scholarly journals Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yang Zhao ◽  
Dumitru Baleanu ◽  
Mihaela Cristina Baleanu ◽  
De-Fu Cheng ◽  
Xiao-Jun Yang
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fourier Transforms ◽  
Special Functions ◽  
Integral Transforms ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Special Integral ◽  
Fractional Differential

The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.

scholarly journals Ulam-Hyers Stability for Cauchy Fractional Differential Equation in the Unit Disk

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Unit Disk ◽  
Fractional Differential Equations ◽  
Fractional Operators ◽  
Owa Operators ◽  
Non Linear ◽  
Fractional Differential

We prove the Ulam-Hyers stability of Cauchy fractional differential equations in the unit disk for the linear and non-linear cases. The fractional operators are taken in sense of Srivastava-Owa operators.

scholarly journals Generalized proportional fractional integral Hermite–Hadamard’s inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tariq A. Aljaaidi ◽  
Deepak B. Pachpatte ◽  
Thabet Abdeljawad ◽  
Mohammed S. Abdo ◽  
Mohammed A. Almalahi ◽  
...  
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Approximation Theory ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Fractional Operators ◽  
Uniqueness Of Solutions ◽  
Special Cases ◽  
Fractional Differential

AbstractThe theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.

Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel

2018 ◽  
Vol 13 (1) ◽  
pp. 13 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Differential Equations ◽  
Fractional Operators ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Derivative Order

Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

Synchronized bioluminescence behavior of a set of fireflies involving fractional operators of Liouville–Caputo type

2018 ◽  
Vol 11 (03) ◽  
pp. 1850041 ◽  
Author(s):  
J. E. Escalante-Martínez ◽  
J. F. Gómez-Aguilar ◽  
C. Calderón-Ramón ◽  
A. Aguilar-Meléndez ◽  
P. Padilla-Longoria
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Biological Parameters ◽  
Fractional Operators ◽  
Alternative Representation ◽  
Fractional Differential

In this paper, a system of fractional differential equations that model the synchronized bioluminescence behavior of a set of fireflies put on two spatial arrangements is presented; the alternative representation of these equations contains fractional operators of Liouville–Caputo type. The objective of the model is to qualitatively recover synchronization and show that it is persistent. It is shown that the effort made by each firefly glow changes with respect to the number of male competitors and the distance between them. The conditions on biological parameters are interpreted.

scholarly journals On Generalized Hyers-Ulam Stability of Admissible Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Banach Spaces ◽  
Unit Disk ◽  
Fractional Differential Equations ◽  
Complex Banach Space ◽  
Ulam Stability ◽  
Fractional Operators ◽  
Fractional Differential ◽  
Admissible Functions

We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk:Dzβf(z)=G(f(z),Dzαf(z),zf'(z);z),0<α<1<β≤2, in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

scholarly journals Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2132
Author(s):  
Yuri Luchko
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Analytic Solutions ◽  
Fractional Integrals ◽  
Function Type ◽  
Fractional Differential ◽  
Derivatives Of

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

scholarly journals The need for the fractional operators

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
E. A. Abdel-Rehim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Time ◽  
Space Time ◽  
Fractional Operators ◽  
Partial Differential ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

AbstractIn this review paper, I focus on presenting the reasons of extending the partial differential equations to space-time fractional differential equations. I believe that extending any partial differential equations or any system of equations to fractional systems without giving concrete reasons has no sense. The experiments agrees with the theoretical studies on extending the first order-time derivative to time-fractional derivative. The simulations of some processes also agrees with the theory of continuous time random walks for extending the second-order space fractional derivative to the Riesz–Feller fractional operators. For this aim, I give a condense review the theory of Brownian motion, Langevin equations, diffusion processes and the continuous time random walk. Some partial differential equations that are successfully extended to space-time-fractional differential equations are also presented.

Some fractional differential equations involving generalized hypergeometric functions

2019 ◽  
Vol 25 (1) ◽  
pp. 37-44 ◽  
Author(s):  
Praveen Agarwal ◽  
Feng Qi ◽  
Mehar Chand ◽  
Gurmej Singh
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Hypergeometric Functions ◽  
Integral Transforms ◽  
Fractional Operator ◽  
Generalized Hypergeometric Functions ◽  
Fractional Differential

Abstract In the paper, using the generalized Marichev–Saigo–Maeda fractional operator, the authors establish some fractional differential equations associated with generalized hypergeometric functions and, by employing integral transforms, present some image formulas of the resulting equations.

Initialization of Fractional Differential Equations: Background and Theory

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Higher Order Derivative ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. This paper provides background on past work in the area and determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. A companion paper in this conference extends the theory to higher order derivative operators and provides application insight.

scholarly journals Existence of Iterative Cauchy Fractional Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Cauchy Type ◽  
Fractional Operators ◽  
Nonexpansive Operators ◽  
Fractional Differential

Our main aim in this paper is to use the technique of nonexpansive operators in more general iterative and noniterative fractional differential equations (Cauchy type). The noninteger case is taken in sense of the Riemann-Liouville fractional operators. Applications are illustrated.

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