Fractal and Fractional

15 pages, 339 KiB

Open AccessArticle

Riemann–Liouville Fractional Newton’s Type Inequalities for Differentiable Convex Functions

by Thanin Sitthiwirattham, Kamsing Nonlaopon, Muhammad Aamir Ali and Hüseyin Budak

Fractal Fract. 2022, 6(3), 175; https://doi.org/10.3390/fractalfract6030175 - 21 Mar 2022

Cited by 32 | Viewed by 2543

In this paper, we prove some new Newton’s type inequalities for differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation. It is also shown that the newly established [...] Read more.

In this paper, we prove some new Newton’s type inequalities for differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation. It is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. Finally, we give some examples with graphs and show the validity of newly established inequalities. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)

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19 pages, 1403 KiB

Open AccessArticle

The Dynamical Analysis of a Biparametric Family of Six-Order Ostrowski-Type Method under the Möbius Conjugacy Map

by Xiaofeng Wang and Xiaohe Chen

Fractal Fract. 2022, 6(3), 174; https://doi.org/10.3390/fractalfract6030174 - 21 Mar 2022

Cited by 5 | Viewed by 1752

Abstract

In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter [...] Read more.

In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter spaces related to the critical points and dynamic planes are used to visualize their dynamic properties. Eventually, we find the most stable member of the biparametric family of six-order Ostrowski-type methods. Some test equations are examined for supporting the theoretical results. Full article

(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)

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11 pages, 505 KiB

Open AccessArticle

New Exact Solutions of Some Important Nonlinear Fractional Partial Differential Equations with Beta Derivative

by Erdogan Mehmet Ozkan

Fractal Fract. 2022, 6(3), 173; https://doi.org/10.3390/fractalfract6030173 - 21 Mar 2022

Cited by 24 | Viewed by 2443

Abstract

In this work, the F-expansion method is used to find exact solutions of the space-time fractional modified Benjamin Bona Mahony equation and the nonlinear time fractional Schrödinger equation with beta derivative. One of the most efficient and significant methods for obtaining new exact [...] Read more.

In this work, the F-expansion method is used to find exact solutions of the space-time fractional modified Benjamin Bona Mahony equation and the nonlinear time fractional Schrödinger equation with beta derivative. One of the most efficient and significant methods for obtaining new exact solutions to nonlinear equations is this method. With the aid of Maple, more exact solutions defined by the Jacobi elliptic function are obtained. Hyperbolic function solutions and some exact solutions expressed by trigonometric functions are gained in the case of m modulus 1 and 0 limits of the Jacobi elliptic function. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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7 pages, 254 KiB

Open AccessArticle

On Geometric Properties of a Certain Analytic Function with Negative Coefficients

by Matthew Olanrewaju Oluwayemi, Esther O. Davids and Adriana Cătaş

Fractal Fract. 2022, 6(3), 172; https://doi.org/10.3390/fractalfract6030172 - 21 Mar 2022

Cited by 1 | Viewed by 1596

Abstract

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications of such functions have played significant roles in geometry function theory as a field of complex analysis. In this work, therefore, a certain subclass of univalent analytic functions [...] Read more.

Various function theorists have successfully defined and investigated different kinds of analytic functions. The applications of such functions have played significant roles in geometry function theory as a field of complex analysis. In this work, therefore, a certain subclass of univalent analytic functions of the form

f (z) = z - \sum_{m = 2}^{t} \frac{[ω (2 + β) + c γ - σ] C_{m}}{[m σ - c ω (2 + β) + c γ] K^{n}} z^{m} - \sum_{k = t + 1}^{\infty} a_{k} z^{k}

is defined using a generalized differential operator. Furthermore, some geometric properties for the class were established. Full article

(This article belongs to the Special Issue New Trends in Geometric Function Theory)

17 pages, 844 KiB

Open AccessArticle

New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator

by Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Muhammad Tariq and Y. S. Hamed

Fractal Fract. 2022, 6(3), 171; https://doi.org/10.3390/fractalfract6030171 - 19 Mar 2022

Cited by 32 | Viewed by 2637

Abstract

In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into [...] Read more.

