp\u003e\u003cb\u003eAUDIENCE\u003c\/b\u003e \u003c\/p\u003e\n\u003cp\u003eMathematicians, applied mathematicians, engineers , physicists, students in mathematics\u003c\/p\u003e\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\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\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematics [\u003ca title=\"See our other books on Mathematics\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematics%20%5BPB%5D%22\"\u003ePB\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley-Scrivener","offers":[{"title":"Brand New","offer_id":52421390106904,"sku":"9781118998366","price":126.86,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/files\/9781118998366.jpg?v=1784594387","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/i-smooth-analysis-theory-and-applications-hardback-9781118998366","provider":"Freshly Printed Books","version":"1.0","type":"link"}