Fractional Differentiation Inequalities

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.

List of contents

Opial#x2013;Type Inequalities for Functions and Their Ordinary and Canavati Fractional Derivatives.- Canavati Fractional Opial#x2013;Type Inequalities and Fractional Differential Equations.- Riemann#x2014;Liouville Opial#x2014;type Inequalities for Fractional Derivatives.- Opial#x2013;type #x2013;Inequalities for Riemann#x2014;Liouville Fractional Derivatives.- Opial#x2013;Type Inequalities Involving Canavati Fractional Derivatives of Two Functions and Applications.- Opial#x2013;Type Inequalities for Riemann#x2014;Liouville Fractional Derivatives of Two Functions with Applications.- Canavati Fractional Opial#x2013;Type Inequalities for Several Functions and Applications.- Riemann#x2014;Liouville Fractional#x2013;Opial Type Inequalities for Several Functions and Applications.- Converse Canavati Fractional Opial#x2013;Type Inequalities for Several Functions.- Converse Riemann#x2014;Liouville Fractional Opial#x2013;Type Inequalities for Several Functions.- Multivariate Canavati Fractional Taylor Formula.- Multivariate Caputo Fractional Taylor Formula.- Canavati Fractional Multivariate Opial#x2013;Type Inequalities on Spherical Shells.- Riemann#x2014;Liouville Fractional Multivariate Opial#x2013;type inequalities over a spherical shell.- Caputo Fractional Multivariate Opial#x2013;Type Inequalities over a Spherical Shell.- Poincar#x00E9;#x2013;Type Fractional Inequalities.- Various Sobolev#x2013;Type Fractional Inequalities.- General Hilbert#x2014;Pachpatte#x2013;Type Integral Inequalities.- General Multivariate Hilbert#x2014;Pachpatte#x2013;Type Integral Inequalities.- Other Hilbert#x2014;Pachpatte#x2013;Type Fractional Integral Inequalities.- Canavati Fractional and Other Approximation of Csiszar#x2019;s #x2013;Divergence.- Caputo and Riemann#x2014;Liouville Fractional Approximation of Csiszar#x2019;s #x2013;Divergence.- Canavati Fractional Ostrowski#x2013;Type Inequalities.- Multivariate Canavati Fractional Ostrowski#x2013;Type Inequalities.- Caputo Fractional Ostrowski#x2013;Type Inequalities.

Summary

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.

Additional text

From the review:"Professor Anastassiou considers three definitions of fractional derivatives. … The list of references runs to four hundred and twelve items. … for a specialist in fractional derivative inequalities it would be indispensible." (Underwood Dudley, The Mathematical Association of America, September, 2009)"In this book, Anastassiou chooses to concentrate on three special cases: operators of Riemann-Liouville type that have been used very intensively by the pure mathematics community, operators of Caputo’s type that have proven to be very important in many applications … and the relatively little-known Canavati operators. For these types of operators, he provides generalizations of the classical differentiation inequalities … . all the chapters are self-contained. … a very useful and easy-to-read reference for readers who are looking for that." (Kai Diethelm, ACM Computing Reviews, November, 2009)“This book is the first edition of the work on a subject which is not dealt with in a text form, as this is, before. … each chapter has almost identical format with detailed proof of theorems, which will be proved fruitful for both young and matured researchers to understand the subject. … References at the end is exhaustive and fruitful. … The present monograph centers its attention mainly in the aspect of the fractional inequalities and contains a wealth of interesting material … .” (P. K. Banerji, Zentralblatt MATH, 2010)“The text will be very useful to researchers in fractional calculus and its applications to the existence and uniqueness problems of fractional differential and partial differential equations.” (R. N. Kalia, Mathematical Reviews, Issue 2010 g)

Report

From the review:

"Professor Anastassiou considers three definitions of fractional derivatives. ... The list of references runs to four hundred and twelve items. ... for a specialist in fractional derivative inequalities it would be indispensible." (Underwood Dudley, The Mathematical Association of America, September, 2009)
"In this book, Anastassiou chooses to concentrate on three special cases: operators of Riemann-Liouville type that have been used very intensively by the pure mathematics community, operators of Caputo's type that have proven to be very important in many applications ... and the relatively little-known Canavati operators. For these types of operators, he provides generalizations of the classical differentiation inequalities ... . all the chapters are self-contained. ... a very useful and easy-to-read reference for readers who are looking for that." (Kai Diethelm, ACM Computing Reviews, November, 2009)
"This book is the first edition of the work on a subject which is not dealt with in a text form, as this is, before. ... each chapter has almost identical format with detailed proof of theorems, which will be proved fruitful for both young and matured researchers to understand the subject. ... References at the end is exhaustive and fruitful. ... The present monograph centers its attention mainly in the aspect of the fractional inequalities and contains a wealth of interesting material ... ." (P. K. Banerji, Zentralblatt MATH, 2010)
"The text will be very useful to researchers in fractional calculus and its applications to the existence and uniqueness problems of fractional differential and partial differential equations." (R. N. Kalia, Mathematical Reviews, Issue 2010 g)

Product details

• Authors: George A Anastassiou, George A. Anastassiou
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 26.07.2011
• EAN: 9781441931061
• ISBN: 978-1-4419-3106-1
• No. of pages: 686
• Weight: 1038 g
• Illustrations: XIV, 686 p.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Analysis, B, Funktionalanalysis und Abwandlungen, Integralrechnung und -gleichungen, Mathematics and Statistics, Functional Analysis, Ordinary Differential Equations, Partial Differential Equations, Differential calculus & equations, Differential equations, Operational calculus, Integral transforms, Integral Transforms and Operational Calculus, Integral Transforms, Operational Calculus, Integral calculus & equations, Functional analysis & transforms