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 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 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 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 On Generalized k-Fractional Derivative Operator

2017 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Shahid Mubeen
Keyword(s):  
Fractional Derivative ◽  
Hypergeometric Functions ◽  
Mellin Transform ◽  
Beta Function ◽  
Fractional Operator ◽  
Derivative Operator ◽  
Definition Of

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.

scholarly journals Iterative approximation of positive solutions for fractional boundary value problem on the half-line

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6177-6187 ◽  
Author(s):  
Mourad Chamekh ◽  
Abdeljabbar Ghanmi ◽  
Samah Horrigue
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solutions ◽  
Boundary Value ◽  
The Other ◽  
Fractional Boundary Value Problem ◽  
Iterative Approximation ◽  
Derivative Operator ◽  
Liouville Fractional Derivative

In this paper, an iterative method is applied to solve some p-Laplacian boundary value problem involving Riemann-Liouville fractional derivative operator. More precisely, we establish the existence of two positive solutions. Moreover, we prove that these solutions are one maximal and the other is minimal. An example is presented to illustrate our main result. Finally, a numerical method to solve this problem is given.

scholarly journals Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 483 ◽  
Author(s):  
Mehmet Ali Özarslan ◽  
Ceren Ustaoğlu
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Recurrence Relations ◽  
Beta Function ◽  
Integral Operators ◽  
Integral Representations ◽  
Fractional Integral Operators ◽  
Derivative Formulas ◽  
Generating Relations ◽  
Transformation Formulas

Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function B y ( x , z ) . With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions.

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 Numerical approach of riemann-liouville fractional derivative operator

2021 ◽  
Vol 11 (6) ◽  
pp. 5367
Author(s):  
Ramzi B. Albadarneh ◽  
Iqbal M. Batiha ◽  
Ahmad Adwai ◽  
Nedal Tahat ◽  
A. K. Alomari
Keyword(s):  
Fractional Derivative ◽  
Nonlinear Problems ◽  
Numerical Approach ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Weighted Mean ◽  
Derivative Operator ◽  
Liouville Fractional Derivative ◽  
Linear And Nonlinear ◽  
Residual Errors

<p>This article introduces some new straightforward and yet powerful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional derivative operator. These formulas are derived by utilizing some of forthright computations, and by utilizing the so-called weighted mean value theorem (WMVT). Undoubtedly, such formulas will be extremely useful in establishing new approaches for several solutions of both linear and nonlinear fractionalorder differential equations. This assertion is confirmed by addressing several linear and nonlinear problems that illustrate the effectiveness and the practicability of the gained findings.</p>

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