George A. Anastassiou

Fractional Differentiation Inequalities (eBook, PDF)

• Format: PDF

Produktbeschreibung

Fractional differentiation inequalities are by themselves an important area of research. They have many applications in pure and applied mathematics and many other applied sciences. One of the most important applications is in establishing the uniqueness of a solution in fractional differential equations and systems and in fractional partial differential equations. They also provide upper bounds to the solutions of the above equations.

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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Produktdetails

• Verlag: Springer-Verlag GmbH
• Seitenzahl: 686
• Erscheinungstermin: 28. Mai 2009
• Englisch
• ISBN-13: 9780387981284
• Artikelnr.: 37289487

Inhaltsangabe

Introduction.- Opial Type Inequalities for Functions and Their Ordinary and Canavati Fractional Derivatives.- Canavati Fractional Opial Type Inequalities and Fractional Differential Equations.- Riemann-Liouville Opial Type Inequalities for Fractional Derivatives.- Opial Type L^p-Inequalities for Riemann-Liouville Fractional Derivatives.- Opial Type Inequalities Involving Canavati Fractional Derivatives of Two Functions and Applications.- Opial Type Inequalities for Riemann-Liouville Fractional Derivatives of Two Functions with Applications.- Canavati Fractional Opial Type Inequalities for Several Functions and Applications.- Riemann-Liouville Fractional Opial Type Inequalities for Several Functions and Applications.- Converse Canavati Fractional Opial Type Inequalities for Several Functions.- Converse Riemann-Liouville Fractional Opial Type Inequalities for Several Functions.- Multivariate Canavati Fractional Taylor Formula.- Multivariate Caputo Fractional Taylor Formula.- Canavati Fractional Multivariate Opial Type Inequalities on Spherical Shells.- Riemann-Liouville Fractional Multivariate Opial Type Inequalities Over a Spherical Shell.- Caputo Fractional Multivariate Opial Type Inequalities Over a Spherical Shell.- Poincaré Type Fractional Inequalities.- Various Sobolev Type Fractional Inequalities.- General Hilbert-Pachpatte Type Inequalities.- General Multivariate Hilbert-Pachpatte Type Integral Inequalties.- Other Hilbert-Pachpatte Type Fractional Interal Inequalities.- Canavati Fractional and Other Approximation of Csiszar's f-Divergence.- Caputo and Riemann-Liouville Fractional Approximation of Csiszar's f-Divergence.- Canavati Fractional Ostrowski Type Inequalities.- Multivariate Canavati Fractional Ostrowski Type Inequalities.- Caputo Fractional Ostrowski Type Inequalities.- Appendix.- References.- List of Symbols.- Index

Rezensionen

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)