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Fractal Fract., Volume 6, Issue 4 (April 2022) – 51 articles

Cover Story (view full-size image): Unmanned aerial vehicles are being increasingly used in different applications in both the military and civilian domains. Their missions include surveillance, target acquisition, highway monitoring, industrial inspection, etc. New requirements in the recent development of UAVs demand robust and adaptive control techniques for different flight conditions, being susceptible to important variations over time in their dynamics. Fractional-order (FO) controllers are recognized for their ability to ensure robustness. Integrating a FO controller into an adaptive control scheme provides robustness and flying adaptability to these processes. This research presents the design of such a control approach for a quadrotor system. The experimental results highlight the efficiency and excellent performance of the proposed method. View this paper
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Article
Dynamic Response and Damage Regularity of Sandstone with Different Moisture States under Cyclic Loading
Fractal Fract. 2022, 6(4), 226; https://doi.org/10.3390/fractalfract6040226 - 18 Apr 2022
Cited by 1 | Viewed by 1096
Abstract
In the process of geotechnical engineering excavation, wet and water-filled rock masses are inevitable. To obtain the mechanical properties of these rocks, indoor tests are required, and most of the rock tests rock tests are dry or nearly dry. They cannot really reflect [...] Read more.
In the process of geotechnical engineering excavation, wet and water-filled rock masses are inevitable. To obtain the mechanical properties of these rocks, indoor tests are required, and most of the rock tests rock tests are dry or nearly dry. They cannot really reflect the true nature of the rock, let alone its nature under a dynamic load. The rock was repeatedly impacted during the blasting excavation process. To determine the mechanical response characteristics and damage evolution of rocks with different moisture states under cyclic dynamic loads, rock samples with three saturation levels were prepared. In the experiment, the Hopkinson pressure bar equipment was utilized to perform five cycles of impact with the same incident energy, and the dynamic response of rocks with different impact times was recorded. Nuclear magnetic resonance technology was employed to obtain the change law of the pores of rock specimens after impact, and the cumulative damage rules of rock were combined with the fractal theory. From the experiments, it can be observed that the stress-strain curves of all rock samples are similar, in that they all have stress addition and unloading stages. The peak stress is proportional to the impact time and moisture content, whereas the opposite is true for the peak strain. After the impact, the small and large pores closed and increased, respectively. The porosity and porosity change rate increased with an increase in the impact time. With an increase in moisture content, this trend is more obvious. It can be observed via magnetic resonance imaging that the internal fractures of the water-bearing rock are obvious after multiple impacts. In particular, the saturated rock specimens exhibited severe damage. Fractal analysis of the NMR figures revealed that after three impact times, the fractal dimension change in the water-bearing rock samples was not obvious. This phenomenon indicated that a macro gap appeared. The fractal dimensions of the dry rock samples continued to increase, and the internal damage was less obvious. Full article
(This article belongs to the Special Issue Fractures and Fragments by Fractal Analysis)
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Perspective
The Fractal Tapestry of Life: III Multifractals Entail the Fractional Calculus
Fractal Fract. 2022, 6(4), 225; https://doi.org/10.3390/fractalfract6040225 - 15 Apr 2022
Viewed by 1253
Abstract
This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively [...] Read more.