In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard

(H - H)

type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)

20 pages, 5639 KiB

Open AccessArticle

Numerical Approximations for the Solutions of Fourth Order Time Fractional Evolution Problems Using a Novel Spline Technique

by Ghazala Akram, Muhammad Abbas, Hira Tariq, Maasoomah Sadaf, Thabet Abdeljawad and Manar A. Alqudah

Fractal Fract. 2022, 6(3), 170; https://doi.org/10.3390/fractalfract6030170 - 19 Mar 2022

Cited by 6 | Viewed by 2347

Abstract

Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline [...] Read more.

Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed. Full article

(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)

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11 pages, 40159 KiB

Open AccessArticle

Synchronization in a Multiplex Network of Nonidentical Fractional-Order Neurons

by Balamurali Ramakrishnan, Fatemeh Parastesh, Sajad Jafari, Karthikeyan Rajagopal, Gani Stamov and Ivanka Stamova

Fractal Fract. 2022, 6(3), 169; https://doi.org/10.3390/fractalfract6030169 - 18 Mar 2022

Cited by 11 | Viewed by 2327

Abstract

Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers [...] Read more.

Fractional-order neuronal models that include memory effects can describe the rich dynamics of the firing of the neurons. This paper studies synchronization problems in a multiple network of Caputo–Fabrizio type fractional order neurons in which the orders of the derivatives in the layers are different. It is observed that the intralayer synchronization state occurs in weaker intralayer couplings when using nonidentical fractional-order derivatives rather than integer-order or identical fractional orders. Furthermore, the needed interlayer coupling strength for interlayer near synchronization decreases for lower fractional orders. The dynamics of the neurons in nonidentical layers are also considered. It is shown that in lower fractional orders, the neurons’ dynamics change to periodic when the near synchronization state occurs. Moreover, decreasing the derivative order leads to incrementing the frequency of the bursts in the synchronization manifold, which is in contrast to the behavior of the single neuron. Full article

(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)

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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)

17 pages, 1035 KiB

Open AccessArticle

A Local Fractional Elzaki Transform Decomposition Method for the Nonlinear System of Local Fractional Partial Differential Equations

by Halil Anac

Fractal Fract. 2022, 6(3), 167; https://doi.org/10.3390/fractalfract6030167 - 18 Mar 2022

Cited by 7 | Viewed by 2530

Abstract

In this paper, the nonlinear system of local fractional partial differential equations is solved via local fractional Elzaki transform decomposition method. The local fractional Elzaki decomposition transform method combines a local fractional Elzaki transform and the Adomian decomposition method. Applications related to the [...] Read more.

In this paper, the nonlinear system of local fractional partial differential equations is solved via local fractional Elzaki transform decomposition method. The local fractional Elzaki decomposition transform method combines a local fractional Elzaki transform and the Adomian decomposition method. Applications related to the nonlinear system of local fractional partial differential equations are presented. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

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13 pages, 389 KiB

Open AccessArticle

New Explicit Solutions of the Extended Double (2+1)-Dimensional Sine-Gorden Equation and Its Time Fractional Form

by Gangwei Wang, Li Li, Qi Wang and Juan Geng

Fractal Fract. 2022, 6(3), 166; https://doi.org/10.3390/fractalfract6030166 - 17 Mar 2022

Cited by 1 | Viewed by 1900

Abstract

In this paper, the extended double (2+1)-dimensional sine-Gorden equation is studied. First of all, using the symmetry method, the corresponding vector fields, Lie algebra and infinitesimal generators are derived. Then, from infinitesimal generators, the symmetry reductions are presented. In addition, these reduced equations [...] Read more.

In this paper, the extended double (2+1)-dimensional sine-Gorden equation is studied. First of all, using the symmetry method, the corresponding vector fields, Lie algebra and infinitesimal generators are derived. Then, from infinitesimal generators, the symmetry reductions are presented. In addition, these reduced equations are converted into the corresponding partial differential equations, which including classical double (1+1)-dimensional sine-Gorden equation. Moreover, based on the Lie symmetry method again, these reduced equations are investigated. Meanwhile, based on traveling wave transformation, some explicit solutions of the extended double (2+1)-dimensional sine-Gorden equation are obtained. Consequently, a conservation law is derived via conservation law multiplier method. Finally, especially with the help of the fractional complex transform, some solutions of double time fractional (2+1)-dimensional sine-Gorden equation are also derived. These results might explain complex nonlinear phenomenon. Full article

