Unification of Fractional Calculi with Applications

Frontmatter

Chapter 1. Progress on Generalized Hilfer Fractional Calculus and Fractional Integral Inequalities

Abstract

After motivation we give a complete background on needed \(\psi \)-Hilfer fractional Calculus. Then we produce \(\psi \)-Hilfer fractional left and right Taylor formulae. We give also important \(\psi \)-Hilfer fractional left and right representation integral formulae regarding \(\psi \)-Hilfer left and right fractional derivatives. Then we give extensive applications of our \(\psi \)-Hilfer fractional results to left and right \(\psi \)-Hilfer fractional Ostrowski, Opial and Poincaré type integral inequalities. We create the space for more future forthcoming results.

George A. Anastassiou

Chapter 2. Landau Generalized Hilfer Fractional Inequalities

Abstract

Here we present a series of Landau type inequalities related to left and right \(\psi \)-Hilfer fractional derivatives. It follows.

George A. Anastassiou

Chapter 3. Iyengar-Hilfer Generalized Fractional Inequalities

Abstract

Here we present Iyengar type integral inequalities. At the univariate level they involve \(\psi \)-Hilfer left and right fractional derivatives. At the multivariate level they involve Hilfer left and right fractional derivatives, and they deal with radial and non-radial functions on the ball and spherical shell. All estimates are with respect to norms \(\left\| \cdot \right\| _{p}\), \(1\le p\le \infty \). At the end we provide an application.

George A. Anastassiou

Chapter 4. Generalized Hilfer-Polya, Hilfer-Ostrowski and Hilfer-Hilbert-Pachpatte Fractional Inequalities

Abstract

Here we present Hilfer-Polya, \(\psi \)-Hilfer Ostrowski and \(\psi \)-Hilfer-Hilbert-Pachpatte types fractional inequalities. They are univariate inequalities involving left and right Hilfer and \(\psi \)-Hilfer fractional derivatives. All estimates are with respect to norms \(\left\| \cdot \right\| _{p}\), \(1\le p\le \infty \). At the end we provide applications. It follows [4].

George A. Anastassiou

Chapter 5. Generalized Hilfer Fractional Approximation of Csiszar’s f-Divergence

Abstract

Here are given tight probabilistic inequalities that provide nearly best estimates for the Csiszar’s f-divergence. These use the right and left \(\psi \)-Hilfer fractional derivatives of the directing function f. Csiszar’s f-divergence or the so called Csiszar’s discrimination is used as a measure of dependence between two random variables which is a very essential aspect of stochastics, we apply our results there. The Csiszar’s discrimination is the most important and general measure for the comparison between two probability measures. We give also other applications. It follows [3].

George A. Anastassiou

Chapter 6. Generalized Hilfer Fractional Self Adjoint Operator Inequalities

Abstract

We give here \(\psi \)-Hilfer and Hilfer fractional self adjoint operator inner product comparison, Poincaré, Sobolev and Opial type inequalities. At first we give right and left \(\psi \)-Hilfer fractional representation formulae in the self adjoint operator sense. Operator inequalities are based in the self adjoint operator order over a Hilbert space. It follows [3].

George A. Anastassiou

Chapter 7. Essential Forward and Reverse Generalized Hilfer-Hardy Fractional Inequalities

Abstract

Here we present forward and reverse left and right \(L_{p}\) fractional integral inequalities of Hardy type. These engage fractional derivatives and fractional integrals from the \(\psi \)-Hilfer and Hilfer fractional calculi. It follows [4].

George A. Anastassiou

Chapter 8. Principles of Generalized Prabhakar-Hilfer Fractional Calculus and Applications

Abstract

Here we introduce the generalized Prabhakar fractional calculus and we also combine it with the generalized Hilfer calculus. We prove that the generalized left and right side Prabhakar fractional integrals preserve continuity and we find tight upper bounds for them. We present several left and right side generalized Prabhakar fractional inequalities of Hardy, Opial and Hilbert-Pachpatte types.

George A. Anastassiou

Chapter 9. Advanced and General Hilfer-Prabhakar-Hardy Fractional Inequalities

Abstract

We present a series of left and right side Hardy type fractional inequalities under convexity in the setting of generalized Hilfer and Prabhakar fractional Calculi.

George A. Anastassiou

Chapter 10. Vectorial Advanced Hilfer-Prabhakar-Hardy Fractional Inequalities

Abstract

We present a variety of univariate and multivariate left and right side Hardy type fractional inequalities, many of them under convexity, and other also of \(L_{p}\) type, \(p\ge 1\), in the setting of generalized Hilfer and Prabhakar fractional Calculi.

George A. Anastassiou

Chapter 11. Vectorial Prabhakar Hardy Advanced Fractional Inequalities Under Convexity

Abstract

We present a detailed great variety of Hardy type fractional inequalities under convexity and \(L_{p}\) norm in the setting of generalized Prabhakar and Hilfer fractional calculi of left and right integrals and derivatives.

George A. Anastassiou

Chapter 12. Advanced Multivariate Prabhakar fractional integrals and inequalities

Abstract

We introduce here the mixed generalized multivariate Prabhakar type left and right fractional integrals and study their basic properties, such as preservation of continuity and their boundedness as positive linear operators.

George A. Anastassiou

Chapter 13. Non Singular Kernel Multiparameter Fractional Differentiation

Abstract

We introduce here Caputo and Riemann-Liouville type non singular kernel very general multi parameter left and right side fractional derivatives and we prove their continuity.

George A. Anastassiou

Chapter 14. Advanced Hilfer Fractional Opial Inequalities

Abstract

Here we present a detailed collection of Hilfer fractional left and right side Opial type inequalities.

George A. Anastassiou

Chapter 15. Exotic Fractional Integral Inequalities

Abstract

Here we present a thorough collection of Opial and Hardy type fractional inequalities involving also convexity, based on Luchko’s generalized fractional calculus, and Prabhakar’s partial and mixed of variable degree multivariate fractional integrals.

George A. Anastassiou

Chapter 16. Fuzzy Fractional Calculus

Abstract

Here we study very general fuzzy fractional integral-differential operators, give their basic properties and derive related Opial and Hardy type inequalities.

George A. Anastassiou

Chapter 17. Conclusion

Abstract

During the last 50 years fractional calculus due to its wide applications to many applied sciences has become a main trend in mathematics. Its predominant kinds are the old Riemann-Liouville fractional calculus and the newer one of Caputo type.

George A. Anastassiou