This is the third essay advocating the use the (non-integer) fractional calculus (FC) to capture the dynamics of complex networks in the twilight of the Newtonian era. Herein, the focus is on drawing a distinction between networks described by monfractal time series extensively discussed in the prequels and how they differ in function from multifractal time series, using physiological phenomena as exemplars. In prequel II, the network effect was introduced to explain how the collective dynamics of a complex network can transform a many-body non-linear dynamical system modeled using the integer calculus (IC) into a single-body fractional stochastic rate equation. Note that these essays are about biomedical phenomena that have historically been improperly modeled using the IC and how fractional calculus (FC) models better explain experimental results. This essay presents the biomedical entailment of the FC, but it is not a mathematical discussion in the sense that we are not concerned with the formal infrastucture, which is cited, but we are concerned with what that infrastructure entails. For example, the health of a physiologic network is characterized by the width of the multifractal spectrum associated with its time series, and which becomes narrower with the onset of certain pathologies. Physiologic time series that have explicitly related pathology to a narrowing of multifractal time series include but are not limited to heart rate variability (HRV), stride rate variability (SRV) and breath rate variability (BRV). The efficiency of the transfer of information due to the interaction between two such complex networks is determined by their relative spectral width, with information being transferred from the network with the broader to that with the narrower width. A fractional-order differential equation, whose order is random, is shown to generate a multifractal time series, thereby providing a FC model of the information exchange between complex networks. This equivalence between random fractional derivatives and multifractality has not received the recognition in the bioapplications literature we believe it warrants. Full article
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Article
Fractional-Order PI Controller Design Based on Reference–to–Disturbance Ratio
Fractal Fract. 2022, 6(4), 224; https://doi.org/10.3390/fractalfract6040224 - 15 Apr 2022
Cited by 1 | Viewed by 981
Abstract
The presence of disturbances in practical control engineering applications is unavoidable. At the same time, they drive the closed-loop system’s response away from the desired behavior. For this reason, the attenuation of disturbance effects is a primary goal of the control loop. Fractional-order [...] Read more.
The presence of disturbances in practical control engineering applications is unavoidable. At the same time, they drive the closed-loop system’s response away from the desired behavior. For this reason, the attenuation of disturbance effects is a primary goal of the control loop. Fractional-order controllers have now been researched intensively in terms of improving the closed-loop results and robustness of the control system, compared to the standard integer-order controllers. In this study, a novel tuning method for fractional-order controllers is developed. The tuning is based on improving the disturbance attenuation of periodic disturbances with an estimated frequency. For this, the reference–to–disturbance ratio is used as a quantitative measure of the control system’s ability to reject disturbances. Numerical examples are included to justify the approach, quantify the advantages and demonstrate the robustness. The simulation results provide for a validation of the proposed tuning method. Full article
(This article belongs to the Special Issue Women’s Special Issue Series: Fractal and Fractional)
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Article
The Sharp Bounds of the Third-Order Hankel Determinant for Certain Analytic Functions Associated with an Eight-Shaped Domain
Fractal Fract. 2022, 6(4), 223; https://doi.org/10.3390/fractalfract6040223 - 14 Apr 2022
Cited by 6 | Viewed by 1058
Abstract
The main focus of this research is to solve certain coefficient-related problems for analytic functions that are subordinated to a unique trigonometric function. For the class Ssin*, with the quantity zf(z)f(z) [...] Read more.
The main focus of this research is to solve certain coefficient-related problems for analytic functions that are subordinated to a unique trigonometric function. For the class Ssin*, with the quantity zf(z)f(z) subordinated to 1+sinz, we obtain an estimate on the initial coefficient a4 and an upper bound of the third Hankel determinant. For functions in the class BTsin, with f(z) lie in an eight-shaped domain in the right-half plane, we prove that its upper bound of third Hankel determinant is 116. All the results are proven to be sharp. Full article
(This article belongs to the Special Issue New Trends in Geometric Function Theory)
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Article
Cluster Analysis on Locally Asymptotically Self-Similar Processes with Known Number of Clusters
Fractal Fract. 2022, 6(4), 222; https://doi.org/10.3390/fractalfract6040222 - 14 Apr 2022
Viewed by 1003
Abstract
We conduct cluster analysis of a class of locally asymptotically self-similar stochastic processes with finite covariance structures, which includes Brownian motion, fractional Brownian motion, and multifractional Brownian motion as paradigmatic examples. Given the true number of clusters, a new covariance-based dissimilarity measure is [...] Read more.
