On a non-local boundary problem for a parabolic–hyperbolic equation involving a Riemann–Liouville fractional differential operator

2012 ◽  
Vol 75 (6) ◽  
pp. 3268-3273 ◽  
Author(s):  
A.S. Berdyshev ◽  
A. Cabada ◽  
E.T. Karimov
Keyword(s):  
Differential Operator ◽  
Hyperbolic Equation ◽  
Boundary Problem ◽  
Local Boundary ◽  
Fractional Differential Operator ◽  
Non Local ◽  
Fractional Differential

On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System

2020 ◽  
pp. 115-132
Author(s):  
Ahmed Ezzat Mohamed Matouk
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Differential Operators ◽  
Equilibrium Points ◽  
Hyperchaotic System ◽  
Fractional Differential Operator ◽  
Quadratic Nonlinearities ◽  
Potential Applications ◽  
Non Local ◽  
Fractional Differential

The non-local fractional differential operators have potential applications in many fields of science and technology but especially in the field of dynamical systems. This chapter introduces a new hyperchaotic dynamical system involving non-local fractional differential operator with singular kernel (the Caputo type). The system involves three quadratic nonlinearities and also three equilibrium points. Existence of chaotic and hyperchaotic attractors has been illustrated. Based on Matouk's stability theory of four-dimensional fractional-order systems, the influence of the fractional differential operator on stabilizing the proposed system to its three steady states has been shown. Numerical results have been provided to verify the theoretical analysis. This kind of study is expected to add useful applications to chaos-based secure communications and text encryption.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

scholarly journals On Caputo modification of Hadamard-type fractional derivative and fractional Taylor series

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Rashida Zafar ◽  
Mujeeb ur Rehman ◽  
Moniba Shams
Keyword(s):  
Differential Operator ◽  
Fractional Derivative ◽  
Taylor Series ◽  
General Framework ◽  
Taylor Expansion ◽  
Fractional Integral ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Fractional Differential Operator ◽  
Fractional Differential

Abstract In this paper a general framework is presented on some operational properties of Caputo modification of Hadamard-type fractional differential operator along with a new algorithm proposed for approximation of Hadamard-type fractional integral using Haar wavelet method. Moreover, a generalized Taylor expansion based on Caputo–Hadamard-type fractional differential operator is also established, and an example is presented for illustration.

scholarly journals On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 315
Author(s):  
Najla M. Alarifi ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Operator ◽  
Analytic Functions ◽  
Integral Operators ◽  
Upper And Lower Bounds ◽  
Fractional Operator ◽  
Convolution Operators ◽  
Linear Convolution ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Special Case

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

scholarly journals One-Phase Problems for Discontinuous Heat Transfer in Fractal Media

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Ming-Sheng Hu ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang
Keyword(s):  
Heat Transfer ◽  
Differential Operator ◽  
Transient Heat Transfer ◽  
Fractal Models ◽  
Fractal Media ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
The One

We first propose the fractal models for the one-phase problems of discontinuous transient heat transfer. The models are taken in sense of local fractional differential operator and used to describe the (dimensionless) melting of fractal solid semi-infinite materials initially at their melt temperatures.

Lie symmetries analysis and conservation laws for the fractional Calogero–Degasperis–Ibragimov–Shabat equation

2018 ◽  
Vol 15 (07) ◽  
pp. 1850110 ◽  
Author(s):  
S. Sahoo ◽  
S. Saha Ray
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Conservation Laws ◽  
Symmetry Analysis ◽  
Lie Symmetries ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Conservation Theorem

The present paper includes the study of symmetry analysis and conservation laws of the time-fractional Calogero–Degasperis–Ibragimov–Shabat (CDIS) equation. The Erdélyi–Kober fractional differential operator has been used here for reduction of time fractional CDIS equation into fractional ordinary differential equation. Also, the new conservation theorem has been used for the analysis of the conservation laws. Furthermore, the new conserved vectors have been constructed for time fractional CDIS equation by means of the new conservation theorem with formal Lagrangian.

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