Research

15 pages, 835 KiB

Open AccessArticle

A Novel Implementation of Dhage’s Fixed Point Theorem to Nonlinear Sequential Hybrid Fractional Differential Equation

by Muath Awadalla, Mohamed Hannabou, Kinda Abuasbeh and Khalid Hilal

Fractal Fract. 2023, 7(2), 144; https://doi.org/10.3390/fractalfract7020144 - 02 Feb 2023

Cited by 4 | Viewed by 753

In this work, the existence and uniqueness of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions were investigated by the generalization of Dhage’s fixed point theorem and Banach contraction mapping, respectively. In addition, the U-H technique is employed to [...] Read more.

In this work, the existence and uniqueness of solutions to a sequential fractional (Hybrid) differential equation with hybrid boundary conditions were investigated by the generalization of Dhage’s fixed point theorem and Banach contraction mapping, respectively. In addition, the U-H technique is employed to verify the stability of this solution. This study ends with two examples illustrating the theoretical findings. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

11 pages, 283 KiB

Open AccessArticle

The Stability of Set-Valued Differential Equations with Different Initial Time in the Sense of Fractional-like Hukuhara Derivatives

by Peiguang Wang and Jiahui Bi

Fractal Fract. 2023, 7(1), 20; https://doi.org/10.3390/fractalfract7010020 - 25 Dec 2022

Cited by 1 | Viewed by 1013

Abstract

This paper investigates set-valued differential equations with fractional-like Hukuhara derivatives. Firstly, a novel comparison principle is given by introducing the upper quasi-monotone increasing functions. Then, the stability criteria of Lipschitz stability and practical stability of such equations with different initial time are obtained [...] Read more.

This paper investigates set-valued differential equations with fractional-like Hukuhara derivatives. Firstly, a novel comparison principle is given by introducing the upper quasi-monotone increasing functions. Then, the stability criteria of Lipschitz stability and practical stability of such equations with different initial time are obtained via the new comparison principle and vector Lyapunov functions. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

16 pages, 336 KiB

Open AccessArticle

Existence and Ulam Type Stability for Impulsive Fractional Differential Systems with Pure Delay

by Chaowen Chen and Mengmeng Li

Fractal Fract. 2022, 6(12), 742; https://doi.org/10.3390/fractalfract6120742 - 15 Dec 2022

Cited by 3 | Viewed by 1104

Abstract

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of [...] Read more.

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

22 pages, 947 KiB

Open AccessArticle

Stability of a Nonlinear Langevin System of ML-Type Fractional Derivative Affected by Time-Varying Delays and Differential Feedback Control

by Kaihong Zhao

Fractal Fract. 2022, 6(12), 725; https://doi.org/10.3390/fractalfract6120725 - 08 Dec 2022

Cited by 27 | Viewed by 1476

Abstract

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or [...] Read more.

The Langevin system is an important mathematical model to describe Brownian motion. The research shows that fractional differential equations have more advantages in viscoelasticity. The exploration of fractional Langevin system dynamics is novel and valuable. Compared with the fractional system of Caputo or Riemann–Liouville (RL) derivatives, the system with Mittag–Leffler (ML)-type fractional derivatives can eliminate singularity such that the solution of the system has better analytical properties. Therefore, we concentrate on a nonlinear Langevin system of ML-type fractional derivatives affected by time-varying delays and differential feedback control in the manuscript. We first utilize two fixed-point theorems proposed by Krasnoselskii and Schauder to investigate the existence of a solution. Next, we employ the contraction mapping principle and nonlinear analysis to establish the stability of types such as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) as well as generalized UH and UHR. Lastly, the theoretical analysis and numerical simulation of some interesting examples are carried out by using our main results and the DDESD toolbox of MATLAB. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

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20 pages, 6139 KiB

Open AccessArticle

Magnetic Field, Variable Thermal Conductivity, Thermal Radiation, and Viscous Dissipation Effect on Heat and Momentum of Fractional Oldroyd-B Bio Nano-Fluid within a Channel

by Muhammad Madssar Kaleem, Muhammad Usman, Muhammad Imran Asjad and Sayed M. Eldin

Fractal Fract. 2022, 6(12), 712; https://doi.org/10.3390/fractalfract6120712 - 30 Nov 2022

Cited by 5 | Viewed by 1344

Abstract

This study deals with the analysis of the heat and velocity profile of the fractional-order Oldroyd-B bio-nanofluid within a bounded channel. The study has a wide range of scope in modern fields of basic science such as medicine, the food industry, electrical appliances, [...] Read more.