We conduct cluster analysis of a class of locally asymptotically self-similar stochastic processes with finite covariance structures, which includes Brownian motion, fractional Brownian motion, and multifractional Brownian motion as paradigmatic examples. Given the true number of clusters, a new covariance-based dissimilarity measure is introduced, based on which we obtain approximately asymptotically consistent algorithms for clustering locally asymptotically self-similar stochastic processes. In the simulation study, clustering data sampled from fractional and multifractional Brownian motions with distinct Hurst parameters illustrates the approximated asymptotic consistency of the proposed algorithms. Clustering global financial markets’ equity indexes returns and sovereign CDS spreads provides a successful real world application. Implementations in MATLAB of the proposed algorithms and the simulation study are publicly shared in GitHub. Full article
(This article belongs to the Special Issue Applications of Fractional Calculus in Economics)
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Article
Synchronization of Incommensurate Fractional-Order Chaotic Systems Based on Linear Feedback Control
Fractal Fract. 2022, 6(4), 221; https://doi.org/10.3390/fractalfract6040221 - 14 Apr 2022
Cited by 6 | Viewed by 1177
Abstract
The dynamics and synchronization of fractional-order (FO) chaotic systems have received much attention in recent years. However, the research are focused mostly on FO commensurate systems. This paper addresses the synchronization of incommensurate FO (IFO) chaotic systems. By employing the comparison principle for [...] Read more.
The dynamics and synchronization of fractional-order (FO) chaotic systems have received much attention in recent years. However, the research are focused mostly on FO commensurate systems. This paper addresses the synchronization of incommensurate FO (IFO) chaotic systems. By employing the comparison principle for FO systems with multi-order and the linear feedback control method, a sufficient condition for ensuring the synchronization of IFO chaotic systems is developed in terms of linear matrix inequalities (LMIs). Such synchronization condition relies just on the system parameters, and is easily verify and implemented. Two typical FO chaotic systems, named the IFO Genesio-Tesi system and Hopfied neural networks are selected to demonstrate the effectiveness and feasibility of the proposed method. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
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Article
Optimal Design of TD-TI Controller for LFC Considering Renewables Penetration by an Improved Chaos Game Optimizer
Fractal Fract. 2022, 6(4), 220; https://doi.org/10.3390/fractalfract6040220 - 13 Apr 2022
Cited by 10 | Viewed by 1172
Abstract
This study presents an innovative strategy for load frequency control (LFC) using a combination structure of tilt-derivative and tilt-integral gains to form a TD-TI controller. Furthermore, a new improved optimization technique, namely the quantum chaos game optimizer (QCGO) is applied to tune the [...] Read more.
This study presents an innovative strategy for load frequency control (LFC) using a combination structure of tilt-derivative and tilt-integral gains to form a TD-TI controller. Furthermore, a new improved optimization technique, namely the quantum chaos game optimizer (QCGO) is applied to tune the gains of the proposed combination TD-TI controller in two-area interconnected hybrid power systems, while the effectiveness of the proposed QCGO is validated via a comparison of its performance with the traditional CGO and other optimizers when considering 23 bench functions. Correspondingly, the effectiveness of the proposed controller is validated by comparing its performance with other controllers, such as the proportional-integral-derivative (PID) controller based on different optimizers, the tilt-integral-derivative (TID) controller based on a CGO algorithm, and the TID controller based on a QCGO algorithm, where the effectiveness of the proposed TD-TI controller based on the QCGO algorithm is ensured using different load patterns (i.e., step load perturbation (SLP), series SLP, and random load variation (RLV)). Furthermore, the challenges of renewable energy penetration and communication time delay are considered to test the robustness of the proposed controller in achieving more system stability. In addition, the integration of electric vehicles as dispersed energy storage units in both areas has been considered to test their effectiveness in achieving power grid stability. The simulation results elucidate that the proposed TD-TI controller based on the QCGO controller can achieve more system stability under the different aforementioned challenges. Full article
(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)
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Article
Mitochondrial DNA Profiling by Fractal Lacunarity to Characterize the Senescent Phenotype as Normal Aging or Pathological Aging
Fractal Fract. 2022, 6(4), 219; https://doi.org/10.3390/fractalfract6040219 - 13 Apr 2022
Viewed by 946
Abstract
Biocomplexity, chaos, and fractality can explain the heterogeneity of aging individuals by regarding longevity as a “secondary product” of the evolution of a dynamic nonlinear system. Genetic-environmental interactions drive the individual senescent phenotype toward normal, pathological, or successful aging. Mitochondrial dysfunctions and mitochondrial [...] Read more.
