Fractal and Fractional

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3 pages, 193 KiB

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Editorial for Special Issue “Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications”

by Omar Bazighifan

Fractal Fract. 2022, 6(9), 503; https://doi.org/10.3390/fractalfract6090503 - 8 Sep 2022

Viewed by 1224

Abstract

Differential equations, both fractional and ordinary, give key tools in understanding the mechanisms of physical systems and solving various problems of nonlinear phenomena [...] Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

Research

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15 pages, 366 KiB

Open AccessArticle

Multivalent Functions and Differential Operator Extended by the Quantum Calculus

by Samir B. Hadid, Rabha W. Ibrahim and Shaher Momani

Fractal Fract. 2022, 6(7), 354; https://doi.org/10.3390/fractalfract6070354 - 24 Jun 2022

Cited by 10 | Viewed by 1704

Abstract

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of [...] Read more.

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of the proposed functions are geometrically investigated. Our approach was influenced by the theory of differential subordination. As a result, we called attention to a few well-known corollaries of our main conclusions. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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11 pages, 367 KiB

Open AccessArticle

Stationary Response of a Kind of Nonlinear Stochastic Systems with Variable Mass and Fractional Derivative Damping

by Shuo Zhang, Lu Liu and Chunhua Wang

Fractal Fract. 2022, 6(6), 342; https://doi.org/10.3390/fractalfract6060342 - 20 Jun 2022

Cited by 2 | Viewed by 1523

Abstract

Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelasticity and variable mass, stationary response of a kind of [...] Read more.

Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelasticity and variable mass, stationary response of a kind of nonlinear stochastic systems with stochastic variable-mass and fractional derivative, damping is investigated in this paper. Firstly, an approximately equivalent system of the studied nonlinear stochastic system is presented according to the Taylor expansion technique. Then, based on stochastic averaging of energy envelope, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced, which gives an approximated analytical solution of stationary response. Finally, a nonlinear oscillator with variable mass and fractional derivative damping is proposed in numerical simulations. The approximated analytical solution is compared with Monte Carlo numerical solution, which could verify the effectiveness of the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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11 pages, 402 KiB

Open AccessArticle

A New Approach to Compare the Strong Convergence of the Milstein Scheme with the Approximate Coupling Method

by Yousef Alnafisah

Fractal Fract. 2022, 6(6), 339; https://doi.org/10.3390/fractalfract6060339 - 17 Jun 2022

Cited by 4 | Viewed by 1800

Abstract

Milstein and approximate coupling approaches are compared for the pathwise numerical solutions to stochastic differential equations (SDE) driven by Brownian motion. These methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the approximate coupling method. We use [...] Read more.

Milstein and approximate coupling approaches are compared for the pathwise numerical solutions to stochastic differential equations (SDE) driven by Brownian motion. These methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the approximate coupling method. We use MATLAB to simulate these methods by applying them to a particular two-dimensional SDE. Then, we analyze the performance of both methods and the amount of time required to obtain the result. This comparison is essential in several areas, such as stochastic analysis, financial mathematics, and some biological applications. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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14 pages, 332 KiB

Open AccessArticle

Contribution of Using Hadamard Fractional Integral Operator via Mellin Integral Transform for Solving Certain Fractional Kinetic Matrix Equations

by Mohamed Abdalla and Mohamed Akel

Fractal Fract. 2022, 6(6), 305; https://doi.org/10.3390/fractalfract6060305 - 31 May 2022

Cited by 15 | Viewed by 1747

Abstract

Recently, the importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in electrodynamics, control systems, economic, physics, geophysics and hydrodynamics. Among the many fractional differential equations are kinetic equations. Fractional-order kinetic [...] Read more.

Recently, the importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in electrodynamics, control systems, economic, physics, geophysics and hydrodynamics. Among the many fractional differential equations are kinetic equations. Fractional-order kinetic Equations (FOKEs) are a unifying tool for the description of load vector behavior in disorderly media. In this article, we employ the Hadamard fractional integral operator via Mellin integral transform to establish the generalization of some fractional-order kinetic equations including extended

(k, τ)

-Gauss hypergeometric matrix functions. Solutions to certain fractional-order kinetic matrix Equations (FOKMEs) involving extended

(k, τ)

-Gauss hypergeometric matrix functions are also introduced. Moreover, several special cases of our main results are archived. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

28 pages, 2516 KiB

Open AccessArticle

Numerical Solutions of Variable-Coefficient Fractional-in-Space KdV Equation with the Caputo Fractional Derivative

by Che Han and Yu-Lan Wang

Fractal Fract. 2022, 6(4), 207; https://doi.org/10.3390/fractalfract6040207 - 7 Apr 2022

Cited by 16 | Viewed by 2337

Abstract

In this paper, numerical solutions of the variable-coefficient Korteweg-De Vries (vcKdV) equation with space described by the Caputo fractional derivative operator is developed. The propagation and interaction of vcKdV equation in different cases, such as breather soliton and periodic suppression soliton, are numerically [...] Read more.

