实例介绍
【实例简介】
分数阶微分方程方面的入门必读数目，有理论方法和应用
This is olume 198 in MATHEMATICS IN SCIENCE AND ENGINEERING By Igor Podlubny, Technical University of Kosice, Slovak Republic A list of recent titles in this series appears on page v of this volume FRACTIONAL DIFFERENTIAL EQUATIONS An Introduction to Fractional Derivatives ractional Differential equations to methods of their Solution and some of their Applications y Igor Podlubny Technical University of Kosice, Slovak Republic A ACADEMIC PRESS San diego· Boston◆ New york london· Sydney· Tokyo· Toront This book is printed on ac id-free paper Copyright. 199 9 by ACADEMIC PRESS All Right. s Reserved No part of this publication may be reproduced or transmit ted in any for by photostat, microfill, or any other means without written permission fron the publishers ACADEMIC PRESS 525B Street Suite 1 900 San diego, California 92101-4495, USA http://www.apnet.com ACADEAIC PRESS LONDON NWI 7DX. UK http://www.hbik.co.lk/ap/ A catalogue record for t.his book is available froin the British library lSBN(-12558840-2 Printed in the United states of america 9900010203MP987654321 Mathematics in Science and Engineering Edited by Willian F. Aes, Georgia Institute of Technology nt. title T.A. BI Volterra Interal and Different al Equatio Ran P. Kanwal, Generalized Function.s: TILe ory and Technique Marc Mangel. Decision and Control in Uncertain Resource Systems Yoshikazu Sawaragi lirot aka Nakayana, and T'ctsuzo Tanino. Theory of Multi-objcctin e Optimization Edward Haug, Kying K. Choi, and Vadin Konko, Design Se71situvity Analysis of Structural Systen.s Yaakoy Bar-Shaloln and Thomas E. Fortmann, Tracking and Data As sociation V B. Kolnanovskii and V R. Nosov, Stability of Functional Differ'ential Equations V. LakshinikanthaIn and D. Trigiantc, Theory of Differenc e equations Applacation s to Numerical Analysis B D. Vujanovic and S.E. Jones. Variational methods in Nonconservative Phenomena C. Rogers and W. F. Ales, Nonlinear boundary value Probleans in science and E7igineeri7g W. F. Amcs and C. Rogers, Nonlincur Equatzo7us i7 the Applied sciences Josip E. pecari, Frank Proschan, and Y L. Tong: Convea: FunctioniS, Partial Orderings. and Statistical Applications E.N. Chukwll, Stability and Time-Optumal Control of Hereditary Systens Viorel Barbu, Analysis and Control of Nonlzmeur Infinite Dimensional Systens Yang Kuang, Delay DifferentiaL Equation.s: Wath Applications inl Popilation Dymamics K.A. Ames and B Straughan, Non-Standard and Improperly Posed Problems Z. Gajic and M.T. QUreshi, The Lyapunov Matri. Equation in Systern Stability and Control D. Straub, Alternate Mathenatical Theory of Non-cquilibrium Phenonilenla B.C. Pachpattc, Ine qualities for DDferen tial and Integral equations I. Podlublly, Fractional Differential Equations To my parents Contents Preface Acknowledgements XX 1 Special Functions of the Fractional Calculus 1.1 Gamma Function 1. 1. 1 Definition of the gamma function 1.1.2 Some Properties of the Gamma Function 1. 1.3 Limit Representation of the gamIna function 1. 1. 4 Beta F 1.1.5 Cont our Integral Representation 10 1.1.6 Contour Integral Representation of 1/r(a) 12 2 Mittag-Lefflo 16 1.2.1 Definition and relation to Some Other functions 17 1. 2,2 The laplace Transform of the mittag-Leffer Function in Two Parameters 20 1.2.3D ag Leffler 1. 