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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.
Front Matter
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 Integral Inequalities
General Multivariate Hilbert—Pachpatte–Type Integral Inequalities
Other Hilbert—Pachpatte–Type Fractional Integral 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
Back Matter