scholarly journals Numerical algorithm based on fast convolution for fractional calculus

2012 ◽  
Vol 16 (2) ◽  
pp. 365-371
Author(s):  
An Chen ◽  
Peng Guo ◽  
Changpin Li
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Numerical Algorithm ◽  
Fractional Integral ◽  
Numerical Algorithms ◽  
Fast Convolution

In this paper, numerical algorithms based on fast convolution for the fractional integral and fractional derivative are proposed. Two examples are also included which show the efficiency of the derived method.

scholarly journals Some New Fractional-Calculus Connections between Mittag–Leffler Functions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 485 ◽  
Author(s):  
Hari M. Srivastava ◽  
Arran Fernandez ◽  
Dumitru Baleanu
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Integral Expression ◽  
Potential Applications ◽  
The One ◽  
Two Parameter

We consider the well-known Mittag–Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag–Leffler function as a fractional derivative of the two-parameter Mittag–Leffler function, which is in turn a fractional integral of the one-parameter Mittag–Leffler function. Hence, we derive an integral expression for the three-parameter one in terms of the one-parameter one. We discuss the importance and applications of all three Mittag–Leffler functions, with a view to potential applications of our results in making certain types of experimental data much easier to analyse.

scholarly journals On Opial-Type Inequalities for Superquadratic Functions and Applications in Fractional Calculus

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Waqas Nazeer ◽  
Ghulam Farid ◽  
Zabidin Salleh ◽  
Ayesha Bibi
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Mean Value ◽  
Mean Value Theorems ◽  
Superquadratic Functions

We have studied the Opial-type inequalities for superquadratic functions proved for arbitrary kernels. These are estimated by applying mean value theorems. Furthermore, by analyzing specific functions, the fractional integral and fractional derivative inequalities are obtained.

A Modified Fractional Calculus Approach to Model Hysteresis

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
Mohammed Rabius Sunny ◽  
Rakesh K. Kapania ◽  
Ronald D. Moffitt ◽  
Amitabh Mishra ◽  
Nakhiah Goulbourne
Keyword(s):  
Experimental Data ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Conductive Polymer ◽  
Polymer Nanocomposite ◽  
Hysteretic Behavior ◽  
Integral Model ◽  
Strain Variation ◽  
Integer Order

This paper describes the development of a fractional calculus approach to model the hysteretic behavior shown by the variation in electrical resistances with strain in conductive polymers. Experiments have been carried out on a conductive polymer nanocomposite sample to study its resistance-strain variation under strain varying with time in a triangular manner. A combined fractional derivative and integer order integral model and a fractional integral model (with two submodels) have been developed to simulate this behavior. The efficiency of these models has been discussed by comparing the results, obtained using these models, with the experimental data. It has been shown that by using only a few parameters, the hysteretic behavior of such materials can be modeled using the fractional calculus with some modifications.

scholarly journals Solution of Fractional Order Equations in the Domain of the Mellin Transform

2019 ◽  
pp. 138-142
Author(s):  
Sunday Emmanuel Fadugba
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Mellin Transform ◽  
Applied Mathematics ◽  
Fundamental Properties ◽  
Sequential Fractional Derivative ◽  
Liouville Fractional Derivative

This paper presents the Mellin transform for the solution of the fractional order equations. The Mellin transform approach occurs in many areas of applied mathematics and technology. The Mellin transform of fractional calculus of different flavours; namely the Riemann-Liouville fractional derivative, Riemann-Liouville fractional integral, Caputo fractional derivative and the Miller-Ross sequential fractional derivative were obtained. Three illustrative examples were considered to discuss the applications of the Mellin transform and its fundamental properties. The results show that the Mellin transform is a good analytical method for the solution of fractional order equations.

scholarly journals The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions

2021 ◽  
Vol 5 (4) ◽  
pp. 210
Author(s):  
Hari M. Srivastava ◽  
Eman S. A. AbuJarad ◽  
Fahd Jarad ◽  
Gautam Srivastava ◽  
Mohammed H. A. AbuJarad
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Bessel Function ◽  
Fractional Integral ◽  
Hadamard Product ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Special Cases ◽  
Fractional Calculus Operators ◽  
Fractional Derivative Operators

The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p,q)-extended Bessel function. The results are expressed as the Hadamard product of the (p,q)-extended Gauss hypergeometric function Fp,q and the Fox-Wright function rΨs(z). Some special cases of our main results are considered. Furthermore, the (p,q)-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the (p,q)-extended Bessel-Wright function is established.

scholarly journals Nabla Fractional Derivative and Fractional Integral on Time Scales

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 317
Author(s):  
Bikash Gogoi ◽  
Utpal Kumar Saha ◽  
Bipan Hazarika ◽  
Delfim F. M. Torres ◽  
Hijaz Ahmad
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Time Scales ◽  
Fractional Integral ◽  
Basic Properties

In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann–Liouville sense. We also introduce the nabla fractional derivative in Grünwald–Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed.

Hadamard’s inequality and its extensions for conformable fractional integrals of any order α>0

2018 ◽  
Vol 27 (2) ◽  
pp. 197-206
Author(s):  
ERHAN SET ◽  
◽  
AHMET OCAK AKDEMIR ◽  
I. MUMCU ◽  
◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Hadamard’S Inequality ◽  
Conformable Fractional Integral ◽  
Definition Of ◽  
Appl Math

Recently the authors Abdeljawad [Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66] and Khalil et al. [Khalil, R., Horani, M. Al., Yousef, A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70] introduced a new and simple well-behaved concept of fractional integral called conformable fractional integral. In this article, we establish Hermite-Hadamard’s inequalities for conformable fractional integral. We also give extensions of Hermite-Hadamard type inequalities for conformable fractional integrals.

A Modified Fractional Derivative Approach to Model Hysteresis

2007 ◽  
Author(s):  
M. Mohammed Rabius Sunny ◽  
Rakesh K. Kapania ◽  
Ronald D. Moffitt ◽  
Amitabh Mishra ◽  
Nakhiah Goulbourne
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Hysteretic Behavior ◽  
Experimental Result ◽  
Integral Model ◽  
Nano Composites ◽  
Strain Resistance ◽  
Integer Order ◽  
Resistance Variation

This paper describes the development of a fractional calculus approach to model the hysteretic behavior shown by the variation of resistance with strain in nano-composites (like MetalRubberOˆ). Experiments have been carried out on MetalRubberOˆ to study the strain-resistance variation of this material under strains varying in a triangular manner. Combined fractional derivative and integer order integral model and fractional integral model (with two sub-models) have been developed to model this behavior. Effieiency of these models has been discussed by comparison of their results with the experimental result.

FRACTIONAL CALCULUS OF COALESCENCE HIDDEN-VARIABLE FRACTAL INTERPOLATION FUNCTIONS

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750019 ◽  
Author(s):  
SRIJANANI ANURAG PRASAD
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Hidden Variable ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Free Parameters ◽  
Interpolation Functions

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order [Formula: see text] of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF.

ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050123
Author(s):  
YONG-SHUN LIANG
Keyword(s):  
Fractal Dimension ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Hölder Condition ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Liouville Fractional Derivative

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

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