Fractional differentiation inequalities

George A. Anastassiou (auth.)0387981276, 9780387981277

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.

Table of contents :
Front Matter….Pages i-xi
Introduction….Pages 1-5
Opial–Type Inequalities for Functions and Their Ordinary and Canavati Fractional Derivatives….Pages 7-22
Canavati Fractional Opial–Type Inequalities and Fractional Differential Equations….Pages 23-39
Riemann—Liouville Opial—type Inequalities for Fractional Derivatives….Pages 41-52
Opial–type L p –Inequalities for Riemann—Liouville Fractional Derivatives….Pages 53-65
Opial–Type Inequalities Involving Canavati Fractional Derivatives of Two Functions and Applications….Pages 67-106
Opial–Type Inequalities for Riemann—Liouville Fractional Derivatives of Two Functions with Applications….Pages 107-147
Canavati Fractional Opial–Type Inequalities for Several Functions and Applications….Pages 149-177
Riemann—Liouville Fractional–Opial Type Inequalities for Several Functions and Applications….Pages 179-203
Converse Canavati Fractional Opial–Type Inequalities for Several Functions….Pages 205-227
Converse Riemann—Liouville Fractional Opial–Type Inequalities for Several Functions….Pages 229-255
Multivariate Canavati Fractional Taylor Formula….Pages 257-268
Multivariate Caputo Fractional Taylor Formula….Pages 269-278
Canavati Fractional Multivariate Opial–Type Inequalities on Spherical Shells….Pages 279-317
Riemann—Liouville Fractional Multivariate Opial–type inequalities over a spherical shell….Pages 319-390
Caputo Fractional Multivariate Opial–Type Inequalities over a Spherical Shell….Pages 391-444
Poincaré–Type Fractional Inequalities….Pages 445-482
Various Sobolev–Type Fractional Inequalities….Pages 483-504
General Hilbert—Pachpatte–Type Integral Inequalities….Pages 505-522
General Multivariate Hilbert—Pachpatte–Type Integral Inequalities….Pages 523-544
Other Hilbert—Pachpatte–Type Fractional Integral Inequalities….Pages 545-561
Canavati Fractional and Other Approximation of Csiszar’s f –Divergence….Pages 563-576
Caputo and Riemann—Liouville Fractional Approximation of Csiszar’s f –Divergence….Pages 577-587
Canavati Fractional Ostrowski–Type Inequalities….Pages 589-594
Multivariate Canavati Fractional Ostrowski–Type Inequalities….Pages 595-613
Caputo Fractional Ostrowski–Type Inequalities….Pages 615-633
Appendix….Pages 635-639
Back Matter….Pages 1-34