scholarly journals On Extended Caputo Fractional Derivative Operator

2017 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Muhammad Arshad
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Elementary Functions ◽  
Derivative Operator ◽  
Mellin Transforms ◽  
Generating Relations

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

scholarly journals A New Extension of Extended Caputo Fractional Derivative Operator

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Shahid Mubeen
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Beta Function ◽  
Elementary Functions ◽  
Research Papers ◽  
Derivative Operator ◽  
Mellin Transforms ◽  
Generating Relations

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.

scholarly journals A new extension and applications of Caputo fractional derivative operator

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Muhammad Arshad
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Special Functions ◽  
Elementary Functions ◽  
Derivative Operator ◽  
Generating Relations

AbstractThe main objective of this paper is to introduce a further extension of the extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Additionally, we investigate the extended fractional derivative of some familiar special functions, the Mellin transform of the newly defined Caputo fractional derivative operator and generating relations for the extensions of extended hypergeometric functions.

scholarly journals An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2238
Author(s):  
Rahul Goyal ◽  
Praveen Agarwal ◽  
Alexandra Parmentier ◽  
Clemente Cesarano
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Derivative Operator ◽  
The Family ◽  
Generating Relations ◽  
Fractional Derivative Operators ◽  
Extended Operator ◽  
Two Parameter

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

scholarly journals Extended Riemann-Liouville type fractional derivative operator with applications

2017 ◽  
Vol 15 (1) ◽  
pp. 1667-1681 ◽  
Author(s):  
P. Agarwal ◽  
Juan J. Nieto ◽  
M.-J. Luo
Keyword(s):  
Fractional Derivative ◽  
Hypergeometric Functions ◽  
Beta Function ◽  
Gauss Hypergeometric Function ◽  
Modified Bessel Function ◽  
Elementary Functions ◽  
Liouville Type ◽  
Derivative Operator ◽  
Generating Relations ◽  
Liouville Fractional Derivative

Abstract The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.

scholarly journals On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

2020 ◽  
Vol 4 (4) ◽  
pp. 48 ◽  
Author(s):  
Övgü Gürel Yılmaz ◽  
Rabia Aktaş ◽  
Fatma Taşdelen
Keyword(s):  
Fractional Derivative ◽  
Hypergeometric Functions ◽  
Close Relation ◽  
Generating Relations ◽  
Transformation Formulas ◽  
Appell Functions

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of F1Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F2 and F3 Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions.

scholarly journals An Investigation of Fractional Bagley–Torvik Equation

Entropy ◽  
2019 ◽  
Vol 22 (1) ◽  
pp. 28 ◽  
Author(s):  
Azhar Ali Zafar ◽  
Grzegorz Kudra ◽  
Jan Awrejcewicz
Keyword(s):  
Fractional Derivative ◽  
Mechanical System ◽  
Caputo Fractional Derivative ◽  
Integral Transform ◽  
Rolling Bearing ◽  
Degree Of Freedom ◽  
Free Oscillations ◽  
Derivative Operator ◽  
Transform Method ◽  
Integral Transform Method

In this article, we will solve the Bagley–Torvik equation by employing integral transform method. Caputo fractional derivative operator is used in the modeling of the equation. The obtained solution is expressed in terms of generalized G function. Further, we will compare the obtained results with other available results in the literature to validate their usefulness. Furthermore, examples are included to highlight the control of the fractional parameters on he dynamics of the model. Moreover, we use this equation in modelling of real free oscillations of a one-degree-of-freedom mechanical system composed of a cart connected with the springs to the support and moving via linear rolling bearing block along a rail.

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