scholarly journals On Existence of Sequences of Weak Solutions of Fractional Systems with Lipschitz Nonlinearity

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Rafik Guefaifia ◽  
Salah Mahmoud Boulaaras ◽  
Adel Abd Elaziz El-Sayed ◽  
Mohamed Abdalla ◽  
Bahri-Belkacem Cherif
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Fractional Order ◽  
Weak Solutions ◽  
Infinitely Many Solutions ◽  
Control Parameters ◽  
Fractional Systems ◽  
New Class ◽  
Lipschitz Nonlinearity

In this article, the variational method together with two control parameters is used for introducing the proof for the existence of infinitely many solutions for a new class of perturbed nonlinear system having p -Laplacian fractional-order differentiation.

scholarly journals Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Rafik Guefaifia ◽  
Salah Mahmoud Boulaaras ◽  
Bahri Cherif ◽  
Taha Radwan
Keyword(s):  
Variational Method ◽  
Control Parameter ◽  
Infinitely Many Solutions ◽  
Differential Systems ◽  
Fractional Systems ◽  
Lipschitz Nonlinearity

The paper deals with the existence of infinitely many solutions of a class of perturbed nonlinear fractional p -Laplacian differential systems using one control parameter combined with the variational method.

scholarly journals Multiplicity of solutions for perturbed nonlinear fractional p-Laplacian boundary value systems related with two control parameters

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2827-2848
Author(s):  
Jiabin Zuo ◽  
Rafik Guefaifia ◽  
Fares Kamache ◽  
Salah Boulaaras
Keyword(s):  
Variational Method ◽  
Critical Points ◽  
Weak Solutions ◽  
Boundary Value ◽  
Value Systems ◽  
Control Parameters ◽  
Differential Systems ◽  
Multiplicity Of Solutions ◽  
Three Critical Points Theorem ◽  
Perturbed Systems

This paper deals with the study of a class of perturbed nonlinear fractional p-Laplacian differential systems, where by using the variational method, two control parameters together with recent three critical points theorem by Bonanno and Candito for differentiable functionals for perturbed systems, the existence of three weak solutions has been proved.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

Analysis of a Nonlinear System with a Random Parameter

2014 ◽  
Vol 721 ◽  
pp. 366-369
Author(s):  
Hong Gang Dang ◽  
Xiao Ya Yang ◽  
Wan Sheng He
Keyword(s):  
Nonlinear System ◽  
Fractional Order ◽  
Approximation Method ◽  
Lorenz System ◽  
Laguerre Polynomial ◽  
Random Parameter ◽  
Order System ◽  
Order Complex ◽  
Ensemble Mean ◽  
Complex Lorenz System

In this paper, a nonlinear system with random parameter, which is called stochastic fractional-order complex Lorenz system, is investigated. The Laguerre polynomial approximation method is used to study the system. Then, the stochastic fractional-order system is reduced into the equivalent deterministic one with Laguerre approximation. The ensemble mean and sample responses of the stochastic system can be obtained.

Vibrational Resonance in an Overdamped System with a Fractional Order Potential Nonlinearity

2018 ◽  
Vol 28 (07) ◽  
pp. 1850082 ◽  
Author(s):  
Jianhua Yang ◽  
Dawen Huang ◽  
Miguel A. F. Sanjuán ◽  
Houguang Liu
Keyword(s):  
Numerical Simulation ◽  
Nonlinear System ◽  
Theoretical Analysis ◽  
Fractional Order ◽  
Low Frequency ◽  
Response Amplitude ◽  
Resonance Phenomenon ◽  
Vibrational Resonance ◽  
Frequency Excitation ◽  
Overdamped System

We investigate the vibrational resonance by the numerical simulation and theoretical analysis in an overdamped system with fractional order potential nonlinearities. The nonlinearity is a fractional power function with deflection, in which the response amplitude presents vibrational resonance phenomenon for any value of the fractional exponent. The response amplitude of vibrational resonance at low-frequency is deduced by the method of direct separation of slow and fast motions. The results derived from the theoretical analysis are in good agreement with those of numerical simulation. The response amplitude decreases with the increase of the fractional exponent for weak excitations. The amplitude of the high-frequency excitation can induce the vibrational resonance to achieve the optimal response amplitude. For the overdamped systems, the nonlinearity is the crucial and necessary condition to induce vibrational resonance. The response amplitude in the nonlinear system is usually not larger than that in the corresponding linear system. Hence, the nonlinearity is not a sufficient factor to amplify the response to the low-frequency excitation. Furthermore, the resonance may be also induced by only a single excitation acting on the nonlinear system. The theoretical analysis further proves the correctness of the numerical simulation. The results might be valuable in weak signal processing.

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850155 ◽  
Author(s):  
Chengwei Dong
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Building Blocks ◽  
Topological Classification ◽  
One Dimensional ◽  
Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

Adaptive Fractional Order Sliding Mode Control for a Nonlinear System

2021 ◽  
Author(s):  
Esraa Mostafa ◽  
Ahmad M. El-Nagar ◽  
Osama Elshazly ◽  
Mohammad El-Bardini
Keyword(s):  
Nonlinear System ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Mode Control
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