In this paper, numerical solutions of the variable-coefficient Korteweg-De Vries (vcKdV) equation with space described by the Caputo fractional derivative operator is developed. The propagation and interaction of vcKdV equation in different cases, such as breather soliton and periodic suppression soliton, are numerically simulated. Especially, the Fourier spectral method is used to solve the fractional-in-space vcKdV equation with breather soliton. From numerical simulations and compared with other methods, it can be easily seen that our method has low computational complexity and higher precision. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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17 pages, 376 KiB

Open AccessArticle

Inequalities for Fractional Integrals of a Generalized Class of Strongly Convex Functions

by Tao Yan, Ghulam Farid, Hafsa Yasmeen, Soo Hak Shim and Chahn Yong Jung

Fractal Fract. 2022, 6(3), 168; https://doi.org/10.3390/fractalfract6030168 - 18 Mar 2022

Cited by 1 | Viewed by 1836

Abstract

Fractional integral operators are useful tools for generalizing classical integral inequalities. Convex functions play very important role in the theory of mathematical inequalities. This paper aims to investigate the Hadamard type inequalities for a generalized class of functions namely strongly [...] Read more.

Fractional integral operators are useful tools for generalizing classical integral inequalities. Convex functions play very important role in the theory of mathematical inequalities. This paper aims to investigate the Hadamard type inequalities for a generalized class of functions namely strongly

(α, h - m)

-p-convex functions by using Riemann–Liouville fractional integrals. The results established in this paper give refinements of various well-known inequalities which have been published in the recent past. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

16 pages, 346 KiB

Open AccessArticle

Fractional p(·)-Kirchhoff Type Problems Involving Variable Exponent Logarithmic Nonlinearity

by Jiabin Zuo, Amita Soni and Debajyoti Choudhuri

Fractal Fract. 2022, 6(2), 106; https://doi.org/10.3390/fractalfract6020106 - 12 Feb 2022

Cited by 3 | Viewed by 1729

Abstract

In this paper, we investigate a fractional

p (\cdot)

-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main [...] Read more.

In this paper, we investigate a fractional

p (\cdot)

-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main aspect and challenges of this paper are the presence of double non-local terms and logarithmic nonlinearity. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

11 pages, 303 KiB

Open AccessArticle

Asymptotic Behavior of Solutions of Even-Order Differential Equations with Several Delays

by Osama Moaaz and Wedad Albalawi

Fractal Fract. 2022, 6(2), 87; https://doi.org/10.3390/fractalfract6020087 - 3 Feb 2022

Cited by 3 | Viewed by 1367

Abstract

The higher-order delay differential equations are used in the describing of many natural phenomena. This work investigates the asymptotic properties of the class of even-order differential equations with several delays. Our main concern revolves around how to simplify and improve the oscillation parameters [...] Read more.

The higher-order delay differential equations are used in the describing of many natural phenomena. This work investigates the asymptotic properties of the class of even-order differential equations with several delays. Our main concern revolves around how to simplify and improve the oscillation parameters of the studied equation. For this, we use an improved approach to obtain new properties of the positive solutions of these equations. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

22 pages, 366 KiB

Open AccessArticle

Some Novel Fractional Integral Inequalities over a New Class of Generalized Convex Function

by Soubhagya Kumar Sahoo, Muhammad Tariq, Hijaz Ahmad, Bibhakar Kodamasingh, Asif Ali Shaikh, Thongchai Botmart and Mohammed A. El-Shorbagy

Fractal Fract. 2022, 6(1), 42; https://doi.org/10.3390/fractalfract6010042 - 13 Jan 2022

Cited by 23 | Viewed by 2599

Abstract

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with [...] Read more.