(This article belongs to the Special Issue The Solutions of Partial Differential Equations and Recent Applications)

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15 pages, 3478 KiB

Open AccessArticle

Influence of the Fractal Distribution of Particle Size on the Critical State Characteristics of Calcareous Sand

by Xue Shen, Yang Shen, Junhong Xu and Hanlong Liu

Fractal Fract. 2022, 6(3), 165; https://doi.org/10.3390/fractalfract6030165 - 17 Mar 2022

Cited by 8 | Viewed by 2259

Abstract

To study the influence of the fractal distribution of particle size on the critical state characteristics of calcareous sand, a type of calcareous sand from a certain reef of the South China Sea was used in this study. For comparison, standard quartz sand [...] Read more.

To study the influence of the fractal distribution of particle size on the critical state characteristics of calcareous sand, a type of calcareous sand from a certain reef of the South China Sea was used in this study. For comparison, standard quartz sand was also used. A series of drained shear tests on the two sands were then conducted to investigate their critical state characteristics. It was demonstrated that the fractal dimension is suitable for characterizing the particle size distribution (PSD) of calcareous sand with different fine sand content. The critical state equation of sand proposed by Li and Wang (1998) is suitable for fitting the critical state line of calcareous sand. In the plane of deviatoric stress versus the effective confining pressure (q–p′ plane) and the plane of void ratio versus (p′/p_a)^α, the critical state lines of calcareous sand are always above those of quartz sand. The critical state lines of calcareous sand with different fractal dimensions in the q–p′ plane are unique. However, in the e–(p′/p_a)^α plane, the critical state lines appear to rotate anticlockwise as the fractal dimension increases. In addition, there is an “intersection” in the e–(p′/p_a)^α plane. Considering the influence of the fractal distribution of particle size, an expression for the critical state line of calcareous sand in the e–(p′/p_a)^α plane was proposed. The related constitutive model was also revised, where a complete set of model parameters suitable for modeling calcareous sand was provided. Full article

(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)

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16 pages, 461 KiB

Open AccessArticle

Robust H_∞ Control of Fractional-Order Switched Systems with Order 0 < α < 1 and Uncertainty

by Bingxin Li, Xiangfei Zhao, Yaowei Liu and Xin Zhao

Fractal Fract. 2022, 6(3), 164; https://doi.org/10.3390/fractalfract6030164 - 16 Mar 2022

Cited by 7 | Viewed by 2621

Abstract

In this paper, robust

H_{\infty}

control for fractional-order switched systems (FOSSs) with uncertainty is studied. Firstly, the fractional-order switching law for FOSSs is proposed. Then,

H_{\infty}

control for FOSSs is proven based on the switching law and linear matrix inequalities (LMIs). [...] Read more.

In this paper, robust

H_{\infty}

control for fractional-order switched systems (FOSSs) with uncertainty is studied. Firstly, the fractional-order switching law for FOSSs is proposed. Then,

H_{\infty}

control for FOSSs is proven based on the switching law and linear matrix inequalities (LMIs). Moreover,

H_{\infty}

control for FOSSs with a state feedback controller is extended. Furthermore, the LMI-based condition of robust

H_{\infty}

control for FOSSs with uncertainty is proven. Furthermore, the condition of robust

H_{\infty}

control is proposed to design the state feedback controller. Finally, four simulation examples verified the effectiveness of the proposed methods. Full article

(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)

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35 pages, 3431 KiB

Open AccessArticle

Application of the Fractional Riccati Equation for Mathematical Modeling of Dynamic Processes with Saturation and Memory Effect

by Dmitriy Tverdyi and Roman Parovik

Fractal Fract. 2022, 6(3), 163; https://doi.org/10.3390/fractalfract6030163 - 16 Mar 2022

Cited by 11 | Viewed by 2292

Abstract

In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed [...] Read more.

