• Title/Summary/Keyword: Mittag-Leffler function

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FOURIER'S TRANSFORM OF FRACTIONAL ORDER VIA MITTAG-LEFFLER FUNCTION AND MODIFIED RIEMANN-LIOUVILLE DERIVATIVE

  • Jumarie, Guy
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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    • pp.1101-1121
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    • 2008
  • One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

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ANALYSIS OF NON-INTEGER ORDER THERMOELASTIC TEMPERATURE DISTRIBUTION AND THERMAL DEFLECTION OF THIN HOLLOW CIRCULAR DISK UNDER THE AXI-SYMMETRIC HEAT SUPPLY

  • KHAVALE, SATISH G.;GAIKWAD, KISHOR R.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.26 no.1
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    • pp.67-75
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    • 2022
  • Analysis of non-integer order thermoelastic temperature distribution and it's thermal deflection of thin hollow circular disk under the axi-symmetric heat supply is investigated. Initially, the disk is kept at zero temperature. For t > 0 the parametric surfaces are thermally insulated and axi-symmetric heat supply on the thickness of the disk. The governing heat conduction equation has been solved by integral transform technique, including Mittag-Leffler function. The results have been computed numerically and illustrated graphically with the help of PTC-Mathcad.

ON THE STABILITY OF DIFFERENTIAL SYSTEMS INVOLVING 𝜓-HILFER FRACTIONAL DERIVATIVE

  • Limpanukorn, Norravich;Ngiamsunthorn, Parinya Sa;Songsanga, Danuruj;Suechoei, Apassara
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.3
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    • pp.513-532
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    • 2022
  • This paper deals with the stability of solutions to 𝜓-Hilfer fractional differential systems. We derive the fundamental solution for the system by using the generalized Laplace transform and the Mittag-Leffler function with two parameters. In addition, we obtained some necessary conditions on the stability of the solutions to linear fractional differential systems for homogeneous, non-homogeneous and non-autonomous cases. Numerical examples are also given to illustrate the behavior of solutions.

CERTAIN NEW FAMILIES FOR BI-UNIVALENT FUNCTIONS DEFINED BY A KNOWN OPERATOR

  • Wanas, Abbas Kareem;Choi, Junesang
    • East Asian mathematical journal
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    • v.37 no.3
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    • pp.319-331
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    • 2021
  • In this paper, we aim to introduce two new families of analytic and bi-univalent functions associated with the Attiya's operator, which is defined by the Hadamard product of a generalized Mittag-Leffler function and analytic functions on the open unit disk. Then we estimate the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we investigate Fekete-Szegö problem for these families. Some relevant connections of certain special cases of the main results with those in several earlier works are also pointed out. Two naturally-arisen problems are given for further investigation.

THE POLYANALYTIC SUB-FOCK REPRODUCING KERNELS WITH CERTAIN POSITIVE INTEGER POWERS

  • Kim, Hyeseon
    • Honam Mathematical Journal
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    • v.44 no.3
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    • pp.447-460
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    • 2022
  • We consider a closed subspace ${\tilde{A}}^{{\alpha},m}_q$ (ℂ) of the Fock space Aα,mq (ℂ) of q-analytic functions with the weight ϕ(z) = -α log |z|2+|z|2m for any positive integer m. We obtain the corresponding reproducing kernel Kα,q,m(z, w) using the weighted Laguerre polynomials and the Mittag-Leffler functions. Finally, we investigate the necessary and sufficient condition on (α, q, m) such that Kα,q,m(z, w) is zero-free.

A NOTE ON A CLASS OF CONVOLUTION INTEGRAL EQUATIONS

  • LUO, MIN-JIE;RAINA, R.K.
    • Honam Mathematical Journal
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    • v.37 no.4
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    • pp.397-409
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    • 2015
  • This paper considers a class of new convolution integral equations whose kernels involve special functions such as the generalized Mittag-Leffler function and the extended Kummer hypergeometric function. Some basic properties of interconnection with the familiar Riemann-Liouville operators are obtained which are used in fiding the solution of the main convolution integral equation. Several consequences are deduced from the main result by incorporating certain extended forms of hypergeometric functions in our present investigation.

ON GENERALIZED WRIGHT'S HYPERGEOMETRIC FUNCTIONS AND FRACTIONAL CALCULUS OPERATORS

  • Raina, R.K.
    • East Asian mathematical journal
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    • v.21 no.2
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    • pp.191-203
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    • 2005
  • In the present paper we first establish some basic results for a substantially more general class of functions defined below. The results include simple differentiation and fractional calculus operators(integration and differentiation of arbitrary orders) for this class of functions. These results are then invoked in determining similar properties for the generalized Wright's hypergeometric functions. Further, norm estimate of a certain class of integral operators whose kernel involves the generalized Wright's hypergeometric function, and its composition(and other related properties) with the fractional calculus operators are also investigated.

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Fractional radioactive decay law and Bateman equations

  • Cruz-Lopez, C.A.;Espinosa-Paredes, G.
    • Nuclear Engineering and Technology
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    • v.54 no.1
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    • pp.275-282
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    • 2022
  • The aim of this work is to develop the fractional Bateman equations, which can model memory effects in successive isotopes transformations. Such memory effects have been previously reported in the alpha decay, which exhibits a non-Markovian behavior. Since there are radioactive decay series with consecutive alpha decays, it is convenient to include the mentioned memory effects, developing the fractional Bateman Equations, which can reproduce the standard ones when the fractional order is equal to one. The proposed fractional model preserves the mathematical shape and the symmetry of the standard equations, being the only difference the presence of the Mittag-Leffler function, instead of the exponential one. This last is a very important result, because allows the implementation of the proposed fractional model in burnup and activation codes in a straightforward way. Numerical experiments show that the proposed equations predict high decay rates for small time values, in comparison with the standard equations, which have high decay rates for large times. This work represents a novelty approach to the theory of successive transformations, and opens the possibility to study properties of the Bateman equation from a fractional approach.

ANALYSIS OF SOLUTIONS OF TIME FRACTIONAL TELEGRAPH EQUATION

  • Joice Nirmala, R.;Balachandran, K.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.18 no.3
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    • pp.209-224
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    • 2014
  • In this paper, the solution of time fractional telegraph equation is obtained by using Adomain decomposition method and compared with various other method to determine the efficiency of Adomain decomposition method. These methods are used to obtain the series solutions. Finally, results are analysed by plotting the solutions for various fractional orders.

ON SEQUENCE OF FUNCTIONS

  • Salehbhai, Ibrahim A.;Prajapati, Jyotindra C.;Shukla, Ajay K.
    • Communications of the Korean Mathematical Society
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    • v.28 no.1
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    • pp.123-134
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    • 2013
  • Operational techniques have drawn the attention of several researchers in the study of sequence of functions and polynomials. An attempt is made to introduce a new sequence of functions by using operational techniques. Some generating relations and finite summation formulae have been obtained. The corresponding MAPLE code for obtaining above sequence of functions for different values of parameters was also discussed.