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Umfang: XIII, 234 S. : Ill., graph. Darst.

ISBN: 0-12-525550-0

Serie: Mathematics in science and engineering 111

Sprache: Englisch

Fachgebiete: Mathematik

Schlagwort(e): Analysis ; Ableitung gebrochener Ordnung ; Spezielle Funktion ; Funktionentheorie

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URL: Verlagsangaben

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Umfang: 1 online resource (252 p.) : , ill.

ISBN: 9780080956206 (e-book) , 9780125255509 (hbk.)

Serie: Mathematics in science and engineering ; v. 111

Inhalt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory and its particular branches, such as optimal filtering and information compression.

Anmerkung: Description based upon print version of record. , 1. Introduction; 1.1 Historical Survey; 1.2 Notation; 1.3 Properties of the Gamma Function; 2. Differentiation and Integration to Integer Order; 2.1 Symbolism; 2.2 Conventional Definitions; 2.3 Composition Rule for Mixed Integer Orders; 2.4 Dependence of Multiple Integrals on Lower Limit; 2.5 Product Rule for Multiple Integrals; 2.6 The Chain Rule for Multiple Derivatives; 2.7 Iterated Integrals; 2.8 Differentiation and Integration of Series; 2.9 Differentiation and Integration of Powers 2.10 Differentiation and Integration of Hypergeometrics; 3. Fractional Derivatives and Integrals: Definitions and Equivalences; 3.1 Differintegrable Functions; 3.2 Fundamental Definitions; 3.3 Identity of Definitions; 3.4 Other General Definitions; 3.5 Other Formulas Applicable to Analytic Functions; 3.6 Summary of Definitions; 4. Differintegration of Simple Functions; 4.1 The Unit Function; 4.2 The Zero Function; 4.3 The Function x - a; 4.4 The Function [x - a]a; 5. General Properties; 5.1 Linearity; 5.2 Differintegration Term by Term; 5.3 Homogeneity; 5.4 Scale Change; 5.5 Leibniz's Rule; 5.6 Chain Rule; 5.7 Composition Rule; 5.8 Dependence on Lower Limit; 5.9 Translation; 5.10 Behavior Near Lower Limit; 5.11 Behavior Far from Lower Limit; 6. Differintegration of More Complex Functions; 6.1 The Binomial Function [C - cx]p; 6.2 The Exponential Function exp(C - cx); 6.3 The Functions xq/[1 - x] and xp/[1 - x] and [1 - X]q-1; 6.4 The Hyperbolic and Trigonometric Functions sinh(vx) and sin (x); 6.5 The Bessel Functions; 6.6 Hypergeometric Functions; 6.7 Logarithms; 6.8 The Heaviside and Dirac Functions; 6.9 The Sawtooth Function; 6.10 Periodic Functions; 6.11 Cyclodifferential Functions6.12 The Function xq - 1 exp[-1/x]; 7. Semiderivatives and Semiintegrals; 7.1 Definitions; 7.2 General Properties; 7.3 Constants and Powers; 7.4 Binomials; 7.5 Exponential and Related Functions; 7.6 Trigonometric and Hyperbolic Functions; 7.7 Bessel and Struve Functions; 7.8 Generalized Hypergeometric Functions; 7.9 Miscellaneous Functions; 8. Techniques in the Fractional Calculus; 8.1 Laplace Transformation; 8.2 Numerical Differintegration; 8.3 Analog Differintegration; 8.4 Extraordinary Differential Equations; 8.5 Semidifferential Equations; 8.6 Series Solutions; 9. Representation of Transcendental Functions; 9.1 Transcendental Functions as Hypergeometrics; 9.2 Hypergeometrics with K 〉 L; 9.3 Reduction of Complex Hypergeometrics; 9.4 Basis Hypergeometrics; 9.5 Synthesis of K = L Transcendentals; 9.6 Synthesis of K = L - 1 Transcendentals; 9.7 Synthesis of K = L - 2 Transcendentals; 10. Applications in the Classical Calculus; 10.1 Evaluation of Definite Integrals and Infinite Sums; 10.2 Abel's Integral Equation; 10.3 Solution of Bessel's Equation; 10.4 Candidate Solutions for Differential Equations; 10.5 Function Families; 11. Applications to Diffusion Problems; 11.1 Transport in a Semiinfinite Medium; 11.2 Planar Geometry; 11.3 Spherical Geometry; 11.4 Incorporation of Sources and Sinks; 11.5 Transport in Finite Media; 11.6 Diffusion on a Curved Surface; References; Index.

Sprache: Englisch

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Umfang: Online Ressource (xiii, 234 pages) , illustrations.

