Erdélyi-Kober Fractional Calculus
From a Statistical Perspective, Inspired by Solar Neutrino Physics
1st Edition
Published on 6. September 2018
XII, 122 pages
978-981-13-1159-8 (ISBN)
Description
This book focuses on Erdélyi-Kober fractional calculus from a statistical perspective inspired by solar neutrino physics. Results of diffusion entropy analysis and standard deviation analysis of data from the Super-Kamiokande solar neutrino experiment lead to the development of anomalous diffusion and reaction in terms of fractional calculus. The new statistical perspective of Erdélyi-Kober fractional operators outlined in this book will have fundamental applications in the theory of anomalous reaction and diffusion processes dealt with in physics.
A major mathematical objective of this book is specifically to examine a new de?nition for fractional integrals in terms of the distributions of products and ratios of statistically independently distributed positive scalar random variables or in terms of Mellin convolutions of products and ratios in the case of real scalar variables. The idea will be generalized to cover multivariable cases as well as matrix variable cases. In the matrix variable case, M-convolutions of products and ratios will be used to extend the ideas. We then give a de?nition for the case of real-valued scalar functions of several matrices.More details
Series
Language
English
Place of publication
Singapore
Singapore
Publishing group
Springer Singapore
Target group
Professional and scholarly
Product notice
Reflowable
Illustrations
XII, 122 p. 6 illus., 3 illus. in color.
File size
1,98 MB
ISBN-13
978-981-13-1159-8 (9789811311598)
DOI
10.1007/978-981-13-1159-8
Schweitzer Classification
Other editions
Additional editions
A. M. Mathai | H. J. Haubold
Erdélyi-Kober Fractional Calculus
From a Statistical Perspective, Inspired by Solar Neutrino Physics
Book
09/2018
Springer
€53.49
Shipment within 15-20 days
Content
- Intro
- Preface
- Contents
- Acronyms
- 1 Solar Neutrinos, Diffusion, Entropy, Fractional Calculus
- References
- 2 Erdélyi-Kober Fractional Integrals in the Real Scalar Variable Case
- 2.1 Introduction
- 2.2 Some Notations
- 2.2.1 Some of the Fractional Integrals and the Notations in the Literature
- 2.3 Fractional Integrals of the First Kind in the Real Scalar Variable Case
- 2.4 A Pathway Generalization of Erdélyi-Kober Fractional Integral Operator of the First Kind
- 2.5 Some Special Cases
- 2.6 Erdélyi-Kober Fractional Integrals of the First Kind and Hypergeometric Series
- 2.7 Mellin Transform of the Generalized Erdélyi-Kober Fractional Integral of the First Kind
- 2.8 Riemann-Liouville Operators as Mellin Convolution
- 2.9 Distribution of a Product and Erdélyi-Kober Operators of the Second Kind
- 2.10 A Pathway Extension of Erdélyi-Kober Operator of the Second Kind
- 2.11 Special Cases
- 2.12 Another Form of Generalization of Erdélyi-Kober Operators of the Second Kind
- 2.13 Mellin Transform of the Generalized Erdélyi-Kober Operator of the Second Kind
- 2.14 A Geometrical and Some Physical Interpretations of Fractional Integrals
- 2.14.1 An Interpretation in Terms of Densities of Sum and Difference
- 2.14.2 Fractional Integrals as Fractions of Total Probabilities
- 2.14.3 A Geometrical Interpretation
- 2.15 A General Definition of Fractional Integrals
- 2.15.1 Mellin Convolution of Product and Second Kind Integrals
- 2.15.2 Mellin Convolution of a Ratio and First Kind Fractional Integrals
- References
- 3 Erdélyi-Kober Fractional Integrals in the Real Matrix-Variante Case
- 3.1 Explicit Evaluations of Matrix-Variate Gammaand Beta Integrals
- 3.1.1 Explicit Evaluation of Real Matrix-Variate Gamma Integral
- 3.1.2 Evaluation of Matrix-Variate Type-1 Beta Integral in the Real Case
- 3.1.3 General Partitions
- 3.1.4 A Method of Avoiding Integration Over the Stiefel Manifold
- 3.2 Erdélyi-Kober Fractional Integral Operator of the Second Kind for the Real Matrix-Variate Case
- 3.3 A Pathway Generalization of Erdélyi-Kober Fractional Integral Operator of the Second Kind in the Real Matrix-Variate Case
- 3.4 M-Transforms of Erdélyi-Kober Fractional Integral of the Second Kind in the Real Matrix-Variate Case
- 3.5 Generalization in Terms of Hypergeometric Series for Erdélyi-Kober Fractional Integral of the Second Kind in the Real Matrix-Variate Case
- 3.6 Erdélyi-Kober Fractional Integral of the First Kind in the Real Matrix-Variate Case
- 3.7 Pathway Extension of Erdélyi-Kober Fractional Integral of the First Kind in the Real Matrix-Variate Case
- 3.8 A General Definition
- 3.8.1 Special Cases
- 3.8.2 Special Cases of First Kind Fractional Integrals
- References
- 4 Erdélyi-Kober Fractional Integrals in the Many Real Scalar Variables Case
- 4.1 Erdélyi-Kober Fractional Integrals of the Second Kind in Multivariate Case as Statistical Densities
- 4.2 A Pathway Generalization of Erdélyi-Kober Fractional Integral Operator of the Second Kind in the Multivariate Case
- 4.3 Mellin Transform in the Multivariate Case for Erdélyi-Kober Fractional Integral of the Second Kind
- 4.4 Erdélyi-Kober Fractional Integral of the First Kind for Multivariate Case
- 4.5 A General Definition for First and Second Kind Fractional Integrals in the Multivariate Case
- 4.5.1 A Special Case of (4.32)
- 4.5.2 Special Case of (4.34)
- Reference
- 5 Erdélyi-Kober Fractional Integrals Involving Many Real Matrices
- 5.1 Second Kind Fractional Integrals in the Many Matrix-variate Case and Statistical Densities
- 5.2 Fractional Integrals of the First Kind in the Case of Many Real Matrix Variables
- 5.3 M-Transforms for the Fractional Integrals in the Many Real Matrix-Variate Case
- References
- 6 Erdélyi-Kober Fractional Integrals in the Complex Domain
- 6.1 Introduction
- 6.2 Explicit Evaluations of Gamma and Beta Integrals in the Complex Domain
- 6.2.1 An Alternate Method Based on Partitioned Matrix
- 6.3 Evaluation of Matrix-Variate Beta Integrals in the Complex Domain
- 6.4 Fractional Integrals in the Matrix-Variate Case in the Complex Domain
- 6.4.1 Erdélyi-Kober Fractional Integral of the Second Kind of Order a
- 6.4.2 The Right-Sided Riemann-Liouville and Weyl Fractional Integrals in the Complex Matrix-Variate Case
- 6.4.3 Saigo and Related Fractional Integrals of the Second Kind
- 6.4.4 A Pathway Generalized Definition of Fractional Integrals of the Second Kind in the Complex Matrix-Variate Case
- 6.5 Fractional Integral of Order a and Parameter ß of the First Kind in the Complex Matrix-variate Case
- 6.5.1 Erdélyi-Kober Fractional Integral of Order a of the First Kind for Complex Matrix-Variate Case
- 6.5.2 Riemann-Liouvile and Weyl Fractional Integrals of the First Kind of Order a for the Complex Matrix-Variate Case
- References
- Index
