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Fractal Fract., Volume 6, Issue 1 (January 2022) – 48 articles

Cover Story (view full-size image): The scientific community has recently seen a fast-growing number of publications tackling the topic of fractional order controllers in general, with a focus on the fractional order PID. Very few recent papers discuss the practical use of such controllers. However, the industrial acceptance of these controllers is still far from being reached. Autotuning methods for such fractional order PIDs could possibly make them more appealing to industrial applications, as well. In this paper, the current autotuning methods for fractional order PIDs are reviewed. The focus is on the most recent findings. A comparison between several autotuning approaches is considered for various types of processes. Numerical examples are given to highlight the practicality of the methods that could be extended to simple industrial processes. View this paper
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8 pages, 261 KiB  
Article
Third Hankel Determinant for a Subclass of Univalent Functions Associated with Lemniscate of Bernoulli
by Najeeb Ullah, Irfan Ali, Sardar Muhammad Hussain, Jong-Suk Ro, Nazar Khan and Bilal Khan
Fractal Fract. 2022, 6(1), 48; https://doi.org/10.3390/fractalfract6010048 - 16 Jan 2022
Cited by 3 | Viewed by 2020
Abstract
This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We find the upper bound of the Hankel determinant H3(1) for this subclass by applying the Carlson–Shaffer operator to [...] Read more.
This paper deals with a new subclass of univalent function associated with the right half of the lemniscate of Bernoulli. We find the upper bound of the Hankel determinant H3(1) for this subclass by applying the Carlson–Shaffer operator to it. The present work also deals with certain properties of this newly defined subclass, such as the upper bound of the Hankel determinant of order 3, coefficient estimates, etc. Full article
(This article belongs to the Special Issue New Trends in Geometric Function Theory)
17 pages, 918 KiB  
Article
A Look-Up Table Based Fractional Order Composite Controller Synthesis Method for the PMSM Speed Servo System
by Weijia Zheng, Runquan Huang, Ying Luo, YangQuan Chen, Xiaohong Wang and Yong Chen
Fractal Fract. 2022, 6(1), 47; https://doi.org/10.3390/fractalfract6010047 - 15 Jan 2022
Cited by 10 | Viewed by 2368
Abstract
Considering the performance requirements in actual applications, a look-up table based fractional order composite control scheme for the permanent magnet synchronous motor speed servo system is proposed. Firstly, an extended state observer based compensation scheme was adopted to suppress the motor parametric uncertainties [...] Read more.
Considering the performance requirements in actual applications, a look-up table based fractional order composite control scheme for the permanent magnet synchronous motor speed servo system is proposed. Firstly, an extended state observer based compensation scheme was adopted to suppress the motor parametric uncertainties and convert the speed servo plant into a double-integrator model. Then, a fractional order proportional-derivative (PDμ) controller was adopted as the speed controller to provide the optimal step response performance for the servo system. A universal look-up table was established to estimate the derivative order of the PDμ controller, according to the optimal samples collected by an improved differential evolution algorithm. With the look-up table, the optimal PDμ controller can be tuned analytically. Simulation and experimental results show that the servo system using the composite control scheme can achieve optimal tracking performance and has robustness to the motor parametric uncertainties and disturbance torques. Full article
(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)
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11 pages, 274 KiB  
Article
On Global Convergence of Third-Order Chebyshev-Type Method under General Continuity Conditions
by Fouad Othman Mallawi, Ramandeep Behl and Prashanth Maroju
Fractal Fract. 2022, 6(1), 46; https://doi.org/10.3390/fractalfract6010046 - 14 Jan 2022
Cited by 2 | Viewed by 1647
Abstract
There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method [...] Read more.
There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method is proposed under the assumptions that Fréchet derivative of first order satisfies continuity condition of the Hölder. Finally, we consider some integral equation and boundary value problem (BVP) in order to illustrate the suitability of theoretical results. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
27 pages, 394 KiB  
Article
A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions
by Ravi P. Agarwal, Hana Al-Hutami and Bashir Ahmad
Fractal Fract. 2022, 6(1), 45; https://doi.org/10.3390/fractalfract6010045 - 14 Jan 2022
Cited by 6 | Viewed by 1833
Abstract
We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the [...] Read more.
