Intelligent computations : abstract fractional calculus, inequalities, approximations

322 Pages·2018·3.51 MB·English

by  Anastassiou

Studies in Computational Intelligence 734
George A. Anastassiou
Intelligent Computations:
Abstract Fractional
Calculus, Inequalities,
Approximations
Studies in Computational Intelligence
Volume 734
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: [email protected]
About this Series
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More information about this series at http://www.springer.com/series/7092
George A. Anastassiou
Intelligent Computations:
Abstract Fractional Calculus,
Inequalities, Approximations
123
George A.Anastassiou
Department ofMathematical Sciences
University of Memphis
Memphis,TN
USA
ISSN 1860-949X ISSN 1860-9503 (electronic)
Studies in Computational Intelligence
ISBN978-3-319-66935-9 ISBN978-3-319-66936-6 (eBook)
DOI 10.1007/978-3-319-66936-6
LibraryofCongressControlNumber:2017951144
©SpringerInternationalPublishingAG2018
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To My Family
Preface
This rather brief monograph is regarding the fractional analysis of Banach space
valued functions of a real domain. It is the natural evolution of recent author’s
researchworkputinabookformforthefirsttime.Theauthoristhefounderofthe
so-called strong theory offractional calculus at abstract level. Usual vector-valued
differentiationresemblesthenumericalone,andintegralshereareofBochnertype.
We apply our theory extensively to abstract analytic inequalities and abstract
Korovkin approximation offunctions, as well as abstract neural network approxi-
mation. By using Bochner integral properties, we present a left and right
Caputo-type, as well as Canavati-type, strong fractional theory which is the direct
analog of the real one, but now dealing with Banach space valued functions.
In the literature, there are very few articles about the weak fractional theory of
Banach space valued functions. However, we found that the weak theory, using
Pettis integral and functionals, is complicated, less clear, difficult, uncomfortable
and unnecessary.
Withthismonograph,wetrytosimplifymattersandputtherelatedtheoryonits
natural grounds and resemble the theory on real numbers.
Here, chapters are self-contained and can be read independently. This concise
monograph is suitable to be used in related graduate classes and many research
projects.
The list of presented topics follows:
strong left fractional calculus for Banach space valued functions,
strong right fractional calculus for Banach space valued functions,
strongmixedandgeneralizedfractionalcalculusforBanachspacevaluedfunctions,
principles of general fractional analysis for Banach space valued functions,
basic Korovkin theory for Banach space valued functions,
vector fractional Korovkin-type approximations,
high-order approximation theory for Banach space valued functions,
vectorial fractional approximation by using linear operators,
vector fractional trigonometric Korovkin approximation,
multivariate and abstract approximation theory for Banach space valued functions,
vii
viii Preface
arctangent function-based Banach space valued neural network approximation,
An extensive list of references is given per chapter.
The book’s results are expected to find applications in many areas of pure and
applied mathematics, especially in fractional ordinary and partial differential
equationsandfractionalnumericalanalysis.Assuch,thismonographissuitablefor
researchers, graduate students and seminars of the above disciplines, also to be in
all science and engineering libraries.
Thepreparationofthisbooktookplaceduringtheacademicyear2016–2017at
the University of Memphis.
The author likes to thank Prof. Alina Lupas of University of Oradea, Romania,
for checking and reading the manuscript.
Memphis, USA George A. Anastassiou
May 2017
Contents
1 A Strong Left Fractional Calculus for Banach Space Valued
Functions. .... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 1
1.2 Main Results.. .... ..... .... .... .... .... .... ..... .... 2
References. .... .... .... ..... .... .... .... .... .... ..... .... 31
2 Strong Right Abstract Fractional Calculus... .... .... ..... .... 33
2.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 33
2.2 Main Results.. .... ..... .... .... .... .... .... ..... .... 34
References. .... .... .... ..... .... .... .... .... .... ..... .... 63
3 Strong Mixed and Generalized Abstract Fractional Calculus . .... 65
3.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 65
3.2 Main Results.. .... ..... .... .... .... .... .... ..... .... 66
References. .... .... .... ..... .... .... .... .... .... ..... .... 85
4 Foundations of General Fractional Analysis for Banach Space
Valued Functions... .... ..... .... .... .... .... .... ..... .... 87
4.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 87
4.2 Auxilliary Results.. ..... .... .... .... .... .... ..... .... 89
4.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 98
4.4 Applications .. .... ..... .... .... .... .... .... ..... .... 141
References. .... .... .... ..... .... .... .... .... .... ..... .... 145
5 Vector Abstract Fractional Korovkin Approximation .. ..... .... 147
5.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 147
5.2 Background .. .... ..... .... .... .... .... .... ..... .... 150
5.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 161
5.4 Application... .... ..... .... .... .... .... .... ..... .... 170
References. .... .... .... ..... .... .... .... .... .... ..... .... 172
ix
x Contents
6 Basic Abstract Korovkin Theory... .... .... .... .... ..... .... 175
6.1 Motivation ... .... ..... .... .... .... .... .... ..... .... 175
6.2 Main Results.. .... ..... .... .... .... .... .... ..... .... 177
References. .... .... .... ..... .... .... .... .... .... ..... .... 188
7 High Approximation for Banach Space Valued Functions.... .... 189
7.1 Motivation ... .... ..... .... .... .... .... .... ..... .... 189
7.2 Main Results.. .... ..... .... .... .... .... .... ..... .... 191
References. .... .... .... ..... .... .... .... .... .... ..... .... 210
8 Vectorial Abstract Fractional Approximation Using Linear
Operators. .... .... .... ..... .... .... .... .... .... ..... .... 211
8.1 Motivation ... .... ..... .... .... .... .... .... ..... .... 211
8.2 Background .. .... ..... .... .... .... .... .... ..... .... 213
8.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 219
8.4 Application... .... ..... .... .... .... .... .... ..... .... 231
References. .... .... .... ..... .... .... .... .... .... ..... .... 233
9 Abstract Fractional Trigonometric Korovkin Approximation . .... 235
9.1 Motivation ... .... ..... .... .... .... .... .... ..... .... 235
9.2 Background .. .... ..... .... .... .... .... .... ..... .... 237
9.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 244
9.4 Application... .... ..... .... .... .... .... .... ..... .... 255
References. .... .... .... ..... .... .... .... .... .... ..... .... 259
10 Multivariate Abstract Approximation for Banach Space Valued
Functions. .... .... .... ..... .... .... .... .... .... ..... .... 261
10.1 Motivation ... .... ..... .... .... .... .... .... ..... .... 261
10.2 Background .. .... ..... .... .... .... .... .... ..... .... 263
10.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 268
10.4 Application... .... ..... .... .... .... .... .... ..... .... 279
References. .... .... .... ..... .... .... .... .... .... ..... .... 281
11 Arctangent Function Based Abstract Neural Network
Approximation. .... .... ..... .... .... .... .... .... ..... .... 283
11.1 Introduction .. .... ..... .... .... .... .... .... ..... .... 283
11.2 Basics... .... .... ..... .... .... .... .... .... ..... .... 284
11.3 Main Results.. .... ..... .... .... .... .... .... ..... .... 293
References. .... .... .... ..... .... .... .... .... .... ..... .... 319