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Presents various fractional differential inequalities
Applications are included along with the theory
Each chapter is self-contained
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.
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