In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed model is applied to the description of solar activity (SA), namely the number of sunspots observed over the past 25 years. It is also used to describe data from Johns Hopkins University on coronavirus infection COVID-19, in particular data on the Russian Federation and the Republic of Uzbekistan. Finally, it is used to study issues related to seismic activity, in particular, the description of data on the volumetric activity of Radon (RVA). The Riccati equation used in the mathematical model was numerically solved by constructing an implicit finite difference scheme (IFDS) and its implementation by the modified Newton method (MNM). The calculated curves obtained in the study are compared with known experimental data. It is shown that if the model parameters are chosen appropriately, the model curves will give results that correlate well with real experimental data. Moreover, with other parameters of the model, it is possible to make some prediction about the possible course of the considered processes. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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16 pages, 757 KiB

Open AccessArticle

Fractional Order Mathematical Model of Serial Killing with Different Choices of Control Strategy

by Mati ur Rahman, Shabir Ahmad, Muhammad Arfan, Ali Akgül and Fahd Jarad

Fractal Fract. 2022, 6(3), 162; https://doi.org/10.3390/fractalfract6030162 - 16 Mar 2022

Cited by 17 | Viewed by 2441

Abstract

The current manuscript describes the dynamics of a fractional mathematical model of serial killing under the Mittag–Leffler kernel. Using the fixed point theory approach, we present a qualitative analysis of the problem and establish a result that ensures the existence of at least [...] Read more.

The current manuscript describes the dynamics of a fractional mathematical model of serial killing under the Mittag–Leffler kernel. Using the fixed point theory approach, we present a qualitative analysis of the problem and establish a result that ensures the existence of at least one solution. Ulam’s stability of the given model is presented by using nonlinear concepts. The iterative fractional-order Adams–Bashforth approach is being used to find the approximate solution. The suggested method is numerically simulated at various fractional orders. The simulation is carried out for various control strategies. Over time, all of the compartments demonstrate convergence and stability. Different fractional orders have produced an excellent comparison outcome, with low fractional orders achieving stability sooner. Full article

(This article belongs to the Special Issue Advances in Fractional and Fractal Boundary Value Problems in Applied Sciences)

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25 pages, 8703 KiB

Open AccessArticle

Dynamic Analysis and Bifurcation Study on Fractional-Order Tri-Neuron Neural Networks Incorporating Delays

by Peiluan Li, Jinling Yan, Changjin Xu and Youlin Shang

Fractal Fract. 2022, 6(3), 161; https://doi.org/10.3390/fractalfract6030161 - 15 Mar 2022

Cited by 10 | Viewed by 2040

Abstract

In this manuscript, we principally probe into a class of fractional-order tri-neuron neural networks incorporating delays. Making use of fixed point theorem, we prove the existence and uniqueness of solution to the fractional-order tri-neuron neural networks incorporating delays. By virtue of a suitable [...] Read more.

In this manuscript, we principally probe into a class of fractional-order tri-neuron neural networks incorporating delays. Making use of fixed point theorem, we prove the existence and uniqueness of solution to the fractional-order tri-neuron neural networks incorporating delays. By virtue of a suitable function, we prove the uniformly boundedness of the solution to the fractional-order tri-neuron neural networks incorporating delays. With the aid of the stability theory and bifurcation knowledge of fractional-order differential equation, a new delay-independent condition to guarantee the stability and creation of Hopf bifurcation of the fractional-order tri-neuron neural networks incorporating delays is established. Taking advantage of the mixed controller that contains state feedback and parameter perturbation, the stability region and the time of onset of Hopf bifurcation of the fractional-order trineuron neural networks incorporating delays are successfully controlled. Software simulation plots are displayed to illustrate the established key results. The obtained conclusions in this article have important theoretical significance in designing and controlling neural networks. Full article

(This article belongs to the Special Issue General Operators of Non-Integer Order in Modelling and Control of Dynamical Systems)

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13 pages, 1057 KiB

Open AccessArticle

A Chebyshev Collocation Approach to Solve Fractional Fisher–Kolmogorov–Petrovskii–Piskunov Equation with Nonlocal Condition

by Dapeng Zhou, Afshin Babaei, Seddigheh Banihashemi, Hossein Jafari, Jehad Alzabut and Seithuti P. Moshokoa

Fractal Fract. 2022, 6(3), 160; https://doi.org/10.3390/fractalfract6030160 - 15 Mar 2022

Cited by 3 | Viewed by 2213

Abstract

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. [...] Read more.