2.4 Differential Equations for t he Mitt. ag- Leffler Funct 2.5 Summation formulas 1.2.6 Integration of the Mittag- Leffler Function 4 1.2.7 Asymptotic expansions 1.3 Wright Fur 37 1.3.1 Definition 37 1.3.2 Integral Representation 37 1.3.3 Relation to other functions 2 Fractional Derivatives and Integrals 41 2.1 The Name of the Game 2.2 Griinwald Letnikoy Fractional derivatives CONTENTS 2.2.1 Unification of Integer-order Derivatives and Integrals 43 2.2.2 Integrals of Arbitrary Order 2.2.3 Derivatives of Arbitrary Order 52 2.2.4 Fractional Derivative of (t -a 55 2.2.5 Coinposition with Integer-order Derivatives... 57 2.2.6 Colnposition with Fract. ional Derivatives 2.3 Riean liouville fract ional derivatives 2.3.1 Unification of Integer-order derivatives and Integrals 2.3.2 integrals of Arbitrary Order 2.3.3 Derivatives of arbitrary order 23 Fractional Derivative of(t-a)b 72 2.3.5 Conposition with Integer-order Derivatives... 73 2.3.6 COIl position with Fract. ional Derivatives 74 2.3.7 Link to the grunwald- Letnikov Approach 2.4 Some Othier Approaches 77 1.1 Caputo's frac ional derivative 2.4.2 Generalized Functions Approach 88 2.5 Sequential Fractional Derivatives 86 2.6 Left and Right Fractional derivative 2.7 Properties of Fractional Derivatives 90 .7 1 linearity 90 2.7.2 The leibniz rule for fractional derivatives.. 91 2.7.3 Fractional Derivative of a Coinposite Function. 97 2.7. Riemann Liouville fractional differentiation of an Integral Depending on a Parameter 98 2. 7. 5 Behaviour near the lower terminal 2.7.6 Behaviour far fron the lower terninal 2. 8 Laplace Transfe of fractional de 103 2. 8. 1 Basic Facts on the L aplac c Transforlll 103 2.8.2 Laplace TransforIll of t.hie Riellanlll Liouville Fractional derivative 2.8.3 Laplace 'T'ransforlll of the Caputo derivative 106 of the Griinwald. Letnikov Fractional d) 2.8.5 Laplace Transform of the Mliller ross Sequential fractional de 2.9 Fourier Trausforins of fractional derivatives 2.9.1 Basic Facts (ll the Fourier fransform 109 CONTENTS 2.9.2 FOurier Transform of Fractional integrals 110 2.3 FOurier I'ransforu of Fractional Derivatives 2.10 Mellin 'ransforins of fractional derivatives 112 2.10.1 Basic Facts oll the Allin transforM 12 2. 10.2 Mellin 'Transform of tI Riemann liouville Fractional lnt.coral 115 2.10.3 Alellin ']ransform of the Riemann-liouville Fractional derivative 115 2.10.1 lellin Transform of the Caputo Fractional derivative 116 2.10.5 Mellin Transform of the Miller Ross F1 117 3 Existence and Uniqueness Theorems 121 3.1 Linear Fractional Diferent ial Equations 122 3.2 Fractional Differential Equation of a General FornL 126 3.3 Exist. ence and Uniqueness Theorem as a Method of Solution 131 3.1 Dependenc c of a Solution on Init ial Conditions 133 4 The Laplace Transform Method 137 4.1 Standard fractional Differential equations .138 4. 1. 1 Ordinary Linear Fractional Differenntial equatiOns 138 4.1.2 Partial Linear fractional Differential equations 140 4.2 Sequential Fractional Differential equations 144 4.2.1 Ordinary Linear Fractional Differential equations 144 4. 2.2 Part ial Linear fractional Differential Equations 146 5 Fractional Green,s Function 149 5. Definition and Some Propertjes 150 5.1.1 Definition 5.1.2 Properti 5.2 One-t.erIn Equation 5.T Equat 154 5.4 Three-ter Ec 155 5.5F( Equat i 156 【实例截图】
【核心代码】
分数阶微分方程方面的入门必读数目，有理论方法和应用