The comprehension of inequalities in convexity is very important for fractional calculus and its effectiveness in many applied sciences. In this article, we handle a novel investigation that depends on the Hermite–Hadamard-type inequalities concerning a monotonic increasing function. The proposed methodology deals with a new class of convexity and related integral and fractional inequalities. There exists a solid connection between fractional operators and convexity because of its fascinating nature in the numerical sciences. Some special cases have also been discussed, and several already-known inequalities have been recaptured to behave well. Some applications related to special means, q-digamma, modified Bessel functions, and matrices are discussed as well. The aftereffects of the plan show that the methodology can be applied directly and is computationally easy to understand and exact. We believe our findings generalise some well-known results in the literature on s-convexity. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

14 pages, 1358 KiB

Open AccessArticle

An Implicit Numerical Approach for 2D Rayleigh Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

by Anam Naz, Umair Ali, Ashraf Elfasakhany, Khadiga Ahmed Ismail, Abdullah G. Al-Sehemi and Ahmed A. Al-Ghamdi

Fractal Fract. 2021, 5(4), 283; https://doi.org/10.3390/fractalfract5040283 - 20 Dec 2021

Cited by 5 | Viewed by 2553

Abstract

In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme [...] Read more.

In this research work, our aim is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) involving Riemann–Liouville time fractional derivative. We suggest the modified implicit scheme formulated in the Riemann–Liouville integral sense and the scheme can be applied to the fractional RSP-HGSGF. Numerical experiments will be conducted, to show that the scheme is stress-free to implement, and the outcomes reveal the ideal execution of the suggested technique. The Fourier series will be used to examine the proposed scheme stability and convergence. The technique is stable, and the approximation solution converges to the exact result. To demonstrate the applicability and viability of the suggested strategy, a numerical demonstration will be provided. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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19 pages, 330 KiB

Open AccessArticle

A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces

by Daliang Zhao

Fractal Fract. 2021, 5(4), 279; https://doi.org/10.3390/fractalfract5040279 - 17 Dec 2021

Cited by 7 | Viewed by 2078

Abstract

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear [...] Read more.

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

15 pages, 423 KiB

Open AccessArticle

Solvability of a New q-Differential Equation Related to q-Differential Inequality of a Special Type of Analytic Functions

by Ibtisam Aldawish and Rabha W. Ibrahim

Fractal Fract. 2021, 5(4), 228; https://doi.org/10.3390/fractalfract5040228 - 17 Nov 2021

Cited by 9 | Viewed by 1700

Abstract

The current study acts on the notion of quantum calculus together with a symmetric differential operator joining a special class of meromorphic multivalent functions in the puncher unit disk. We formulate a quantum symmetric differential operator and employ it to investigate the geometric [...] Read more.

The current study acts on the notion of quantum calculus together with a symmetric differential operator joining a special class of meromorphic multivalent functions in the puncher unit disk. We formulate a quantum symmetric differential operator and employ it to investigate the geometric properties of a class of meromorphic multivalent functions. We illustrate a set of differential inequalities based on the theory of subordination and superordination. In this real case study, we found the analytic solutions of q-differential equations. We indicate that the solutions are given in terms of confluent hypergeometric function of the second type and Laguerre polynomial. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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11 pages, 8705 KiB

Open AccessArticle

Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

by A. Othman Almatroud

Fractal Fract. 2021, 5(4), 202; https://doi.org/10.3390/fractalfract5040202 - 5 Nov 2021

Cited by 10 | Viewed by 1926

Abstract

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model [...] Read more.

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model are experimentally investigated via the maximum Lyapunov exponent, phase portraits, and bifurcation diagrams. Numerical simulation demonstrates that the new network has various types of coexisting attractors. Moreover, it is of note that the interesting phenomena of extreme multistability is discovered, i.e., the coexistence of symmetric multiple attractors. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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21 pages, 372 KiB

Open AccessArticle

Analytical Study of Two Nonlinear Coupled Hybrid Systems Involving Generalized Hilfer Fractional Operators

by Mohammed A. Almalahi, Omar Bazighifan, Satish K. Panchal, S. S. Askar and Georgia Irina Oros

Fractal Fract. 2021, 5(4), 178; https://doi.org/10.3390/fractalfract5040178 - 22 Oct 2021

Cited by 25 | Viewed by 1628

Abstract

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving

ϕ

-Hilfer fractional derivatives. The existence results are established [...] Read more.