We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
19 pages, 1645 KiB  
Article
Maximum Likelihood Estimation for Mixed Fractional Vasicek Processes
by Chun-Hao Cai, Yin-Zhong Huang, Lin Sun and Wei-Lin Xiao
Fractal Fract. 2022, 6(1), 44; https://doi.org/10.3390/fractalfract6010044 - 14 Jan 2022
Cited by 3 | Viewed by 1844
Abstract
In this paper, we consider the problem of estimating the drift parameters in the mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model. Using the fundamental martingale and the Laplace transform, both the strong consistency and the asymptotic [...] Read more.
In this paper, we consider the problem of estimating the drift parameters in the mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model. Using the fundamental martingale and the Laplace transform, both the strong consistency and the asymptotic normality of the maximum likelihood estimators are studied for all H(0,1), H1/2. On the other hand, we present that the MLE can be simulated when the Hurst parameter H>1/2. Full article
(This article belongs to the Special Issue Stochastic Calculus for Fractional Brownian Motion)
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11 pages, 877 KiB  
Article
Consensus of Julia Sets
by Weihua Sun and Shutang Liu
Fractal Fract. 2022, 6(1), 43; https://doi.org/10.3390/fractalfract6010043 - 13 Jan 2022
Cited by 3 | Viewed by 1913
Abstract
The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus [...] Read more.
The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods. Full article
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22 pages, 366 KiB  
Article
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 22 | Viewed by 2238
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
21 pages, 404 KiB  
Article
On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations
by Ravshan Ashurov and Yusuf Fayziev
Fractal Fract. 2022, 6(1), 41; https://doi.org/10.3390/fractalfract6010041 - 12 Jan 2022
Cited by 13 | Viewed by 2266
Abstract
The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<tT), [...] Read more.
The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<tT), u(ξ)=αu(0)+φ (α is a constant and 0<ξT), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
20 pages, 4656 KiB  
Article
Influence of MgO on the Hydration and Shrinkage Behavior of Low Heat Portland Cement-Based Materials via Pore Structural and Fractal Analysis
by Lei Wang, Xiao Lu, Lisheng Liu, Jie Xiao, Ge Zhang, Fanxing Guo and Li Li
Fractal Fract. 2022, 6(1), 40; https://doi.org/10.3390/fractalfract6010040 - 12 Jan 2022
Cited by 65 | Viewed by 3526
Abstract
Currently, low heat Portland (LHP) cement is widely used in mass concrete structures. The magnesia expansion agent (MgO) can be adopted to reduce the shrinkage of conventional Portland cement-based materials, but very few studies can be found that investigate the influence of MgO [...] Read more.
Currently, low heat Portland (LHP) cement is widely used in mass concrete structures. The magnesia expansion agent (MgO) can be adopted to reduce the shrinkage of conventional Portland cement-based materials, but very few studies can be found that investigate the influence of MgO on the properties of LHP cement-based materials. In this study, the influences of two types of MgO on the hydration, as well as the shrinkage behavior of LHP cement-based materials, were studied via pore structural and fractal analysis. The results indicate: (1) The addition of reactive MgO (with a reactivity of 50 s and shortened as M50 thereafter) not only extends the induction stage of LHP cement by about 1–2 h, but also slightly increases the hydration heat. In contrast, the addition of weak reactive MgO (with a reactivity of 300 s and shortened as M300 thereafter) could not prolong the induction stage of LHP cement. (2) The addition of 4 wt.%–8 wt.% MgO (by weight of binder) lowers the mechanical property of LHP concrete. Higher dosages of MgO and stronger reactivity lead to a larger reduction in mechanical properties at all of the hydration times studied. M300 favors the strength improvement of LHP concrete at later ages. (3) M50 effectively compensates the shrinkage of LHP concrete at a much earlier time than M300, whereas M300 compensates the long-term shrinkage more effectively than M50. Thus, M300 with an optimal dosage of 8 wt.% is suggested to be applied in mass LHP concrete structures. (4) The addition of M50 obviously refines the pore structures of LHP concrete at 7 days, whereas M300 starts to refine the pore structure at around 60 days. At 360 days, the concretes containing M300 exhibits much finer pore structures than those containing M50. (5) Fractal dimension is closely correlated with the pore structure of LHP concrete. Both pore structure and fractal dimension exhibit weak (or no) correlations with shrinkage of LHP concrete. Full article
(This article belongs to the Special Issue Fractal and Fractional in Cement-based Materials)
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23 pages, 10450 KiB  
Article
Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations
by Christoph Bandt and Dmitry Mekhontsev
Fractal Fract. 2022, 6(1), 39; https://doi.org/10.3390/fractalfract6010039 - 12 Jan 2022
Viewed by 3555
Abstract
Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and [...] Read more.