We provide a detailed description of a numerical approach that makes use of the shifted Chebyshev polynomials of the sixth kind to approximate the solution of some fractional order differential equations. Specifically, we choose the fractional Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) to describe this method. We write our approximate solution in the product form, which consists of unknown coefficients and shifted Chebyshev polynomials. To compute the numerical values of coefficients, we use the initial and boundary conditions and the collocation technique to create a system of equations whose number matches the unknowns. We test the applicability and accuracy of this numerical approach using two examples. Full article

(This article belongs to the Special Issue Importance of Fractional Order Derivatives in Real-World Applications: New Aspects and Understanding the Natural Phenomena)

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14 pages, 444 KiB

Open AccessArticle

Robust Control for Variable-Order Fractional Interval Systems Subject to Actuator Saturation

by Ri Liu, Zhe Wang, Xuefeng Zhang, Jianxu Ren and Qinglong Gui

Fractal Fract. 2022, 6(3), 159; https://doi.org/10.3390/fractalfract6030159 - 15 Mar 2022

Cited by 6 | Viewed by 1866

Abstract

In this paper, a class of variable-order fractional interval systems (VO-FIS) in which the system matrices are affected by the fractional order is investigated. Firstly, the sufficient conditions for robust stability of a VO-FIS with a unified order range of [...] Read more.

In this paper, a class of variable-order fractional interval systems (VO-FIS) in which the system matrices are affected by the fractional order is investigated. Firstly, the sufficient conditions for robust stability of a VO-FIS with a unified order range of

ν (σ) \in (0, 2)

are proposed. Secondly, the stabilization conditions of a VO-FIS subject to actuator saturation are derived in terms of linear matrix inequalities (LMIs). Then, by using the proposed algorithm through an optimization problem, the stability region is estimated. To summarize, the paper gives a stabilization criterion for VO-FIS subject to actuator saturation. Finally, three numerical examples are proposed to verify the effectiveness of our results. Full article

(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)

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13 pages, 333 KiB

Open AccessArticle

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

by Noureddine Djenina, Adel Ouannas, Taki-Eddine Oussaeif, Giuseppe Grassi, Iqbal M. Batiha, Shaher Momani and Ramzi B. Albadarneh

Fractal Fract. 2022, 6(3), 158; https://doi.org/10.3390/fractalfract6030158 - 14 Mar 2022

Cited by 11 | Viewed by 1991

Abstract

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor [...] Read more.

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

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17 pages, 435 KiB

Open AccessArticle

Stability Analysis and Optimal Control of a Fractional Cholera Epidemic Model

by Yanyan He and Zhen Wang

Fractal Fract. 2022, 6(3), 157; https://doi.org/10.3390/fractalfract6030157 - 14 Mar 2022

Cited by 6 | Viewed by 2426

Abstract

In this paper, a fractional model for the transmission dynamics of cholera was developed. In invariant regions of the model, solutions were generated. Disease-free and endemic equilibrium points were obtained. The basic reproduction number was evaluated, and the sensitivity analysis was performed. Under [...] Read more.

In this paper, a fractional model for the transmission dynamics of cholera was developed. In invariant regions of the model, solutions were generated. Disease-free and endemic equilibrium points were obtained. The basic reproduction number was evaluated, and the sensitivity analysis was performed. Under the support of Pontryagin’s maximum principle, the fractional order optimal control was obtained. Furthermore, an optimal strategy was discussed, which minimized the total number of infected individuals and the costs associated with control. Treatment, vaccination, and awareness programs were regarded as three means to reduce the number of infected. Finally, numerical simulations and cost-effectiveness analysis were presented to show the result that the best strategy was the combination of treatment and awareness programs. Full article

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12 pages, 1191 KiB

Open AccessArticle

The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation

by Farah M. Al-Askar, Wael W. Mohammed, Abeer M. Albalahi and Mahmoud El-Morshedy

Fractal Fract. 2022, 6(3), 156; https://doi.org/10.3390/fractalfract6030156 - 13 Mar 2022

Cited by 13 | Viewed by 2040

Abstract

We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the [...] Read more.

