Identities on fractional integrals and various integral transforms (original) (raw)
Some identities on fractional integrals and integral transforms
Hacettepe Journal of Mathematics and Statistics
In this paper, we introduce various theorems that associate the generalized Riemann-Liouville fractional integral operator and the generalized Weyl fractional integral operator with some well-known integral transforms including generalized Laplace transform, Widder potential transform, generalized Widder transform, Hankel transform and Bessel transform. We evaluate certain integrals of some elementary functions and some special functions as applications of these theorems and their results.
A New Integral Transform With Applications to Fractional Calculus
Communications in Mathematics and Applications, 2022
In this paper, an integral transform with the kernel being the Mittag-Leffler function in two parameters is introduced. Some properties of this integral transform are discussed. Also, its formulae for derivatives of the function are derived. The new integral transform is applied to derive the exact formula for the Laplace transform of fractional derivatives.
On Novel Fractional Integral and Differential Operators and Their Properties
Journal of Mathematics, 2023
Te main goal of this paper is to describe the new version of extended Bessel-Maitland function and discuss its special cases. Ten, using the aforementioned function as their kernels, we develop the generalized fractional integral and diferential operators. Te convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann-Liouville fractional operators are explored. Te integral transforms of newly defned and generalized fractional operators in terms of the generalized Fox-Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.
Note on Integral Transform of Fractional Calculus
https://www.ijrrjournal.com/IJRR\_Vol.6\_Issue.2\_Feb2019/Abstract\_IJRR0016.html, 2019
In recent years Fractional Calculus is highly growing field in research because of its wide applicability and interdisciplinary approach. In this article we study various integral transform particularly Laplace Transform, Mellin Transform, of Fractional calculus i.e. Fractional derivative and Fractional Integral particularly of Riemann-Liouville Fractional derivative, Riemann-Liouville Fractional integral, Caputo’s Fractional derivative and their properties.
Study of Generalized Integral Transforms their Properties and Relations
2017
In this article, we study the basic theoretical properties of Mellin-type and Weyl fractional integrals and fractional derivatives. Furthermore, we prove some properties of Weyl fractional transform. Also, we study fractional Mellin transform and we prove relation between fractional Mellin transform and Fourier fractional Mellin transform. AMS subject classification:
Taiwanese Journal of Mathematics, 2004
In this paper we establish a very general and useful theorem which interconnects the Laplace transform and the generalized Weyl fractional integral operator involving the multivariable H-function of related functions of several variables. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to these sequences, one can easily evaluate the generalized Weyl fractional integral operator of special functions of several variables. We have illustrated it for Srivastava-Daoust multivariable hypergeometric function. On account of general nature of this function a number of results involving special functions of one or more variables can be obtained merely by specializing the parameters.
Generalized Operational Relations and Properties of Fractional Hankel Transform
Namias 4 had defined fractionalization of conventional Hankel transform, using the method of eigen values and studied and open the door for the research in fractional integral transform. This paper studies the fractionalization of generalized Hankel transform, as given by Zemanian 5 . We referred it as fractional Hankel transform. First we introduce fractional Hankel transform in the generalized sense. Generalized operational relations are derived that can be used to solve certain classes of ordinary and partial differential equations. Lastly the values of fractional Hankel transform are obtained for some special functions.
Extensions and results from a method for evaluating fractional integrals
arXiv (Cornell University), 1994
We present a method derived from Laplace transform theory that enables the evaluation of fractional integrals. This method is adapted and extended in a variety of ways to demonstrate its utility in deriving alternative representations for other classes of integrals. We also use the method in conjunction with several different techniques to derive many results that have not appeared in tables of integrals.
New fractional integral unifying six existing fractional integrals
2016
In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erd\'elyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form \[ \left({}^{\rho}\mathcal{I}^{\alpha, \beta}_{a+;\eta, \kappa}f\right)(x)=\frac{\rho^{1-\beta}x^{\kappa}}{\Gamma(\alpha)}\int_a^x \frac{\tau^{\rho \eta +\rho-1}}{(x^\rho-\tau^\rho)^{1-\alpha}}f(\tau)\text{d}\tau, \quad 0\leq a < x < b \leq \infty. \] A similar generalization is not possible with the Erd\'elyi-Kober operator though there is a close resemblance with the operator in question. We also give semigroup, boundedness, shift and integration-by-parts formulas for completeness.
Mellin Transforms of the Generalized Fractional Integrals and Derivatives
Arxiv preprint arXiv:1112.6031, 2015
We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann-Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized δ r,m operators with generalized Stirling numbers and Lah numbers. For example, we show that δ 1,1 corresponds to the Stirling numbers of the 2 nd kind and δ 2,1 corresponds to the unsigned Lah numbers. Further, we show that the two operators δ r,m and δ m,r , r, m ∈ N, generate the same sequence given by the recurrence relation