Basic Definitions and Properties of Fractional Integrals and Derivatives

2014 ◽  
pp. 17-47
Author(s):  
Teodor M. Atanacković ◽  
Stevan Pilipović ◽  
Bogoljub Stanković ◽  
Dušan Zorica
Keyword(s):  
Fractional Integrals ◽  
Fractional Integrals And Derivatives

Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

2020 ◽  
Vol 23 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Latif A-M. Hanna ◽  
Maryam Al-Kandari ◽  
Yuri Luchko
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Operational Method ◽  
Initial Value ◽  
Right Hand ◽  
Fractional Differential ◽  
Left And Right ◽  
Fractional Integrals And Derivatives

AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

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.

scholarly journals Some fractional proportional integral inequalities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Gauhar Rahman ◽  
Thabet Abdeljawad ◽  
Aftab Khan ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Exponential Functions ◽  
Proportional Integral ◽  
Conformable Derivatives ◽  
Fractional Integrals And Derivatives ◽  
Decreasing Functions

Abstract In the last few years, various researchers studied the so-called conformable integrals and derivatives. Based on that notion some authors used modified conformable derivatives (proportional derivatives) to generate nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, which contain exponential functions in their kernels. Our aim in this paper is to establish some new integral inequalities by utilizing the fractional proportional-integral operators. In fact, certain new classes of integral inequalities for a class of n ($n\in \mathbb{N}$ n ∈ N ) positive continuous and decreasing functions on $[a,b]$ [ a , b ] are presented. The inequalities presented in this paper are more general than the existing classical inequalities.

scholarly journals Two index laws for fractional integrals and derivatives

1972 ◽  
Vol 14 (4) ◽  
pp. 385-410 ◽  
Author(s):  
E. R. Love
Keyword(s):  
Integral Equations ◽  
Fractional Derivatives ◽  
Addition Theorem ◽  
Positive Real Part ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Positive Real ◽  
Integrable Functions ◽  
Hypergeometric Integral ◽  
Fractional Integrals And Derivatives

SummaryThe first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

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).

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