We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the

exp (- Φ (η))

-expansion method and sine–cosine method. Since this equation is used to explain the hydrodynamic model of shallow-water waves, the wave of leading fluid flow, and plasma physics, scientists will be able to characterize a wide variety of fascinating physical phenomena with these solutions. Furthermore, we evaluate the influence of noise on the behavior of the acquired solutions using 2D and 3D graphical representations. Full article

(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)

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16 pages, 347 KiB

Open AccessArticle

On Certain Integrals Related to Saran’s Hypergeometric Function F_K

by Minjie Luo, Minghui Xu and Ravinder Krishna Raina

Fractal Fract. 2022, 6(3), 155; https://doi.org/10.3390/fractalfract6030155 - 13 Mar 2022

Cited by 4 | Viewed by 1857

Abstract

In the present paper, we establish two Erdélyi-type integrals for Saran’s hypergeometric function

F_{K}

, which has applications in specific branches of applied physics and statistics (see below). We employ methods based on the k-dimensional fractional integration by parts to obtain [...] Read more.

In the present paper, we establish two Erdélyi-type integrals for Saran’s hypergeometric function

F_{K}

, which has applications in specific branches of applied physics and statistics (see below). We employ methods based on the k-dimensional fractional integration by parts to obtain our main integral identities. The first integral generalizes Koschmieder’s result and the second integral extends one of Erdélyi’s classical hypergeometric integral. Some useful special cases and important remarks are also discussed. Full article

(This article belongs to the Section General Mathematics, Analysis)

18 pages, 353 KiB

Open AccessArticle

Nonlocal Boundary Value Problems for Hilfer Generalized Proportional Fractional Differential Equations

by Jessada Tariboon, Ayub Samadi and Sotiris K. Ntouyas

Fractal Fract. 2022, 6(3), 154; https://doi.org/10.3390/fractalfract6030154 - 13 Mar 2022

Cited by 8 | Viewed by 1942

Abstract

In this paper, we discuss the existence and uniqueness of solutions for boundary value problems for Hilfer generalized proportional fractional differential equations with multi-point boundary conditions. Firstly, we consider the scalar case for which the uniqueness result is proved by using Banach’s fixed [...] Read more.

In this paper, we discuss the existence and uniqueness of solutions for boundary value problems for Hilfer generalized proportional fractional differential equations with multi-point boundary conditions. Firstly, we consider the scalar case for which the uniqueness result is proved by using Banach’s fixed point theorem and the existence results are established via Krasnosel’skiĭ’s fixed point theorem and Leray–Schauder nonlinear alternative. We then establish an existence result in the Banach space case based on Mönch’s fixed point theorem and the technique of the measure of noncompactness. Examples are constructed to illustrate the application of the main results. We emphasize that, in this paper, we initiate the study of Hilfer generalized proportional fractional boundary value problems of order in (1, 2]. Full article

(This article belongs to the Section General Mathematics, Analysis)

17 pages, 33572 KiB

Open AccessArticle

A Fractal Permeability Model of Tight Oil Reservoirs Considering the Effects of Multiple Factors

by Zhongwei Wu, Chuanzhi Cui, Yong Yang, Chuanbao Zhang, Jian Wang and Xin Cai

Fractal Fract. 2022, 6(3), 153; https://doi.org/10.3390/fractalfract6030153 - 11 Mar 2022

Cited by 2 | Viewed by 2358

Abstract

The prediction of permeability and the evaluation of tight oil reservoirs are very important to extract tight oil resources. Tight oil reservoirs contain enormous micro/nanopores, in which the fluid flow exhibits micro/nanoscale flow and has a slip length. Furthermore, the porous size distribution [...] Read more.

The prediction of permeability and the evaluation of tight oil reservoirs are very important to extract tight oil resources. Tight oil reservoirs contain enormous micro/nanopores, in which the fluid flow exhibits micro/nanoscale flow and has a slip length. Furthermore, the porous size distribution (PSD), stress sensitivity, irreducible water, and pore wall effect must also be taken into consideration when conducting the prediction and evaluation of tight oil permeability. Currently, few studies on the permeability model of tight oil reservoirs have simultaneously taken the above factors into consideration, resulting in low reliability of the published models. To fill this gap, a fractal permeability model of tight oil reservoirs based on fractal geometry theory, the Hagen–Poiseuille equation (H–P equation), and Darcy’s formula is proposed. Many factors, including the slip length, PSD, stress sensitivity, irreducible water, and pore wall effect, were coupled into the proposed model, which was verified through comparison with published experiments and models, and a sensitivity analysis is presented. From the work, it can be concluded that a decrease in the porous fractal dimension indicates an increase in the number of small pores, thus decreasing the permeability. Similarly, a large tortuous fractal dimension represents a complex flow channel, which results in a decrease in permeability. A decrease in irreducible water or an increase in slip length results in an increase in flow space, which increases permeability. The permeability decreases with an increase in effective stress; moreover, when the mechanical properties of rock (elastic modulus and Poisson’s ratio) increase, the decreasing rate of permeability with effective stress is reduced. Full article

