• Title/Summary/Keyword: fractional derivative and integral operators

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FRACTIONAL CALCULUS AND INTEGRAL TRANSFORMS OF INCOMPLETE τ-HYPERGEOMETRIC FUNCTION

  • Pandey, Neelam;Patel, Jai Prakash
    • Communications of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.127-142
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    • 2018
  • In the present article, authors obtained certain fractional derivative and integral formulas involving incomplete ${\tau}$-hypergeometric function introduced by Parmar and Saxena [14]. Some interesting special cases and consequences of our main results are also considered.

Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments

  • Srivastava, Hari Mohan
    • Kyungpook Mathematical Journal
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    • v.60 no.1
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    • pp.73-116
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    • 2020
  • The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.

SOME APPLICATIONS AND PROPERTIES OF GENERALIZED FRACTIONAL CALCULUS OPERATORS TO A SUBCLASS OF ANALYTIC AND MULTIVALENT FUNCTIONS

  • Lee, S.K.;Khairnar, S.M.;More, Meena
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.127-145
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    • 2009
  • In this paper we introduce a new subclass $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ of analytic and multivalent functions with negative coefficients using fractional calculus operators. Connections to the well known and some new subclasses are discussed. A necessary and sufficient condition for a function to be in $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ is obtained. Several distortion inequalities involving fractional integral and fractional derivative operators are also presented. We also give results for radius of starlikeness, convexity and close-to-convexity and inclusion property for functions in the subclass. Modified Hadamard product, application of class preserving integral operator and other interesting properties are also discussed.

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ERTAIN k-FRACTIONAL CALCULUS OPERATORS AND IMAGE FORMULAS OF GENERALIZED k-BESSEL FUNCTION

  • Agarwal, P.;Suthar, D.L.;Tadesse, Hagos;Habenom, Haile
    • Honam Mathematical Journal
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    • v.43 no.2
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    • pp.167-181
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    • 2021
  • In this paper, the Saigo's k-fractional integral and derivative operators involving k-hypergeometric function in the kernel are applied to the generalized k-Bessel function; results are expressed in term of k-Wright function, which are used to present image formulas of integral transforms including beta transform. Also special cases related to fractional calculus operators and Bessel functions are considered.

SOME FAMILIES OF INFINITE SERIES SUMMABLE VIA FRACTIONAL CALCULUS OPERATORS

  • Tu, Shih-Tong;Wang, Pin-Yu;Srivastava, H.M.
    • East Asian mathematical journal
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    • v.18 no.1
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    • pp.111-125
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    • 2002
  • Many different families of infinite series were recently observed to be summable in closed forms by means of certain operators of fractional calculus(that is, calculus of integrals and derivatives of any arbitrary real or complex order). In this sequel to some of these recent investigations, the authors present yet another instance of applications of certain fractional calculus operators. Alternative derivations without using these fractional calculus operators are shown to lead naturally a family of analogous infinite sums involving hypergeometric functions.

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CERTAIN RESULTS INVOLVING FRACTIONAL OPERATORS AND SPECIAL FUNCTIONS

  • Aghili, Arman
    • Korean Journal of Mathematics
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    • v.27 no.2
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    • pp.487-503
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    • 2019
  • In this study, the author provided a discussion on one dimensional Laplace and Fourier transforms with their applications. It is shown that the combined use of exponential operators and integral transforms provides a powerful tool to solve space fractional partial differential equation with non - constant coefficients. The object of the present article is to extend the application of the joint Fourier - Laplace transform to derive an analytical solution for a variety of time fractional non - homogeneous KdV. Numerous exercises and examples presented throughout the paper.

FRACTIONAL DIFFERENTIATIONS AND INTEGRATIONS OF QUADRUPLE HYPERGEOMETRIC SERIES

  • Bin-Saad, Maged G.;Nisar, Kottakkaran S.;Younis, Jihad A.
    • Communications of the Korean Mathematical Society
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    • v.36 no.3
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    • pp.495-513
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    • 2021
  • The hypergeometric series of four variables are introduced and studied by Bin-Saad and Younis recently. In this line, we derive several fractional derivative formulas, integral representations and operational formulas for new quadruple hypergeometric series.

SOME FAMILIES OF INFINITE SUMS DERIVED BY MEANS OF FRACTIONAL CALCULUS

  • Romero, Susana Salinas De;Srivastava, H.M.
    • East Asian mathematical journal
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    • v.17 no.1
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    • pp.135-146
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    • 2001
  • Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

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ON SOME WEIGHTED HARDY-TYPE INEQUALITIES INVOLVING EXTENDED RIEMANN-LIOUVILLE FRACTIONAL CALCULUS OPERATORS

  • Iqbal, Sajid;Pecaric, Josip;Samraiz, Muhammad;Tehmeena, Hassan;Tomovski, Zivorad
    • Communications of the Korean Mathematical Society
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    • v.35 no.1
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    • pp.161-184
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    • 2020
  • In this article, we establish some new weighted Hardy-type inequalities involving some variants of extended Riemann-Liouville fractional derivative operators, using convex and increasing functions. As special cases of the main results, we obtain the results of [18,19]. We also prove the boundedness of the k-fractional integral operator on Lp[a, b].

SUBCLASSES OF k-UNIFORMLY CONVEX AND k-STARLIKE FUNCTIONS DEFINED BY SĂLĂGEAN OPERATOR

  • Seker, Bilal;Acu, Mugur;Eker, Sevtap Sumer
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.169-182
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    • 2011
  • The main object of this paper is to introduce and investigate new subclasses of normalized analytic functions in the open unit disc $\mathbb{U}$, which generalize the familiar class of k-starlike functions. The various properties and characteristics for functions belonging to these classes derived here include (for example) coefficient inequalities, distortion theorems involving fractional calculus, extreme points, integral operators and integral means inequalities.