Self-similar sets with the open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity, were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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28 pages, 596 KiB  
Article
Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects
by Ridhwan Reyaz, Ahmad Qushairi Mohamad, Yeou Jiann Lim, Muhammad Saqib and Sharidan Shafie
Fractal Fract. 2022, 6(1), 38; https://doi.org/10.3390/fractalfract6010038 - 12 Jan 2022
Cited by 13 | Viewed by 2327
Abstract
Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations [...] Read more.
Fractional derivatives have been proven to showcase a spectrum of solutions that is useful in the fields of engineering, medical, and manufacturing sciences. Studies on the application of fractional derivatives on fluid flow are relatively new, especially in analytical studies. Thus, geometrical representations for fractional derivatives in the mechanics of fluid flows are yet to be discovered. Nonetheless, theoretical studies will be useful in facilitating future experimental studies. Therefore, the aim of this study is to showcase an analytical solution on the impact of the Caputo-Fabrizio fractional derivative for a magnethohydrodynamic (MHD) Casson fluid flow with thermal radiation and chemical reaction. Analytical solutions are obtained via Laplace transform through compound functions. The obtained solutions are first verified, then analysed. It is observed from the study that variations in the fractional derivative parameter, α, exhibits a transitional behaviour of fluid between unsteady state and steady state. Numerical analyses on skin friction, Nusselt number, and Sherwood number were also analysed. Behaviour of these three properties were in agreement of that from past literature. Full article
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25 pages, 6256 KiB  
Review
A Review of Recent Developments in Autotuning Methods for Fractional-Order Controllers
by Cristina I. Muresan, Isabela Birs, Clara Ionescu, Eva H. Dulf and Robin De Keyser
Fractal Fract. 2022, 6(1), 37; https://doi.org/10.3390/fractalfract6010037 - 11 Jan 2022
Cited by 35 | Viewed by 3661
Abstract
The scientific community has recently seen a fast-growing number of publications tackling the topic of fractional-order controllers in general, with a focus on the fractional order PID. Several versions of this controller have been proposed, including different tuning methods and implementation possibilities. Quite [...] Read more.
The scientific community has recently seen a fast-growing number of publications tackling the topic of fractional-order controllers in general, with a focus on the fractional order PID. Several versions of this controller have been proposed, including different tuning methods and implementation possibilities. Quite a few recent papers discuss the practical use of such controllers. However, the industrial acceptance of these controllers is still far from being reached. Autotuning methods for such fractional order PIDs could possibly make them more appealing to industrial applications, as well. In this paper, the current autotuning methods for fractional order PIDs are reviewed. The focus is on the most recent findings. A comparison between several autotuning approaches is considered for various types of processes. Numerical examples are given to highlight the practicality of the methods that could be extended to simple industrial processes. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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28 pages, 1099 KiB  
Article
Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach
by Pratap Anbalagan, Raja Ramachandran, Jehad Alzabut, Evren Hincal and Michal Niezabitowski
Fractal Fract. 2022, 6(1), 36; https://doi.org/10.3390/fractalfract6010036 - 11 Jan 2022
Cited by 14 | Viewed by 1552
Abstract
This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust [...] Read more.
This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results. Full article
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15 pages, 559 KiB  
Article
Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative
by Ndolane Sene
Fractal Fract. 2022, 6(1), 35; https://doi.org/10.3390/fractalfract6010035 - 11 Jan 2022
Cited by 35 | Viewed by 1987
Abstract
This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and [...] Read more.
This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators. Full article
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12 pages, 372 KiB  
Article
Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions
by Ravi Agarwal, Snezhana Hristova and Donal O’Regan
Fractal Fract. 2022, 6(1), 34; https://doi.org/10.3390/fractalfract6010034 - 10 Jan 2022
Cited by 24 | Viewed by 2271
Abstract
In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as [...] Read more.