This study deals with the analysis of the heat and velocity profile of the fractional-order Oldroyd-B bio-nanofluid within a bounded channel. The study has a wide range of scope in modern fields of basic science such as medicine, the food industry, electrical appliances, nuclear as well as industrial cooling systems, reducing pollutants, fluids used in the brake systems of vehicles, etc. Oldroyd-B fluid is taken as a bio-nanofluid composed of base fluid (blood) and copper as nanoparticles. Using the fractional-order Oldroyd-B parameter, the governing equation is generalized from an integer to a non-integer form. A strong approach, i.e., a finite difference scheme, is applied to discretize the model, because the fractional approach can well address the physical phenomena and memory effect of the flow regime. Therefore, a Caputo fractional differentiation operator is used for the purpose. The transformations for the channel flow are utilized to transfigure the fractional-order partial differential equations (PDEs) into non-dimension PDEs. The graphical outcomes for non-integer ordered Oldroyd-B bio-nanofluid dynamics and temperature profiles are navigated using the numerical technique. These results are obtained under some very important physical conditions applied as a magnetic field effect, variable thermal conductivity, permeable medium, and heat source/sink. The results show that the addition of (copper) nanoparticles to (blood) base fluids enhances the thermal conductivity. For a comparative study, the obtained results are compared with the built-in results using the mathematical software MAPLE 2016. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

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23 pages, 513 KiB

Open AccessArticle

Lyapunov Direct Method for Nonlinear Hadamard-Type Fractional Order Systems

by Changping Dai and Weiyuan Ma

Fractal Fract. 2022, 6(8), 405; https://doi.org/10.3390/fractalfract6080405 - 22 Jul 2022

Cited by 2 | Viewed by 1356

Abstract

In this paper, a rigorous Lyapunov direct method (LDM) is proposed to analyze the stability of fractional non-linear systems involving Hadamard or Caputo–Hadamard derivatives. Based on the characteristics of Hadamard-type calculus, several new inequalities are derived for different definitions. By means of the [...] Read more.

In this paper, a rigorous Lyapunov direct method (LDM) is proposed to analyze the stability of fractional non-linear systems involving Hadamard or Caputo–Hadamard derivatives. Based on the characteristics of Hadamard-type calculus, several new inequalities are derived for different definitions. By means of the developed inequalities and modified Laplace transform, the sufficient conditions can be derived to guarantee the Hadamard–Mittag–Leffler (HML) stability of the systems. Lastly, two illustrative examples are given to show the effectiveness of our proposed results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

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16 pages, 357 KiB

Open AccessArticle

On the Finite-Time Boundedness and Finite-Time Stability of Caputo-Type Fractional Order Neural Networks with Time Delay and Uncertain Terms

by Bandana Priya, Ganesh Kumar Thakur, M. Syed Ali, Gani Stamov, Ivanka Stamova and Pawan Kumar Sharma

Fractal Fract. 2022, 6(7), 368; https://doi.org/10.3390/fractalfract6070368 - 30 Jun 2022

Cited by 6 | Viewed by 1345

Abstract

This study investigates the problem of finite-time boundedness of a class of neural networks of Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to ensure that the addressed fractional-order uncertain neural networks [...] Read more.

This study investigates the problem of finite-time boundedness of a class of neural networks of Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to ensure that the addressed fractional-order uncertain neural networks are finite-time stable. Criteria for finite-time boundedness of the considered fractional-order uncertain models are also achieved. The obtained results are based on a newly developed property of Caputo fractional derivatives, properties of Mittag–Leffler functions and Laplace transforms. In addition, examples are developed to manifest the usefulness of our theoretical results. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

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17 pages, 10399 KiB

Open AccessArticle

Thermophysical Study of Oldroyd-B Hybrid Nanofluid with Sinusoidal Conditions and Permeability: A Prabhakar Fractional Approach

by Juan Zhang, Ali Raza, Umair Khan, Qasim Ali, Aurang Zaib, Wajaree Weera and Ahmed M. Galal

Fractal Fract. 2022, 6(7), 357; https://doi.org/10.3390/fractalfract6070357 - 26 Jun 2022

Cited by 16 | Viewed by 1320

Abstract

The functional implications of substances, such as retardation and relaxation, can be studied for magnetized diffusion coefficient based on the relative increase throughout magnetization is a well-known realization. In this context, we have explored the Oldroyd-B hybrid nanofluid flowing through a pored oscillating [...] Read more.

The functional implications of substances, such as retardation and relaxation, can be studied for magnetized diffusion coefficient based on the relative increase throughout magnetization is a well-known realization. In this context, we have explored the Oldroyd-B hybrid nanofluid flowing through a pored oscillating plate along with an inclined applied magnetics effect. The slipping effect and sinusoidal heating conditions are also supposed to be under consideration. An innovative and current classification of fractional derivatives, i.e., Prabhakar fractional derivative and Laplace transform, are implemented for the result of transformed leading equations. The graphical representation is also described to understand the physical implementation of all effecting parameters. In order to justify and physically examine the considered problem, some limiting cases, the rate of heat and mass transfer, and friction factors are also analyzed. As a result, we have concluded that the thermal enhancement can be improved more progressively with the interaction of silver-water-based nanofluid suspension compared to copper-nanoparticles mixed nanofluid. Furthermore, It has examined the impact of both parameters, i.e., time relaxation

Ω_{1}

and retardation

Ω_{2}

is opposite of the momentum field. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Stability Analysis and Applications)

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