Ausgabe: Online-Ausg. [S.l.] HathiTrust Digital Library Online-Ausg. [S.l.] : HathiTrust Digital Library

ISBN: 9780080956206 , 0080956203 , 9780125255509 , 9781282702899

Serie: Mathematics in science and engineering v. 111

Inhalt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank

Anmerkung: Includes bibliographical references (pages 219-223) and index. - Print version record , Print version record , Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002 , Front Cover; THE FRACTIONAL CALCULUS; Copyright Page; CONTENTS; Preface; Acknowledgments; Chapter 1. INTRODUCTION; 1.1 Historical Survey; 1.2 Notation; 1.3 Properties of the Gamma Function; Chapter 2. DIFFERENTIATION AND INTEGRATION TO INTEGER ORDER; 2.1 Symbolism; 2.2 Conventional Definitions; 2.3 Composition Rule for Mixed Integer Orders; 2.4 Dependence of Multiple Integrals on Lower Limit; 2.5 Product Rule for Multiple Integrals; 2.6 The Chain Rule for Multiple Derivatives; 2.7 Iterated Integrals; 2.8 Differentiation and Integration of Series; 2.9 Differentiation and Integration of Powers , 2.10 Differentiation and Integration of HypergeometricsChapter 3. FRACTIONAL DERIVATIVES AND INTEGRALS: DEFINITIONS AND EQUIVALENCES; 3.1 Differintegrable Functions; 3.2 Fundamental Definitions; 3.3 Identity of Definitions; 3.4 Other General Definitions; 3.5 Other Formulas Applicable to Analytic Functions; 3.6 Summary of Definitions; Chapter 4. DIFFERINTEGRATION OF SIMPLE FUNCTlONS; 4.1 The Unit Function; 4.2 The Zero Function; 4.3 The Function x - a; 4.4 The Function [x - a]a; Chapter 5. GENERAL PROPERTIES; 5.1 Linearity; 5.2 Differintegration Term by Term; 5.3 Homogeneity; 5.4 Scale Change , 5.5 Leibniz's Rule5.6 Chain Rule; 5.7 Composition Rule; 5.8 Dependence on Lower Limit; 5.9 Translation; 5.10 Behavior Near Lower Limit; 5.11 Behavior Far from Lower Limit; Chapter 6. DIFFERINTEGRATION OF MORE COMPLEX FUNCTIONS; 6.1 The Binomial Function [C - cx]p; 6.2 The Exponential Function exp(C - cx); 6.3 The Functions xq/[1 - x] and xp/[1 - x] and [1 - X]q-1; 6.4 The Hyperbolic and Trigonometric Functions sinh(vx) and sin (x); 6.5 The Bessel Functions; 6.6 Hypergeometric Functions; 6.7 Logarithms; 6.8 The Heaviside and Dirac Functions; 6.9 The Sawtooth Function; 6.10 Periodic Functions , 6.11 Cyclodifferential Functions6.12 The Function xq - 1 exp[-1/x]; Chapter 7. SEMIDERIVATIVES AND SEMIINTEG RALS; 7.1 Definitions; 7.2 Genera1 Properties; 7.3 Constants and Powers; 7.4 Binomials; 7.5 Exponential and Related Functions; 7.6 Trigonometric and Hyperbolic Functions; 7.7 Bessel and Struve Functions; 7.8 Generalized Hypergeometric Functions; 7.9 Miscellaneous Functions; Chapter 8. TECHNIQUES IN THE FRACTIONAL CALCULUS; 8.1 Laplace Transformation; 8.2 Numerical Differintegration; 8.3 Analog Differintegration; 8.4 Extraordinary Differential Equations; 8.5 Semidifferential Equations , 8.6 Series SolutionsChapter 9. REPRESENTATION OF TRANSCENDENTAL FUNCTIONS; 9.1 Transcendental Functions as Hypergeometrics; 9.2 Hypergeometrics with K 〉 L; 9.3 Reduction of Complex Hypergeometrics; 9.4 Basis Hypergeometrics; 9.5 Synthesis of K = L Transcendentals; 9.6 Synthesis of K = L - 1 Transcendentals; 9.7 Synthesis of K = L - 2 Transcendentals; Chapter 10. APPLICATIONS IN THE CLASSICAL CALCULUS; 10.1 Evaluation of Definite Integrals and Infinite Sums; 10.2 Abel's Integral Equation; 10.3 Solution of Bessel's Equation; 10.4 Candidate Solutions for Differential Equations , 10.5 Function Families , Online-Ausg. [S.l.] : HathiTrust Digital Library , Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.

Weitere Ausg.: ISBN 0125255500

Weitere Ausg.: Erscheint auch als Druck-Ausgabe Oldham, Keith B Fractional calculus New York, Academic Press, 1974

Sprache: Englisch

Schlagwort(e): Ableitung gebrochener Ordnung ; Electronic books ; Electronic books

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