scholarly journals Novel improved fractional operators and their scientific applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abd-Allah Hyder ◽  
M. A. Barakat
Keyword(s):  
Fractional Integral ◽  
Differential Operators ◽  
Fractional Operators ◽  
Scientific Applications ◽  
Scientific Application ◽  
Electrical Circuits ◽  
Engineering Problems ◽  
Convolution Kernels ◽  
Composition Properties ◽  
Parameter Values

AbstractThe motivation of this research is to introduce some new fractional operators called “the improved fractional (IF) operators”. The originality of these fractional operators comes from the fact that they repeat the method on general forms of conformable integration and differentiation rather than on the traditional ones. Hence the convolution kernels correlating with the IF operators are served in conformable abstract forms. This extends the scientific application scope of their fractional calculus. Also, some results are acquired to guarantee that the IF operators have advantages analogous to the familiar fractional integral and differential operators. To unveil the inverse and composition properties of the IF operators, certain function spaces with their characterizations are presented and analyzed. Moreover, it is remarkable that the IF operators generalize some fractional and conformable operators proposed in abundant preceding works. As scientific applications, the resistor–capacitor electrical circuits are analyzed under some IF operators. In the case of constant and periodic sources, this results in novel voltage forms. In addition, the overall influence of the IF operators on voltage behavior is graphically simulated for certain selected fractional and conformable parameter values. From the standpoint of computation, the usage of new IF operators is not limited to electrical circuits; they could also be useful in solving scientific or engineering problems.

A Universal Predictor-corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations

2021 ◽  
Author(s):  
Zaid Odibat
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Integral ◽  
Fractional Differential ◽  
Predictor Corrector

Abstract This study introduces some remarks on generalized fractional integral and differential operators, that generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor-corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

scholarly journals Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
K. Jangid ◽  
R. K. Parmar ◽  
R. Agarwal ◽  
Sunil D. Purohit
Keyword(s):  
Fractional Calculus ◽  
General Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Differential Operators ◽  
Integral Transforms ◽  
Parameter Selection ◽  
Fractional Operators ◽  
Polynomial Class ◽  
Math Phys

Abstract Motivated by a recent study on certain families of the incomplete H-functions (Srivastava et al. in Russ. J. Math. Phys. 25(1):116–138, 2018), we aim to investigate and develop several interesting properties related to product of a more general polynomial class together with incomplete Fox–Wright hypergeometric functions ${}_{p}\Psi _{q}^{(\gamma )}(\mathfrak{t})$ Ψ q ( γ ) p ( t ) and ${}_{p}\Psi _{q}^{(\Gamma )}(\mathfrak{t})$ Ψ q ( Γ ) p ( t ) including Marichev–Saigo–Maeda (M–S–M) fractional integral and differential operators, which contain Saigo hypergeometric, Riemann–Liouville, and Erdélyi–Kober fractional operators as particular cases regarding different parameter selection. Furthermore, we derive several integral transforms such as Jacobi, Gegenbauer (or ultraspherical), Legendre, Laplace, Mellin, Hankel, and Euler’s beta transforms.

Calculus of the fractional order operators in a discrete time domain

2020 ◽  
Author(s):  
Konrad Hawron ◽  
Maciej Siwczyński ◽  
Zuzanna Siwczyńska
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Differential Operators ◽  
Elementary Theory ◽  
Fractional Operators ◽  
Impulse Responses ◽  
Electrical Circuits ◽  
Distributed Parameters ◽  
The Stability ◽  
Recursive Digital Filters

Abstract The article presents the elementary theory of differential and integral operators of the fractional order in a discrete time approach. A notion of a simple proper fraction operator has been introduced. It has been done for the time equivalent by applying the Taylor series. On this basis a new theory of a certain complexity operators has been formed which includes differential operators of the fractional order. Somewhat more general approach has been presented in the later part of the article by introducing a rational power of the convolution operator. Both approaches to the fractional operators are realized by non–recursive digital filters of infinite impulse responses. The stability of such filters is also being considered. The article also contains the application to the distributed parameters electrical circuits theorem.