(This article belongs to the Special Issue Applications of Fractal Geometry Theory in Porous Media)

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15 pages, 536 KiB

Open AccessArticle

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

by Wafa Shammakh, A. George Maria Selvam, Vignesh Dhakshinamoorthy and Jehad Alzabut

Fractal Fract. 2022, 6(3), 152; https://doi.org/10.3390/fractalfract6030152 - 10 Mar 2022

Cited by 14 | Viewed by 1899

Abstract

Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, [...] Read more.

Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, using the Hilfer operator. Hybrid fixed point theorem is considered to study the existence of solutions, and the uniqueness of the solution is proved using Banach contraction theorem. Stability results in the sense of Ulam and Hyers, and its generalized form of stability for the considered initial value problem are established and we depict numerical simulations to demonstrate the impact of the fractional order on stability. Full article

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14 pages, 3602 KiB

Open AccessArticle

On Z-Intuitionistic Fuzzy Fractional Valuations for Medical Diagnosis: An Intuitionistic Fuzzy Knowledge-Based Expert System

by Nitesh Dhiman, Madan M. Gupta, Dhan Pal Singh, Vandana, Vishnu Narayan Mishra and Mukesh K. Sharma

Fractal Fract. 2022, 6(3), 151; https://doi.org/10.3390/fractalfract6030151 - 10 Mar 2022

Cited by 5 | Viewed by 2069

Abstract

In an uncertain situation, data may present in continuous form or discrete form. We have various techniques to deal with continuous data in a realistic situation. However, when data are in discrete form, the existing techniques are inadequate to deal with these situations, [...] Read more.

In an uncertain situation, data may present in continuous form or discrete form. We have various techniques to deal with continuous data in a realistic situation. However, when data are in discrete form, the existing techniques are inadequate to deal with these situations, and these techniques cannot provide the proper modulation for adequate analysis of the system. In order to provide the proper acceleration to discrete data, we need an appropriate modulation technique that can help us to handle unconditional boundedness on the technique and will operate like the techniques used for continuous data with fractional variables. In this work, we developed an intuitionistic fuzzy fractional knowledge-based expert system using unconditional and qualified fuzzy propositions based on the Z-intuitionistic fuzzy fractional valuation probability density function. In this proposed method, the discrete fractional variables will be converted into intuitionistic fuzzy fractional numbers and then be used in our algorithm. The proposed Z-intuitionistic fuzzy fractional valuation knowledge-based system can easily be applied in the medical field for the diagnosis of diseases in a vague environment due to the ordered-pair characteristics of the Z-intuitionistic fuzzy fractional valuation. In this study, we collected data of dengue patients, which included seven clinical findings: Temperature, sugar, Pulse Rate (PR), age, cough, and Blood Pressure (BP). A numerical example was also carried out to elaborate on the present technique. In addition, a comparative study is discussed in this work. We also provide the managerial implications of the data, with the limitations of the proposed technique presented at the end of this work. Full article

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14 pages, 1041 KiB

Open AccessArticle

A Numerical Method for Simulating Viscoelastic Plates Based on Fractional Order Model

by Suhua Jin, Jiaquan Xie, Jingguo Qu and Yiming Chen

Fractal Fract. 2022, 6(3), 150; https://doi.org/10.3390/fractalfract6030150 - 10 Mar 2022

Cited by 9 | Viewed by 2200

Abstract

In this study, an efficacious method for solving viscoelastic dynamic plates in the time domain is proposed for the first time. The differential operator matrices of different orders of Bernstein polynomials algorithm are adopted to approximate the ternary displacement function. The approximate results [...] Read more.

