Fractional Differentiation Inequalities (eBook)

George A. Anastassiou (Autor)

eBook Download: PDF

2009 | 2009
XIV, 686 Seiten
Springer New York (Verlag)
978-0-387-98128-4 (ISBN)

Infos

In this book the author presents the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. Results for the above are derived using three different types of fractional derivatives, namely by Canavati, Riemann-Liouville and Caputo. The univariate and multivariate cases are both examined. Each chapter is self-contained. The theory is presented systematically along with the applications. The application to information theory is also examined.

This monograph is suitable for researchers and graduate students in pure mathematics. Applied mathematicians, engineers, and other applied scientists will also find this book useful.

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In this book the author presents the Opial, Poincare, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. Results for the above are derived using three different types of fractional derivatives, namely by Canavati, Riemann-Liouville and Caputo. The univariate and multivariate cases are both examined. Each chapter is self-contained. The theory is presented systematically along with the applications. The application to information theory is also examined.This monograph is suitable for researchers and graduate students in pure mathematics. Applied mathematicians, engineers, and other applied scientists will also find this book useful.

Inhalt

Contents 6
Preface 11
Introduction 13
Opial-Type Inequalities for Functions and Their Ordinary and Canavati Fractional Derivatives 18
Preliminaries 18
Main Results 22
Applications 29
Canavati Fractional Opial-Type Inequalities and Fractional Differential Equations 34
Introduction 34
Preliminaries 35
Main Results 37
Applications 45
Other Fractional Differential Equations 49
Riemann--Liouville Opial-Type Inequalities for Fractional Derivatives 51
Introduction and Preliminaries 51
Main Results 54
Applications 58
Opial-Type Lp-Inequalities for Riemann--LiouvilleFractional Derivatives 63
Introduction and Preliminaries 63
Main Results 66
Opial-Type Inequalities Involving Canavati FractionalDerivatives of Two Functions and Applications 76
Introduction 76
Preliminaries 78
Main Results 81
Applications 105
Opial-Type Inequalities for Riemann--Liouville Fractional Derivatives of Two Functions with Applications 116
Introduction 116
Background 117
Main Results 118
Applications 147
Canavati Fractional Opial-Type Inequalities for SeveralFunctions and Applications 157
Introduction 157
Preliminaries 158
Main Results 160
Applications 177
Riemann--Liouville Fractional Opial-Type Inequalitiesfor Several Functions and Applications 186
Introduction 186
Background 187
Main Results 188
Applications 202
Converse Canavati Fractional Opial-Type Inequalitiesfor Several Functions 211
Introduction 211
Preliminaries 212
Main Results 215
Results Involving Two Functions 215
Results Involving Several Functions 226
Converse Riemann--Liouville Fractional Opial-TypeInequalities for Several Functions 234
Introduction 234
Background 235
Main Results 236
Results Involving Two Functions 236
Results Involving Several Functions 249
Results with Respect to Generalized Riemann --Liouville Fractional Derivative 256
Multivariate Canavati Fractional Taylor Formula 261
Introduction 261
Results 262
Multivariate Caputo Fractional Taylor Formula 273
Background 273
Results 274
Canavati Fractional Multivariate Opial-TypeInequalities on Spherical Shells 283
Introduction 283
Results 284
Riemann--Liouville Fractional Multivariate Opial-TypeInequalities over a Spherical Shell 322
Introduction 322
Background---I 323
Background---II 326
Background---III 332
Main Results 337
Riemann--Liouville Fractional Opial-TypeInequalities Involving One Function 337
Riemann--Liouville Fractional Opial-TypeInequalities Involving Two Functions 353
Riemann--Liouville Fractional Opial-TypeInequalities Involving Several Functions 372
Caputo Fractional Multivariate Opial-Type Inequalitiesover a Spherical Shell 394
Introduction 394
Background---I 395
Main Results 400
Results Involving One Function 400
Results Involving Two Functions 405
Results Involving Several Functions 414
Background---II 422
Main Results on a Spherical Shell 427
Results Involving One Function 427
Results Involving Two Functions 430
Results Involving Several Functions 434
Applications 439
Poincaré-Type Fractional Inequalities 448
Introduction 448
Fractional Poincaré Inequalities Results 449
Applications of Fractional Poincaré Inequalities 460
Fractional Mean Poincaré Inequalities 476
Applications of Fractional Mean Poincaré Inequalities 482
Various Sobolev-Type Fractional Inequalities 486
Introduction 486
Various Univariate Sobolev-Type Fractional Inequalities 487
Applications 506
General Hilbert--Pachpatte-Type Integral Inequalities 508
Introduction 508
Main Results 509
General Multivariate Hilbert--Pachpatte-TypeIntegral Inequalities 526
Introduction 526
Symbols and Basics 527
Main Results 530
Other Hilbert--Pachpatte-Type Fractional IntegralInequalities 548
Background 548
Univariate Results 553
Multivariate Results 556
Canavati Fractional and Other Approximationof Csiszar's f-Divergence 565
Preliminaries 565
Main Results 570
Caputo and Riemann--Liouville FractionalApproximation of Csiszar's f-Divergence 579
Preliminaries 579
Results 583
Canavati Fractional Ostrowski-Type Inequalities 590
Background 590
Results 592
Multivariate Canavati Fractional Ostrowski-TypeInequalities 596
Background 596
Results 599
Caputo Fractional Ostrowski-Type Inequalities 615
Background 615
Univariate Results 618
Multivariate Results 622
Appendix 634
Conversion Formulae for Different Kinds of FractionalDerivatives 634
Some Basic Fractional Derivatives 637
References 639
List of Symbols 668
Index 670

Produkt-Details

• Erscheint lt. Verlag: 28.5.2009
• Zusatzinfo: XIV, 686 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• Schlagworte: Approximation • Derivative • Derivatives • Differential • differential equation • Equation • Form • Function • Functions • Information • Information Theory • Integral • Mathematics • Ordinary differential equations • Partial differential equations • Types
• ISBN-10: 0-387-98128-4 / 0387981284
• ISBN-13: 978-0-387-98128-4 / 9780387981284