George A. Anastassiou - Generalized Fractional Calculus: New Advancements and Applications

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Computational and Fractional Analysis nowadays are more and more in the center of mathematics and of other related sciences either by themselves because of their rapid development, which is based on very old foundations, or because they cover a great variety of applications in the real world. In this monograph, all presented is original work by the author given at a very general level to cover a maximum number of cases in various applications. We apply generalized fractional differentiation techniques of Caputo, Canavati, and conformable types to various kinds of inequalities such as Ostrowski, Grss, Iyengar, Polya, Poincar, Sobolev, Hilbert-Pachpatte, and Opial type inequalities. Some of these are extended to Banach space valued functions. Many of these are studying the average of a function as compared to values of the on hand function. These inequalities also have a great impact on numerical analysis, stochastics, and fractional differential equations. We continue with generalized fractional approximation by positive sublinear operators including the case of iterated one which derives from the newly presented Korovkin type inequalities. The linear case is extended to Banach space valued functions. It follows for the first time multivariate complex Korovkin quantitative approximation theory.

We also present -fractional integral inequalities of Ostrowski and Polya types. The M-fractional derivative is based on Mittag-Leffler special function and possesses all nice properties of ordinary derivative.

Our results are weighted so they provide a great variety of cases and applications. The last eight chapters of the book deal for the first time with the quantitative fractional Korovkin type approximation of stochastic processes. We lay there the foundations of stochastic fractional calculus. We consider both Caputo and Conformable fractional directions and we derive regular and trigonometric results. Our positive linear operators can be expectation operator commutative or not. The case of non-commutative operators is especially elaborate and interesting.

This monograph is the natural and expected evolution of recent authors research work put in a book form for the first time. The presented approaches are original, and the chapters are self-contained and can be read independently. This monograph is suitable to be used in related graduate classes and research projects. We exhibit to the maximum our computational, estimating and approximation methods to all possible directions.

The motivation to write this monograph came by the following: various issues related to the modeling and analysis of ordinary and fractional order systems have gained an increased popularity, as witnessed by many books and volumes in Springers program:

http://www.springer.com/gp/search?query=fractional&submit=Prze%C5%9Blij and the purpose of our book is to capture at a very general level a deeper formal analysis on some issues that are relevant to many areas for instance: decision making, complex processes, systems modeling and control, and related areas.

The above are deeply embedded in the fields of mathematics, engineering, computer science, physics, economics, social and life sciences.

The complete list of presented topics follows:

Mixed Caputo weighted-fractional Ostrowski type Inequalities,