In this study, an efficacious method for solving viscoelastic dynamic plates in the time domain is proposed for the first time. The differential operator matrices of different orders of Bernstein polynomials algorithm are adopted to approximate the ternary displacement function. The approximate results are simulated by code. In addition, it is proved that the proposed method is feasible and effective through error analysis and mathematical examples. Finally, the effects of external load, side length of plate, thickness of plate and boundary condition on the dynamic response of square plate are studied. The numerical results illustrate that displacement and stress of the plate change with the change of various parameters. It is further verified that the Bernstein polynomials algorithm can be used as a powerful tool for numerical solution and dynamic analysis of viscoelastic plates. Full article

(This article belongs to the Special Issue Applications of Fractional Operator in Image Processing and Stability of Control Systems)

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21 pages, 13086 KiB

Open AccessArticle

Fractional Modeling Applied to the Dynamics of the Action Potential in Cardiac Tissue

by Sergio Adriani David, Carlos Alberto Valentim and Amar Debbouche

Fractal Fract. 2022, 6(3), 149; https://doi.org/10.3390/fractalfract6030149 - 10 Mar 2022

Cited by 10 | Viewed by 2566

Abstract

We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some [...] Read more.

We investigate a class of fractional time-partial differential equations describing the dynamics of the fast action potential process in contractile myocytes. The system is explored in both one and two dimensional cases. Homogeneous and nonhomogeneous solutions are derived. We also numerically simulate some of the proposed fractional solutions to provide a different modeling perspective on distinct phases of cardiac membrane potential. Results indicate that the fractional diffusion-wave equation may be employed to model membrane potential dynamics with the fractional order working as an extra asset to modulate electricity conduction, particularly for lower fractional order values. Full article

(This article belongs to the Special Issue Dynamical Systems and Their Applications (DSTA) — in Memory of Prof. Dr. José A. Tenreiro Machado)

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11 pages, 560 KiB

Open AccessArticle

The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations

by Jun-Sheng Duan, Li-Xia Jing and Ming Li

Fractal Fract. 2022, 6(3), 148; https://doi.org/10.3390/fractalfract6030148 - 9 Mar 2022

Cited by 6 | Viewed by 2199

Abstract

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval

0 \leq x \leq 1

was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary [...] Read more.

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval

0 \leq x \leq 1

was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential–integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations)

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19 pages, 6497 KiB

Open AccessArticle

Consensus of Fractional-Order Double-Integral Multi-Agent System in a Bounded Fluctuating Potential

by Xi Chen, Maokang Luo and Lu Zhang

Fractal Fract. 2022, 6(3), 147; https://doi.org/10.3390/fractalfract6030147 - 7 Mar 2022

Cited by 2 | Viewed by 1978

Abstract

At present, the consensus problem of fractional complex systems has received more attention. However, there is little literature on the consensus problem of fractional-order complex systems under noise disturbance. In this paper, we present a fractional-order double-integral multi-agent system affected by a common [...] Read more.

At present, the consensus problem of fractional complex systems has received more attention. However, there is little literature on the consensus problem of fractional-order complex systems under noise disturbance. In this paper, we present a fractional-order double-integral multi-agent system affected by a common bounded fluctuating potential, where the protocol term consists of both the relative position and velocity information of neighboring agents. The consensus conditions of the presented system in the absence of noise are analytically given and verified by a numerical simulation algorithm. Then, the influences of the system order and other system parameters on the consensus of the presented system in the presence of bounded noise are also analyzed. It is found that when compared with the classical integer-order system, the presented fractional-order system has a larger range of consensus parameters and has more rich dynamic characteristics under the action of random noise. Especially, the bounded noise has a promoting effect on the consensus of the presented fractional-order system, while there is no similar phenomenon in the corresponding integer-order system. Full article

(This article belongs to the Special Issue Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations)

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13 pages, 325 KiB

Open AccessArticle

Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

by Choukri Derbazi, Zidane Baitiche, Mohammed S. Abdo, Kamal Shah, Bahaaeldin Abdalla and Thabet Abdeljawad

Fractal Fract. 2022, 6(3), 146; https://doi.org/10.3390/fractalfract6030146 - 7 Mar 2022

Cited by 18 | Viewed by 2391

Abstract

The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which [...] Read more.

The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which exercises the procedure of upper and lower solutions. Two sequences of extremal solutions are generated in which one converges to the upper and the other to the corresponding lower solution. The method does not need any prior discretization or collocation for generating the aforesaid two sequences for upper and lower solutions. Further, the aforesaid techniques produce a fruitful combination of upper and lower solutions. To demonstrate our results, we provide some pertinent examples. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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Fractal Fract., Volume 6, Issue 3 (March 2022) – 51 articles

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