In this research paper, we dedicate our interest to an investigation of the sufficient conditions for the existence of solutions of two new types of a coupled systems of hybrid fractional differential equations involving

ϕ

-Hilfer fractional derivatives. The existence results are established in the weighted space of functions using Dhage’s hybrid fixed point theorem for three operators in a Banach algebra and Dhage’s helpful generalization of Krasnoselskii fixed- point theorem. Finally, simulated examples are provided to demonstrate the obtained results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

14 pages, 2016 KiB

Open AccessArticle

An Efficient Stochastic Numerical Computing Framework for the Nonlinear Higher Order Singular Models

by Zulqurnain Sabir, Hafiz Abdul Wahab, Shumaila Javeed and Haci Mehmet Baskonus

Fractal Fract. 2021, 5(4), 176; https://doi.org/10.3390/fractalfract5040176 - 20 Oct 2021

Cited by 52 | Viewed by 1896

Abstract

The focus of the present study is to present a stochastic numerical computing framework based on Gudermannian neural networks (GNNs) together with the global and local search genetic algorithm (GA) and active-set approach (ASA), i.e., GNNs-GA-ASA. The designed computing framework GNNs-GA-ASA is tested [...] Read more.

The focus of the present study is to present a stochastic numerical computing framework based on Gudermannian neural networks (GNNs) together with the global and local search genetic algorithm (GA) and active-set approach (ASA), i.e., GNNs-GA-ASA. The designed computing framework GNNs-GA-ASA is tested for the higher order nonlinear singular differential model (HO-NSDM). Three different nonlinear singular variants based on the (HO-NSDM) have been solved by using the GNNs-GA-ASA and numerical solutions have been compared with the exact solutions to check the exactness of the designed scheme. The absolute errors have been performed to check the precision of the designed GNNs-GA-ASA scheme. Moreover, the aptitude of GNNs-GA-ASA is verified on precision, stability and convergence analysis, which are enhanced through efficiency, implication and dependability procedures with statistical data to solve the HO-NSDM. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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27 pages, 5257 KiB

Open AccessArticle

A Deep Learning BiLSTM Encoding-Decoding Model for COVID-19 Pandemic Spread Forecasting

by Ahmed I. Shahin and Sultan Almotairi

Fractal Fract. 2021, 5(4), 175; https://doi.org/10.3390/fractalfract5040175 - 19 Oct 2021

Cited by 23 | Viewed by 3351

Abstract

The COVID-19 pandemic has widely spread with an increasing infection rate through more than 200 countries. The governments of the world need to record the confirmed infectious, recovered, and death cases for the present state and predict the cases. In favor of future [...] Read more.

The COVID-19 pandemic has widely spread with an increasing infection rate through more than 200 countries. The governments of the world need to record the confirmed infectious, recovered, and death cases for the present state and predict the cases. In favor of future case prediction, governments can impose opening and closing procedures to save human lives by slowing down the pandemic progression spread. There are several forecasting models for pandemic time series based on statistical processing and machine learning algorithms. Deep learning has been proven as an excellent tool for time series forecasting problems. This paper proposes a deep learning time-series prediction model to forecast the confirmed, recovered, and death cases. Our proposed network is based on an encoding–decoding deep learning network. Moreover, we optimize the selection of our proposed network hyper-parameters. Our proposed forecasting model was applied in Saudi Arabia. Then, we applied the proposed model to other countries. Our study covers two categories of countries that have witnessed different spread waves this year. During our experiments, we compared our proposed model and the other time-series forecasting models, which totaled fifteen prediction models: three statistical models, three deep learning models, seven machine learning models, and one prophet model. Our proposed forecasting model accuracy was assessed using several statistical evaluation criteria. It achieved the lowest error values and achieved the highest R-squared value of 0.99. Our proposed model may help policymakers to improve the pandemic spread control, and our method can be generalized for other time series forecasting tasks. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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13 pages, 311 KiB

Open AccessArticle

The Solutions of Some Riemann–Liouville Fractional Integral Equations

by Karuna Kaewnimit, Fongchan Wannalookkhee, Kamsing Nonlaopon and Somsak Orankitjaroen

Fractal Fract. 2021, 5(4), 154; https://doi.org/10.3390/fractalfract5040154 - 6 Oct 2021

Cited by 10 | Viewed by 2970

Abstract

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form [...] Read more.

In this paper, we propose the solutions of nonhomogeneous fractional integral equations of the form

I_{0^{+}}^{3 σ} y (t) + a \cdot I_{0^{+}}^{2 σ} y (t) + b \cdot I_{0^{+}}^{σ} y (t) + c \cdot y (t) = f (t),

where

I_{0^{+}}^{σ}

is the Riemann–Liouville fractional integral of order

σ = 1 / 3, 1, f (t) = t^{n}, t^{n} e^{t}, n \in N \cup {0}, t \in R^{+},

and

a, b, c

are constants, by using the Laplace transform technique. We obtain solutions in the form of Mellin–Ross function and of exponential function. To illustrate our findings, some examples are exhibited. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations, Delay Differential Equations and Their Applications)

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