The Generalized Fractional Exponential Function: The R ‐Function and Other Functions for the Fractional Calculus

2016 ◽  
pp. 19-46
Keyword(s):  
Fractional Calculus ◽  
Exponential Function ◽  
Fractional Exponential Function

scholarly journals Some Applications of the Wright Function in Continuum Physics: A Survey

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 198
Author(s):  
Yuriy Povstenko
Keyword(s):  
Heat Conduction ◽  
Fractional Calculus ◽  
Exponential Function ◽  
Nonlocal Elasticity ◽  
Bessel Functions ◽  
Wright Function ◽  
Mainardi Function ◽  
Integral Relations

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

scholarly journals A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1677-1681 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Mahmoud Abdel-Aty ◽  
Carlo Cattani
Keyword(s):  
Heat Transfer ◽  
Fractional Order ◽  
Exponential Function ◽  
Power Law ◽  
Transfer Model ◽  
Mathematical Tool ◽  
Fractional Exponential Function ◽  
Anomalous Heat Transfer ◽  
First Time ◽  
Anomalous Heat

In this paper, we consider a general fractional-order derivataive of the Liouville-Caputo type with the non-singular kernel of the Rabotnov fractional-exponential function for the first time. A new general fractional-order derivataive heat transfer model is discussed in detail. The general fractional-order derivataive formula is a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

scholarly journals The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the V -Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
S. Chandak ◽  
Biniyam Shimelis ◽  
Nigussie Abeye ◽  
A. Padma
Keyword(s):  
Fractional Calculus ◽  
Bessel Function ◽  
Hypergeometric Function ◽  
Exponential Function ◽  
Special Functions ◽  
Generalized Hypergeometric Function ◽  
Generalized Bessel Function ◽  
Fractional Calculus Operators

In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with V -function as the kernel. In addition, on account of V -function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.

The Number of Real Zeros of the Single Parameter Mittag-Leffler Function for Parameter Values Between 1 and 2

2005 ◽  
Author(s):  
John W. Hanneken ◽  
David M. Vaught ◽  
B. N. Narahari Achar
Keyword(s):  
Fractional Calculus ◽  
Finite Number ◽  
Exponential Function ◽  
Significant Role ◽  
Single Parameter ◽  
Real Zeros ◽  
Iteration Formula ◽  
Physical Problems ◽  
Number Of Real Zeros ◽  
Parameter Values

The single parameter Mittag-Leffler function, which is a generalization of the exponential function, occurs naturally in the solution of physical problems involving fractional calculus and its zeros play a significant role in the dynamic solutions. It is known that the Mittag-Leffler function has a finite number of real zeros in the range of parameter values between 1 and 2, which is quite relevant for many physical problems. However, the number of real zeros for a given parameter value in this range is not known. An iteration formula for calculating the number of real zeros of the Mittag-Leffler function for any value of the parameter in the range between 1 and 2 is presented.

scholarly journals FRACTIONAL CALCULUS OF THERMOELASTIC p-WAVES REFLECTION UNDER INFLUENCE OF GRAVITY AND ELECTROMAGNETIC FIELDS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040037 ◽  
Author(s):  
HAMMAD ALOTAIBI ◽  
S. M. ABO-DAHAB ◽  
H. R. ABDLRAHIM ◽  
A. A. KILANY
Keyword(s):  
Fractional Calculus ◽  
Function Method ◽  
Elastic Half Space ◽  
Harmonic Waves ◽  
P Waves ◽  
Reflected Waves ◽  
Gravity Fields ◽  
Fractional Exponential Function ◽  
Exponential Function Method ◽  
Incident Waves

In this paper, we discussed the longitudinal harmonic waves reflection from a solid elastic half-space with electromagnetic and gravity fields influence, considering a fractional order via fractional exponential function method. The clarifications are required for the reflection amplitudes ratios (i.e. the ratios between the reflected waves amplitude and the incident waves amplitude). The results obtained were calculated analytically and displayed by graphs to show the physical meaning of the phenomenon. A comparison has been made between the fractional and integer derivatives. The results of this paper demonstrate the rigor and effectiveness of the considered fractional technique.

scholarly journals Composition Formulas of Bessel-Struve Kernel Function

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
K. S. Nisar ◽  
S. R. Mondal ◽  
P. Agarwal
Keyword(s):  
Fractional Calculus ◽  
Kernel Function ◽  
Exponential Function ◽  
Integral Representations ◽  
Generalized Fractional Calculus ◽  
Fractional Calculus Operators ◽  
Generalized Fractional Calculus Operators

The object of this paper is to study and develop the generalized fractional calculus operators involving Appell’s functionF3(·)due to Marichev-Saigo-Maeda. Here, we establish the generalized fractional calculus formulas involving Bessel-Struve kernel functionSαλz,  λ,z∈Cto obtain the results in terms of generalized Wright functions. The representations of Bessel-Struve kernel function in terms of exponential function and its relation with Bessel and Struve function are also discussed. The pathway integral representations of Bessel-Struve kernel function are also given in this study.

Sign in / Sign up

Export Citation Format

Share Document