In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided. Full article
(This article belongs to the Special Issue Fractional Derivatives and Their Applications)
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10 pages, 284 KiB  
Article
Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas
by Sabah Iftikhar, Samet Erden, Muhammad Aamir Ali, Jamel Baili and Hijaz Ahmad
Fractal Fract. 2022, 6(1), 33; https://doi.org/10.3390/fractalfract6010033 - 10 Jan 2022
Cited by 4 | Viewed by 1649
Abstract
Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on [...] Read more.
Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
26 pages, 2841 KiB  
Article
Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations
by Hegagi Mohamed Ali, Hijaz Ahmad, Sameh Askar and Ismail Gad Ameen
Fractal Fract. 2022, 6(1), 32; https://doi.org/10.3390/fractalfract6010032 - 10 Jan 2022
Cited by 7 | Viewed by 1779
Abstract
In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM [...] Read more.
In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions. Full article
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21 pages, 4996 KiB  
Article
Discretization, Bifurcation, and Control for a Class of Predator-Prey Interactions
by Asifa Tassaddiq, Muhammad Sajjad Shabbir, Qamar Din and Humera Naaz
Fractal Fract. 2022, 6(1), 31; https://doi.org/10.3390/fractalfract6010031 - 6 Jan 2022
Cited by 20 | Viewed by 1888
Abstract
The present study focuses on the dynamical aspects of a discrete-time Leslie-Gower predator-prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability [...] Read more.
The present study focuses on the dynamical aspects of a discrete-time Leslie-Gower predator-prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark-Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings. Full article
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23 pages, 384 KiB  
Article
On Starlike Functions of Negative Order Defined by q-Fractional Derivative
by Sadia Riaz, Ubaid Ahmed Nisar, Qin Xin, Sarfraz Nawaz Malik and Abdul Raheem
Fractal Fract. 2022, 6(1), 30; https://doi.org/10.3390/fractalfract6010030 - 6 Jan 2022
Cited by 13 | Viewed by 1430
Abstract
In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α(3,1] [...] Read more.
In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α(3,1], q(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α[1,1], q(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included. Full article
14 pages, 1417 KiB  
Article
A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model
by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Thongchai Botmart and Wajaree Weera
Fractal Fract. 2022, 6(1), 29; https://doi.org/10.3390/fractalfract6010029 - 4 Jan 2022
Cited by 19 | Viewed by 1329
Abstract
In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and [...] Read more.
In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and local search genetic algorithms (GAs) and the active-set method (ASM), i.e., ANN-GAASM. The novel NLE-PDSM was derived from the standard LE and the PDSM along with the details of singular points, prediction terms and shape factors. The modeling strength of ANN was implemented to create a merit function based on the second kind of NLE-PDSM using the mean squared error, and optimization was performed through the GAASM. The corroboration, validation and excellence of the ANN-GAASM for three distinct problems were established through relative studies from exact solutions on the basis of stability, convergence and robustness. Furthermore, explanations through statistical investigations confirmed the worth of the proposed scheme. Full article
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15 pages, 336 KiB  
Article
On Hadamard Type Fractional Inequalities for Riemann–Liouville Integrals via a Generalized Convexity
by Tao Yan, Ghulam Farid, Hafsa Yasmeen and Chahn Yong Jung
Fractal Fract. 2022, 6(1), 28; https://doi.org/10.3390/fractalfract6010028 - 3 Jan 2022
Cited by 2 | Viewed by 1176
Abstract
In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define [...] Read more.
In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,hm)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions. Full article
31 pages, 1308 KiB  
Review
Recent Advances and Applications of Spiral Dynamics Optimization Algorithm: A Review
by Madiah Binti Omar, Kishore Bingi, B Rajanarayan Prusty and Rosdiazli Ibrahim
Fractal Fract. 2022, 6(1), 27; https://doi.org/10.3390/fractalfract6010027 - 2 Jan 2022
Cited by 14 | Viewed by 3404
Abstract
This paper comprehensively reviews the spiral dynamics optimization (SDO) algorithm and investigates its characteristics. SDO algorithm is one of the most straightforward physics-based optimization algorithms and is successfully applied in various broad fields. This paper describes the recent advances of the SDO algorithm, [...] Read more.