scholarly journals On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 118
Author(s):  
Muhammad Samraiz ◽  
Muhammad Umer ◽  
Artion Kashuri ◽  
Thabet Abdeljawad ◽  
Sajid Iqbal ◽  
...  
Keyword(s):  
Kinetic Equation ◽  
Laplace Transform ◽  
Fractional Integral ◽  
Differential Operators ◽  
Laplace Transforms ◽  
The Novel ◽  
Fractional Operators ◽  
Fractional Kinetic Equation

In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms.

scholarly journals Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 47
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz ◽  
U. Iturrarán-Viveros ◽  
R. Caballero-Cruz
Keyword(s):  
Fractional Derivative ◽  
Iterative Methods ◽  
Fractional Integral ◽  
Order Of Convergence ◽  
Fractional Operators ◽  
Speed Of Convergence ◽  
Newton Raphson ◽  
Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

scholarly journals On integral inequalities related to the weighted and the extended Chebyshev functionals involving different fractional operators

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Barış Çelik ◽  
Mustafa Ç. Gürbüz ◽  
M. Emin Özdemir ◽  
Erhan Set
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Fractional Operators ◽  
Fractional Integral Operators

AbstractThe role of fractional integral operators can be found as one of the best ways to generalize classical inequalities. In this paper, we use different fractional integral operators to produce some inequalities for the weighted and the extended Chebyshev functionals. The results are more general than the available classical results in the literature.

scholarly journals Fractional Hermite–Hadamard–Fejer Inequalities for a Convex Function with Respect to an Increasing Function Involving a Positive Weighted Symmetric Function

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1503 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Artion Kashuri
Keyword(s):  
Fractional Calculus ◽  
Convex Function ◽  
Symmetric Function ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Important Direction ◽  
Increasing Function

There have been many different definitions of fractional calculus presented in the literature, especially in recent years. These definitions can be classified into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite–Hadamard–Fejer (HHF) inequalities and related results. Here we consider some HHF fractional integral inequalities and related results for a class of fractional operators (namely, the weighted fractional operators), which apply to function of convex type with respect to an increasing function involving a positive weighted symmetric function. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities.

scholarly journals Electrical circuits RC and RL involving fractional operators with bi-order

2017 ◽  
Vol 9 (6) ◽  
pp. 168781401770713 ◽  
Author(s):  
JF Gómez-Aguilar ◽  
RF Escobar-Jiménez ◽  
VH Olivares-Peregrino ◽  
MA Taneco-Hernández ◽  
GV Guerrero-Ramírez
Keyword(s):  
Fractional Operators ◽  
Electrical Circuits

scholarly journals New Modified Conformable Fractional Integral Inequalities of Hermite–Hadamard Type with Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Thabet Abdeljawad ◽  
Pshtiwan Othman Mohammed ◽  
Artion Kashuri
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Operators ◽  
Special Means ◽  
Conformable Fractional Integral ◽  
Fractional Parameter

In this study, a few inequalities of Hermite–Hadamard type are constructed via the conformable fractional operators so that the normal version is recovered in its limit for the conformable fractional parameter. Finally, we present some examples to demonstrate the usefulness of conformable fractional inequalities in the context of special means of the positive numbers.

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Humberto Prado ◽  
Margarita Rivero ◽  
Juan J. Trujillo ◽  
M. Pilar Velasco
Keyword(s):  
Fractional Integral ◽  
Fractional Laplacian ◽  
Singular Integrals ◽  
Differential Operators ◽  
Riesz Potential ◽  
Fractional Power ◽  
Potential Operators ◽  
Relevant Role ◽  
Non Local ◽  
Fractional Power Of Operators

AbstractThe non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

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