Constructive Fractional Analysis with Applications (Studies in Systems, Decision and Control, 362)

✍ Scribed by George A. Anastassiou

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 523
• Category: Library

✦ Synopsis

This book includes constructive approximation theory; it presents ordinary and fractional approximations by positive sublinear operators, and high order approximation by multivariate generalized Picard, Gauss–Weierstrass, Poisson–Cauchy and trigonometric singular integrals. Constructive and Computational Fractional Analysis recently is more and more in the center of mathematics because of their great applications in the real world. In this book, all presented is original work by the author given at a very general level to cover a maximum number of cases in various applications. The author applies generalized fractional differentiation techniques of Riemann–Liouville, Caputo and Canavati types and of fractional variable order to various kinds of inequalities such as of Opial, Hardy, Hilbert–Pachpatte and on the spherical shell. He continues with E. R. Love left- and right-side fractional integral inequalities. They follow fractional Landau inequalities, of left and right sides, univariate and multivariate, including ones for Semigroups. These are developed to all possible directions, and right-side multivariate fractional Taylor formulae are proven for the purpose. It continues with several Gronwall fractional inequalities of variable order. This book results are expected to find applications in many areas of pure and applied mathematics. As such this book is suitable for researchers, graduate students and seminars of the above disciplines, also to be in all science and engineering libraries.

✦ Table of Contents

Preface
Contents
1 Variable Order General Fractional Integral Inequalities
1.1 Background
1.2 Main Results
References
2 Variable Order Fractional Integral Inequalities for Spherical Shell
2.1 Background
2.2 Main Results
References
3 Left Fractional Integral Inequalities of E.R. Love Type
3.1 Introduction
3.2 Main Results
References
4 Right Side Fractional Integral Inequalities of E.R. Love Type
4.1 Introduction
4.2 Main Results
4.2.1 Part I
4.2.2 Part II
References
5 General Fractional Landau Inequalities
5.1 Introduction
5.2 Main Results
References
6 Abstract Fractional Landau Inequalities
6.1 Introduction
6.2 Main Results
References
7 Fractional Landau Inequalities of Riemann–Liouville Type
7.1 Introduction
7.2 Main Results
7.3 Appendix
References
8 Generalized Canavati Fractional Landau Inequalities
8.1 Introduction
8.2 Main Results
References
9 Sequential Left Abstract Fractional Landau Inequalities
9.1 Introduction
9.2 Main Results
References
10 Iterated Left Abstract Generalized Fractional Landau Inequalities
10.1 Introduction
10.2 Main Results
References
11 Sequential General Right Side Fractional Landau Inequalities
11.1 Introduction
11.2 Main Results
References
12 Iterated Generalized Right Side Fractional Landau Inequalities
12.1 Introduction
12.2 Main Results
References
13 High Order Generalized Landau Inequalities
13.1 Introduction
13.2 Main Results
13.3 Appendix
References
14 Multidimensional Caputo Left Side Fractional Landau Inequalities
14.1 Introduction
14.2 Main Results
References
15 Multidimensional Left Canavati Fractional Landau Inequalities
15.1 Introduction
15.2 Main Results
References
16 Multidimensional Right Caputo Fractional Taylor Formula and Landau Inequalities
16.1 Introduction
16.2 Background
16.3 Main Results
References
17 Multidimensional Generalized Right Fractional Taylor Formula and Landau Inequalities
17.1 Introduction
17.2 Background
17.3 Main Results
References
18 Landau's Inequality for Semigroups
18.1 Introduction
18.2 Background
18.3 Main Results
18.4 Application
References
19 Fractional Variable Order Gronwall Inequality
19.1 Introduction
19.2 Main Results
References
20 A Study of Gronwall Inequalities of Fractional Variable Order
20.1 Introduction
20.2 Main Results
References
21 General Ordinary and Fractional Approximation with Positive Sublinear Operators
21.1 Background—I
21.2 Background—II ch2121.4
21.3 Background—III
21.4 Background—IV
21.5 Main Results
21.5.1 Ordinary Approximation
21.5.2 Fractional Approximation
21.5.3 Iterated Fractional Approximation
References
22 Complete Approximations with Multivariate Generalized Picard Singular Integrals
22.1 Introduction
22.2 Auxiliary Essential Results
22.3 Main Results for Pr,n[ m]
22.3.1 Uniform Approximation
22.3.2 Lp Approximation for Pr,n[ m]
22.3.3 Global Smoothness Preservation and Simultaneous Approximation of Pr,n[ m]
22.3.4 Voronovskaya Asymptotic Expansions for Pr,n[ m]
22.3.5 Simultaneous Approximation by Multivariate Complex Pr,n [ m]
References
23 High Order Approximation with Multivariate Generalized Gauss–Weierstrass Singular Integrals
23.1 Introduction
23.2 Auxiliary Essential Results
23.3 Main Results for Wr,n[ m]
23.3.1 Uniform Approximation
23.3.2 Lp Approximation for Wr,n[ m]
23.3.3 Global Smoothness Preservation and Simultaneous Approximation of Wr,n[ m]
23.3.4 Voronovskaya Asymptotic Expansions for Wr,n[ m]
23.3.5 Simultaneous Approximation by Multivariate Complex Wr,n [ m]
References
24 Complete Approximations with Multivariate Generalized Poisson–Cauchy Type Singular Integral Operators
24.1 Introduction
24.2 Auxiliary Essential Results
24.3 Main Results for Ur,n[ m]
24.3.1 Uniform Approximation
24.3.2 Lp Approximation for Ur,n[ m]
24.3.3 Global Smoothness Preservation and Simultaneous Approximation of Ur,n[ m]
24.3.4 Voronovskaya Asymptotic Expansions for Ur,n[ m]
24.3.5 Simultaneous Approximation by Multivariate Complex Ur,n [ m]
References
25 High Order Approximation with Multivariate Generalized Trigonometric Type Singular Integral Operators
25.1 Introduction
25.2 Auxiliary Essential Results
25.3 Main Results for Tr,n[ m]
25.3.1 Uniform Approximation
25.3.2 Lp Approximation for Tr,n[ m]
25.3.3 Global Smoothness Preservation and Simultaneous Approximation Of Tr,n[ m]
25.3.4 Voronovskaya Asymptotic Expansions for Tr,n[ m]
25.3.5 Simultaneous Approximation by Multivariate Complex Tr,n [ m]
References
26 Concluding Remarks