This paper comprehensively reviews the spiral dynamics optimization (SDO) algorithm and investigates its characteristics. SDO algorithm is one of the most straightforward physics-based optimization algorithms and is successfully applied in various broad fields. This paper describes the recent advances of the SDO algorithm, including its adaptive, improved, and hybrid approaches. The growth of the SDO algorithm and its application in various areas, theoretical analysis, and comparison with its preceding and other algorithms are also described in detail. A detailed description of different spiral paths, their characteristics, and the application of these spiral approaches in developing and improving other optimization algorithms are comprehensively presented. The review concludes the current works on the SDO algorithm, highlighting its shortcomings and suggesting possible future research perspectives. Full article
(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)
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22 pages, 4157 KiB  
Article
The Role of the Discount Policy of Prepayment on Environmentally Friendly Inventory Management
by Shirin Sultana, Abu Hashan Md Mashud, Yosef Daryanto, Sujan Miah, Adel Alrasheedi and Ibrahim M. Hezam
Fractal Fract. 2022, 6(1), 26; https://doi.org/10.3390/fractalfract6010026 - 2 Jan 2022
Cited by 10 | Viewed by 2212
Abstract
Nowadays, more and more consumers consider environmentally friendly products in their purchasing decisions. Companies need to adapt to these changes while paying attention to standard business systems such as payment terms. The purpose of this study is to optimize the entire profit function [...] Read more.
Nowadays, more and more consumers consider environmentally friendly products in their purchasing decisions. Companies need to adapt to these changes while paying attention to standard business systems such as payment terms. The purpose of this study is to optimize the entire profit function of a retailer and to find the optimal selling price and replenishment cycle when the demand rate depends on the price and carbon emission reduction level. This study investigates an economic order quantity model that has a demand function with a positive impact of carbon emission reduction besides the selling price. In this model, the supplier requests payment in advance on the purchased cost while offering a discount according to the payment in the advanced decision. Three different types of payment-in-advance cases are applied: (1) payment in advance with equal numbers of instalments, (2) payment in advance with a single instalment, and (3) the absence of payment in advance. Numerical examples and sensitivity analysis illustrate the proposed model. Here, the total profit increases for all three cases with higher values of carbon emission reduction level. Further, the study finds that the profit becomes maximum for case 2, whereas the selling price and cycle length become minimum. This study considers the sustainable inventory model with payment-in-advance settings when the demand rate depends on the price and carbon emission reduction level. From the literature review, no researcher has undergone this kind of study in the authors’ knowledge. Full article
(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)
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16 pages, 337 KiB  
Article
Solvability of Some Nonlocal Fractional Boundary Value Problems at Resonance in ℝn
by Yizhe Feng and Zhanbing Bai
Fractal Fract. 2022, 6(1), 25; https://doi.org/10.3390/fractalfract6010025 - 1 Jan 2022
Cited by 5 | Viewed by 1566
Abstract
In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable xRn and the effect of the coefficient matrices matrices B and C of boundary [...] Read more.
In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable xRn and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained. Full article
11 pages, 5298 KiB  
Article
Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations
by Muhammad Shakeel, Nehad Ali Shah and Jae Dong Chung
Fractal Fract. 2022, 6(1), 24; https://doi.org/10.3390/fractalfract6010024 - 1 Jan 2022
Cited by 11 | Viewed by 1522
Abstract
In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential [...] Read more.
In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions. Full article
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27 pages, 1128 KiB  
Article
Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation
by Dmitriy Tverdyi and Roman Parovik
Fractal Fract. 2022, 6(1), 23; https://doi.org/10.3390/fractalfract6010023 - 31 Dec 2021
Cited by 11 | Viewed by 1530
Abstract
The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, [...] Read more.
