Erdélyi-Kober Fractional Integrals in the Real Scalar Variable Case

2018 ◽  
pp. 13-42
Author(s):  
A. M. Mathai ◽  
H. J. Haubold
Keyword(s):  
Fractional Integrals ◽  
The Real ◽  
Real Scalar ◽  
Scalar Variable

scholarly journals Mathematical Aspects of Krätzel Integral and Krätzel Transform

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 526 ◽  
Author(s):  
Arak M. Mathai ◽  
Hans J. Haubold
Keyword(s):  
Integral Transform ◽  
Statistical Distributions ◽  
The Real ◽  
Real Scalar ◽  
Mathematical Properties ◽  
Scalar Variable ◽  
Krätzel Integral

A real scalar variable integral is known in the literature by different names in different disciplines. It is basically a Bessel integral called specifically Krätzel integral. An integral transform with this Krätzel function as kernel is known as Krätzel transform. This article examines some mathematical properties of Krätzel integral, its connection to Mellin convolutions and statistical distributions, its computable representations, and its extensions to multivariate and matrix-variate cases, in both the real and complex domains. An extension in the pathway family of functions is also explored.

M-convolutions of products and ratios, statistical distributions and fractional calculus

Analysis ◽  
2016 ◽  
Vol 36 (1) ◽  
Author(s):  
Arakaparampil M. Mathai
Keyword(s):  
Fractional Calculus ◽  
Random Variables ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Positive Definite Matrices ◽  
The Real ◽  
Real Scalar ◽  
Scalar Variable ◽  
The Matrix

AbstractIt is shown that Mellin convolutions of products and ratios in the real scalar variable case can be considered as densities of products and ratios of two independently distributed real scalar positive random variables. It is also shown that these are also connected to Krätzel integrals and to the Krätzel transform in applied analysis, to reaction-rate probability integrals in astrophysics and to other related aspects when the random variables have gamma or generalized gamma densities, and to fractional calculus when one of the variables has a type-1 beta density and the other variable has an arbitrary density. Matrix-variate analogues are also discussed. In the matrix-variate case, the M-convolutions introduced by the author are shown to be directly connected to densities of products and ratios of statistically independently distributed positive definite matrix random variables in the real case and to Hermitian positive definite matrices in the complex domain. These M-convolutions reduce to Mellin convolutions in the scalar variable case.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

The real scalar field in extreme RNdS space

2005 ◽  
Vol 37 (7) ◽  
pp. 1323-1330 ◽  
Author(s):  
Guanghai Guo ◽  
Yuanxing Gui ◽  
Jianxiang Tian
Keyword(s):  
Scalar Field ◽  
The Real ◽  
Real Scalar

scholarly journals Numerical Approaches to Fractional Integrals and Derivatives: A Review

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 43 ◽  
Author(s):  
Min Cai ◽  
Changpin Li
Keyword(s):  
Fractional Calculus ◽  
Real World ◽  
Mathematical Concept ◽  
Fractional Integrals ◽  
Numerical Approximations ◽  
The Real ◽  
Potential Applications ◽  
Fractional Integrals And Derivatives ◽  
Almost All ◽  
Numerical Approaches

Fractional calculus, albeit a synonym of fractional integrals and derivatives which have two main characteristics—singularity and nonlocality—has attracted increasing interest due to its potential applications in the real world. This mathematical concept reveals underlying principles that govern the behavior of nature. The present paper focuses on numerical approximations to fractional integrals and derivatives. Almost all the results in this respect are included. Existing results, along with some remarks are summarized for the applied scientists and engineering community of fractional calculus.

Parametric instability of the real scalar pulsons

2005 ◽  
Vol 336 (1) ◽  
pp. 31-36 ◽  
Author(s):  
V.A. Koutvitsky ◽  
E.M. Maslov
Keyword(s):  
Parametric Instability ◽  
The Real ◽  
Real Scalar

Simulating the scalar field using the fuzzy sphere

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2389-2396 ◽  
Author(s):  
XAVIER MARTIN
Keyword(s):  
Scalar Field ◽  
Numerical Scheme ◽  
Approximation Scheme ◽  
Fuzzy Sphere ◽  
Continuous Space ◽  
The Real ◽  
Matrix Algebras ◽  
Real Scalar ◽  
Fuzzy Space ◽  
Commutative Matrix

Fuzzy spaces provide a new approximation scheme using (non–commutative) matrix algebras to approximate the algebra of function of the continuous space. This paper describes how to implement a numerical scheme based on a fuzzy space approximation. In this first attempt, the simplest fuzzy space and field theory, respectively the fuzzy two–sphere and the real scalar field, are used to simulate the real scalar field on the plane. Along the way, this method is compared to its traditional lattice discretisation equivalent.

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