This is olume 198 in MATHEMATICS IN SCIENCE AND ENGINEERING By Igor Podlubny, Technical University of Kosice, Slovak Republic A list of recent titles in this series appears on page v of this volume FRACTIONAL DIFFERENTIAL EQUATIONS An Introduction to Fractional Derivatives ractional Differential equations to methods of their Solution and some of their Applications y Igor Podlubny Technical University of Kosice, Slovak Republic A ACADEMIC PRESS San diego· Boston◆ New york london· Sydney· Tokyo· Toront This book is printed on ac id-free paper Copyright. 199 9 by ACADEMIC PRESS All Right. s Reserved No part of this publication may be reproduced or transmit ted in any for by photostat, microfill, or any other means without written permission fron the publishers ACADEMIC PRESS 525B Street Suite 1 900 San diego, California 92101-4495, USA http://www.apnet.com ACADEAIC PRESS LONDON NWI 7DX. UK http://www.hbik.co.lk/ap/ A catalogue record for t.his book is available froin the British library lSBN(-12558840-2 Printed in the United states of america 9900010203MP987654321 Mathematics in Science and Engineering Edited by Willian F. Aes, Georgia Institute of Technology nt. title T.A. BI Volterra Interal and Different al Equatio Ran P. Kanwal, Generalized Function.s: TILe ory and Technique Marc Mangel. Decision and Control in Uncertain Resource Systems Yoshikazu Sawaragi lirot aka Nakayana, and T'ctsuzo Tanino. Theory of Multi-objcctin e Optimization Edward Haug, Kying K. Choi, and Vadin Konko, Design Se71situvity Analysis of Structural Systen.s Yaakoy Bar-Shaloln and Thomas E. Fortmann, Tracking and Data As sociation V B. Kolnanovskii and V R. Nosov, Stability of Functional Differ'ential Equations V. LakshinikanthaIn and D. Trigiantc, Theory of Differenc e equations Applacation s to Numerical Analysis B D. Vujanovic and S.E. Jones. Variational methods in Nonconservative Phenomena C. Rogers and W. F. Ales, Nonlinear boundary value Probleans in science and E7igineeri7g W. F. Amcs and C. Rogers, Nonlincur Equatzo7us i7 the Applied sciences Josip E. pecari, Frank Proschan, and Y L. Tong: Convea: FunctioniS, Partial Orderings. and Statistical Applications E.N. Chukwll, Stability and Time-Optumal Control of Hereditary Systens Viorel Barbu, Analysis and Control of Nonlzmeur Infinite Dimensional Systens Yang Kuang, Delay DifferentiaL Equation.s: Wath Applications inl Popilation Dymamics K.A. Ames and B Straughan, Non-Standard and Improperly Posed Problems Z. Gajic and M.T. QUreshi, The Lyapunov Matri. Equation in Systern Stability and Control D. Straub, Alternate Mathenatical Theory of Non-cquilibrium Phenonilenla B.C. Pachpattc, Ine qualities for DDferen tial and Integral equations I. Podlublly, Fractional Differential Equations To my parents Contents Preface Acknowledgements XX 1 Special Functions of the Fractional Calculus 1.1 Gamma Function 1. 1. 1 Definition of the gamma function 1.1.2 Some Properties of the Gamma Function 1. 1.3 Limit Representation of the gamIna function 1. 1. 4 Beta F 1.1.5 Cont our Integral Representation 10 1.1.6 Contour Integral Representation of 1/r(a) 12 2 Mittag-Lefflo 16 1.2.1 Definition and relation to Some Other functions 17 1. 2,2 The laplace Transform of the mittag-Leffer Function in Two Parameters 20 1.2.3D ag Leffler 1. 