scholarly journals General Fractional Integrals and Derivatives with the Sonine Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 594
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Power Function ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Integrals And Derivatives ◽  
Fundamental Theorems ◽  
Special Classes

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n-fold general fractional integrals and derivatives and study their properties.

scholarly journals General Fractional Integrals and Derivatives of Arbitrary Order

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 755
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Basic Properties ◽  
Suitable Generalization ◽  
Fractional Integrals And Derivatives ◽  
Fundamental Theorems ◽  
Derivatives Of

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

Extended Mittag-Leffler function and associated fractional calculus operators

2020 ◽  
Vol 27 (2) ◽  
pp. 199-209 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh K. Parmar ◽  
Purnima Chopra
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Fractional Integrals ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Appl Math

AbstractMotivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

A distributional theory of fractional calculus

1985 ◽  
Vol 99 (3-4) ◽  
pp. 347-357 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Fractional Calculus ◽  
Spectral Properties ◽  
Fractional Integration ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Fractional Powers ◽  
Fractional Powers Of Operators ◽  
Fractional Integrals And Derivatives

SynopsisIn this paper, a theory of fractional calculus is developed for certain spacesD′p,μof generalised functions. The theory is based on the construction of fractionalpowers of certain simple differential and integral operators. With the parameter μ suitably restricted, these fractional powers are shown to coincide with the Riemann-Liouville and Weyl operators of fractional integration and differentiation. Standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators; see [6], [7]. Some spectral properties are also obtained.

Some generalized fractional calculus operators and their applications in integral equations

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Om Agrawal
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Integral Equations ◽  
Systematic Approach ◽  
Fractional Integrals ◽  
Generalized Fractional Calculus ◽  
Fractional Calculus Operators ◽  
Fractional Integrals And Derivatives ◽  
Generalized Fractional Calculus Operators

AbstractIn this paper, we survey some generalizations of fractional integrals and derivatives and present some of their properties. Using these properties, we show that many integral equations can be solved in a much elegant way. We believe that this will blur the distinction between the integral and differential equations, and provide a systematic approach for the two of these classes.

scholarly journals Katugampola Fractional Calculus With Generalized k−Wright Function

2021 ◽  
Vol 1 (1) ◽  
pp. 34-44
Author(s):  
Ahmad Y. A. Salamooni ◽  
D. D. Pawar
Keyword(s):  
Fractional Calculus ◽  
Fractional Integrals ◽  
Wright Function ◽  
Fractional Integrals And Derivatives ◽  
Derivatives Of

In this article, we present some properties of the Katugampola fractional integrals and derivatives. Also, we study the fractional calculus properties involving Katugampola Fractional integrals and derivatives of generalized k−Wright function nΦkm(z).

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.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals Fractional derivatives and the fundamental theorem of fractional calculus

2020 ◽  
Vol 23 (4) ◽  
pp. 939-966 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fractional Calculus ◽  
Fundamental Theorem ◽  
Fractional Derivatives ◽  
Finite Interval ◽  
Fractional Integrals ◽  
Important Class ◽  
Left Inverse ◽  
Caputo Fractional Derivatives ◽  
Fractional Integrals And Derivatives ◽  
The One

AbstractIn this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition follows from the fundamental theorem of the Fractional Calculus, i.e., they are introduced as the left-inverse operators to the Riemann-Liouville fractional integrals. Until now, three families of such derivatives were suggested in the literature: the Riemann-Liouville fractional derivatives, the Caputo fractional derivatives, and the Hilfer fractional derivatives. We clarify the interconnections between these derivatives on different spaces of functions and provide some of their properties including the formulas for their projectors and the Laplace transforms. However, it turns out that there exist infinitely many other families of the fractional derivatives that are the left-inverse operators to the Riemann-Liouville fractional integrals. In this paper, we focus on an important class of these fractional derivatives and discuss some of their properties.

scholarly journals Fractional Sums and Differences with Binomial Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Fahd Jarad ◽  
Ravi P. Agarwal
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Binomial Coefficients ◽  
Discrete Version ◽  
Binomial Theorem ◽  
Left And Right ◽  
Fractional Integrals And Derivatives ◽  
Dual Identities

In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

scholarly journals General Fractional Calculus: Multi-Kernel Approach

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1501
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Arbitrary Order ◽  
Fractional Integrals ◽  
Extended Formulation ◽  
Different Types ◽  
Kernel Approach ◽  
Definition Of ◽  
Fundamental Theorems ◽  
First Time

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved.

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