The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid. Full article
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23 pages, 5917 KiB  
Article
Effect of Chemical Composition of Fine Aggregate on the Frictional Behavior of Concrete–Soil Interface under Sulfuric Acid Environment
by Jie Xiao, Zhenming Xu, Yikang Murong, Lei Wang, Bin Lei, Lijing Chu, Haibo Jiang and Wenjun Qu
Fractal Fract. 2022, 6(1), 22; https://doi.org/10.3390/fractalfract6010022 - 31 Dec 2021
Cited by 24 | Viewed by 2749
Abstract
Through direct shear tests, this paper aimed to research the effect of fine marble aggregate on the shear strength and fractal dimension of the interface between soil and concrete corroded by sulfuric acid. More realistic concrete rough surfaces than the artificially roughened surfaces [...] Read more.
Through direct shear tests, this paper aimed to research the effect of fine marble aggregate on the shear strength and fractal dimension of the interface between soil and concrete corroded by sulfuric acid. More realistic concrete rough surfaces than the artificially roughened surfaces were formed by immersing four concrete plates in plastic buckets filled with sulfuric acid for different periods of time. The sand was adopted to imitate the soil. 3D laser scanner was employed to obtain the digital shapes of concrete plates subjected to sulfuric acid, and the rough surfaces were evaluated by fractal dimension. Large direct shear experiments were performed to obtain the curves of the interface shear stress and shear displacement between sand and corroded concrete plate. The method of data fitting was adopted to calculate the parameters of shear strength (i.e., friction angle and the cohesive) and the parameters of the Clough–Duncan hyperbolic model. The results indicated that as the corrosion days increased, the surface of the concrete plate became rougher, the surface fractal dimensions of the concrete corroded by sulfuric acid became bigger, and the interface friction angle became greater. The friction angle of the interface and the fractal dimensions of the surface of the concrete plate containing crushed gravel and marble sand were smaller than that of the concrete plate containing crushed gravel and river sand. Full article
(This article belongs to the Special Issue Fractal and Fractional in Cement-based Materials)
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9 pages, 2006 KiB  
Article
Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations
by Valentina Anna Lia Salomoni and Nico De Marchi
Fractal Fract. 2022, 6(1), 21; https://doi.org/10.3390/fractalfract6010021 - 31 Dec 2021
Cited by 7 | Viewed by 2340
Abstract
Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: [...] Read more.
Background: solute transport in highly heterogeneous media and even neutron diffusion in nuclear environments are among the numerous applications of fractional differential equations (FDEs), being demonstrated by field experiments that solute concentration profiles exhibit anomalous non-Fickian growth rates and so-called “heavy tails”. Methods: a nonlinear-coupled 3D fractional hydro-mechanical model accounting for anomalous diffusion (FD) and advection–dispersion (FAD) for solute flux is described, accounting for a Riesz derivative treated through the Grünwald–Letnikow definition. Results: a long-tailed solute contaminant distribution is displayed due to the variation of flow velocity in both time and distance. Conclusions: a finite difference approximation is proposed to solve the problem in 1D domains, and subsequently, two scenarios are considered for numerical computations. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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12 pages, 300 KiB  
Article
Application of the Pick Function in the Lieb Concavity Theorem for Deformed Exponentials
by Guozeng Yang, Yonggang Li, Jing Wang and Huafei Sun
Fractal Fract. 2022, 6(1), 20; https://doi.org/10.3390/fractalfract6010020 - 31 Dec 2021
Cited by 1 | Viewed by 1139
Abstract
The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity [...] Read more.
The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity theorem for deformed exponentials. In this paper, the Pick function is used to obtain a generalization of the Lieb concavity theorem for deformed exponentials, and some corollaries associated with exterior algebra are obtained. Full article
(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)
19 pages, 518 KiB  
Article
Shifted Fractional-Order Jacobi Collocation Method for Solving Variable-Order Fractional Integro-Differential Equation with Weakly Singular Kernel
by Mohamed A. Abdelkawy, Ahmed Z. M. Amin, António M. Lopes, Ishak Hashim and Mohammed M. Babatin
Fractal Fract. 2022, 6(1), 19; https://doi.org/10.3390/fractalfract6010019 - 30 Dec 2021
Cited by 8 | Viewed by 1811
Abstract
We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving [...] Read more.
We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fractional integral and derivative and fractional-order shifted Jacobi polynomials, the approximate solutions of VO-FIDE-WSK are derived by solving systems of algebraic equations. The superior accuracy of the method is illustrated through several numerical examples. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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