2.4 Differential Equations for t he Mitt. ag- Leffler Funct 2.5 Summation formulas 1.2.6 Integration of the Mittag- Leffler Function 4 1.2.7 Asymptotic expansions 1.3 Wright Fur 37 1.3.1 Definition 37 1.3.2 Integral Representation 37 1.3.3 Relation to other functions 2 Fractional Derivatives and Integrals 41 2.1 The Name of the Game 2.2 Griinwald Letnikoy Fractional derivatives CONTENTS 2.2.1 Unification of Integer-order Derivatives and Integrals 43 2.2.2 Integrals of Arbitrary Order 2.2.3 Derivatives of Arbitrary Order 52 2.2.4 Fractional Derivative of (t -a 55 2.2.5 Coinposition with Integer-order Derivatives... 57 2.2.6 Colnposition with Fract. ional Derivatives 2.3 Riean liouville fract ional derivatives 2.3.1 Unification of Integer-order derivatives and Integrals 2.3.2 integrals of Arbitrary Order 2.3.3 Derivatives of arbitrary order 23 Fractional Derivative of(t-a)b 72 2.3.5 Conposition with Integer-order Derivatives... 73 2.3.6 COIl position with Fract. ional Derivatives 74 2.3.7 Link to the grunwald- Letnikov Approach 2.4 Some Othier Approaches 77 1.1 Caputo's frac ional derivative 2.4.2 Generalized Functions Approach 88 2.5 Sequential Fractional Derivatives 86 2.6 Left and Right Fractional derivative 2.7 Properties of Fractional Derivatives 90 .7 1 linearity 90 2.7.2 The leibniz rule for fractional derivatives.. 91 2.7.3 Fractional Derivative of a Coinposite Function. 97 2.7. Riemann Liouville fractional differentiation of an Integral Depending on a Parameter 98 2. 7. 5 Behaviour near the lower terminal 2.7.6 Behaviour far fron the lower terninal 2. 8 Laplace Transfe of fractional de 103 2. 8. 1 Basic Facts on the L aplac c Transforlll 103 2.8.2 Laplace TransforIll of t.hie Riellanlll Liouville Fractional derivative 2.8.3 Laplace 'T'ransforlll of the Caputo derivative 106 of the Griinwald. Letnikov Fractional d) 2.8.5 Laplace Transform of the Mliller ross Sequential fractional de 2.9 Fourier Trausforins of fractional derivatives 2.9.1 Basic Facts (ll the Fourier fransform 109 CONTENTS 2.9.2 FOurier Transform of Fractional integrals 110 2.3 FOurier I'ransforu of Fractional Derivatives 2.10 Mellin 'ransforins of fractional derivatives 112 2.10.1 Basic Facts oll the Allin transforM 12 2. 10.2 Mellin 'Transform of tI Riemann liouville Fractional lnt.coral 115 2.10.3 Alellin ']ransform of the Riemann-liouville Fractional derivative 115 2.10.1 lellin Transform of the Caputo Fractional derivative 116 2.10.5 Mellin Transform of the Miller Ross F1 117 3 Existence and Uniqueness Theorems 121 3.1 Linear Fractional Diferent ial Equations 122 3.2 Fractional Differential Equation of a General FornL 126 3.3 Exist. ence and Uniqueness Theorem as a Method of Solution 131 3.1 Dependenc c of a Solution on Init ial Conditions 133 4 The Laplace Transform Method 137 4.1 Standard fractional Differential equations .138 4. 1. 1 Ordinary Linear Fractional Differenntial equatiOns 138 4.1.2 Partial Linear fractional Differential equations 140 4.2 Sequential Fractional Differential equations 144 4.2.1 Ordinary Linear Fractional Differential equations 144 4. 2.2 Part ial Linear fractional Differential Equations 146 5 Fractional Green,s Function 149 5. Definition and Some Propertjes 150 5.1.1 Definition 5.1.2 Properti 5.2 One-t.erIn Equation 5.T Equat 154 5.4 Three-ter Ec 155 5.5F( Equat i 156 【实例截图】
【核心代码】
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