scholarly journals Estimates of Classes of Generalized Special Functions and Their Application in the Fractional k , s -Calculus Theory

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
S. Chandak ◽  
D. L. Suthar ◽  
S. AL-Omari ◽  
S. Gulyaz-Ozyurt
Keyword(s):  
Fractional Integral ◽  
Differential Operators ◽  
Special Functions ◽  
Technical Problems ◽  
Set Up ◽  
Scientific Engineering

In this article, we aim to develop new k , s -fractional integral and differential operators containing S -functions as kernels in a form of generalized k -Mittag-Leffer functions. We also set up various properties of such operators. Furthermore, we consider a variety of implications of the major outcomes that will be very useful in the implementation of scientific, engineering, and technical 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.

Dynamical behavior of fractionalized simply supported beam: An application of fractional operators to Bernoulli-Euler theory

2021 ◽  
Vol 10 (1) ◽  
pp. 231-239
Author(s):  
Kashif Ali Abro ◽  
Abdon Atangana ◽  
Ali Raza Khoso
Keyword(s):  
Differential Operators ◽  
Special Functions ◽  
Dynamical Behavior ◽  
Analytic Solutions ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Parametric Resonances ◽  
Fractional Differential ◽  
Fractional Differential Operators ◽  
Graphical Illustration

Abstract The complex structures usually depend upon unconstrained and constrained simply supported beams because the passive damping is applied to control vibrations or dissipate acoustic energies involved in aerospace and automotive industries. This manuscript aims to present an analytic study of a simply supported beam based on the modern fractional approaches namely Caputo-Fabrizio and Atanagna-Baleanu fractional differential operators. The governing equation of motion is fractionalized for knowing the vivid effects of principal parametric resonances. The powerful techniques of Laplace and Fourier sine transforms are invoked for investigating the exact solutions with fractional and non-fractional approaches. The analytic solutions are presented in terms of elementary as well as special functions and depicted for graphical illustration based on embedded parameters. Finally, effects of the amplitude of vibrations and the natural frequency are discussed based on the sensitivities of dynamic characteristics of simply supported beam.

Boundary Value Problems for Harmonic Functions on the Heisenberg Group

1986 ◽  
Vol 38 (2) ◽  
pp. 478-512 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Heisenberg Group ◽  
Harmonic Functions ◽  
Poisson Kernel ◽  
Differential Operators ◽  
Special Functions ◽  
Continuous Functions ◽  
Group Representations ◽  
Hypoelliptic Operators ◽  
Analysis Group ◽  
Partial Differential Operators

Analysis on the Heisenberg group has become an important area with strong connections to Fourier analysis, group representations, and partial differential operators. We propose to show in this work that special functions methods can also play a significant part in this theory. There is a one-parameter family of second-order hypoelliptic operators Lγ, (γ ∊ C), associated to the Laplacian L0 (also called the subelliptic or Kohn Laplacian). These operators are closely related to the unit ball for reasons of homogeneity and unitary group invariance. The associated Dirichlet problem is to find functions with specified boundary values and annihilated by Lγ inside the ball (that is, Lγ-harmonic). This is the topic of this paper.Gaveau [9] proved the first positive result, showing that continuous functions on the boundary can be extended to L0-harmonic functions in the ball, by use of diffusion-theoretic methods. Jerison [15] later gave another proof of the L0-result. Hueber [14] has recently obtained some results dealing with special values of the Poisson kernel for L0.

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.

scholarly journals Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Formulas ◽  
Special Cases

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving theFp(α,β)(·). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functionsFp(α,β,m)(·). Some interesting special cases of our main results are also considered.

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.

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