Biocomplexity, chaos, and fractality can explain the heterogeneity of aging individuals by regarding longevity as a “secondary product” of the evolution of a dynamic nonlinear system. Genetic-environmental interactions drive the individual senescent phenotype toward normal, pathological, or successful aging. Mitochondrial dysfunctions and mitochondrial DNA (mtDNA) mutations represent a possible mechanism shared by disease(s) and the aging process. This study aims to characterize the senescent phenotype and discriminate between normal (nA) and pathological (pA) aging by mtDNA mutation profiling. MtDNA sequences from hospitalized and non-hospitalized subjects (age-range: 65–89 years) were analyzed and compared to the revised Cambridge Reference Sequence (rCRS). Fractal properties of mtDNA sequences were displayed by chaos game representation (CGR) method, previously modified to deal with heteroplasmy. Fractal lacunarity analysis was applied to characterize the senescent phenotype on the basis of mtDNA sequence mutations. Lacunarity parameter β, from our hyperbola model function, was statistically different (p < 0.01) between the nA and pA groups. Parameter β cut-off value at 1.26 × 10−3 identifies 78% nA and 80% pA subjects. This also agrees with the presence of MT-CO gene variants, peculiar to nA (C9546m, 83%) and pA (T9900w, 80%) mtDNA, respectively. Fractal lacunarity can discriminate the senescent phenotype evolving as normal or pathological aging by individual mtDNA mutation profile. Full article
(This article belongs to the Special Issue Fractals in Aging and Disease)
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Article
Design and High-Order Precision Numerical Implementation of Fractional-Order PI Controller for PMSM Speed System Based on FPGA
Fractal Fract. 2022, 6(4), 218; https://doi.org/10.3390/fractalfract6040218 - 12 Apr 2022
Cited by 4 | Viewed by 1113
Abstract
In this paper, the design of a fractional-order proportional integral (FOPI) controller and integer-order (IOPI) controller are compared for the permanent magnet synchronous motor (PMSM) speed regulation system. A high-precision implementation method of a fractional-order proportional integral (FOPI) controller is proposed in this [...] Read more.
In this paper, the design of a fractional-order proportional integral (FOPI) controller and integer-order (IOPI) controller are compared for the permanent magnet synchronous motor (PMSM) speed regulation system. A high-precision implementation method of a fractional-order proportional integral (FOPI) controller is proposed in this work. Three commonly used numerical implementation methods of fractional operators are investigated and compared for comprehensively evaluating the numerical implementation performance in this work. Furthermore, for the impulse response invariant implementation method, the effects of different discretization orders on the control performance of the system are compared. The high-order fractional-order controller can be implemented accurately in a control system with the field-programmable gate array (FPGA) with the capability of parallel calculation. The simulation and experimental results show that the high-precision numerical implementation method of the designed high-order FOPI controller has better performance than the ordinary precision fractional operation implementation method and traditional order integer order PI controller. Full article
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Article
The Traveling Wave Solutions in a Mixed-Diffusion Epidemic Model
Fractal Fract. 2022, 6(4), 217; https://doi.org/10.3390/fractalfract6040217 - 11 Apr 2022
Cited by 1 | Viewed by 841
Abstract
In this paper, we study the traveling wave solution of an epidemic model with mixed diffusion. First, we give two definitions of the minimum wave speeds and prove that they are equivalent. Second, the existence, decaying behavior, and uniqueness of traveling wave fronts [...] Read more.
In this paper, we study the traveling wave solution of an epidemic model with mixed diffusion. First, we give two definitions of the minimum wave speeds and prove that they are equivalent. Second, the existence, decaying behavior, and uniqueness of traveling wave fronts are obtained. Third, the signs of minimum wave speeds are studied, and further, in two specific cases of the dispersal kernel, we show how to identify the signs of minimum wave speeds. Full article
Article
Generalization of k-Uniformly Starlike and Convex Functions Using q-Difference Operator
Fractal Fract. 2022, 6(4), 216; https://doi.org/10.3390/fractalfract6040216 - 11 Apr 2022
Cited by 1 | Viewed by 783
Abstract
In this article we have defined two new subclasses of analytic functions kSq[A,B] and kKq[A,B] by using q-difference operator in an open unit disk. Furthermore, the [...] Read more.
In this article we have defined two new subclasses of analytic functions kSq[A,B] and kKq[A,B] by using q-difference operator in an open unit disk. Furthermore, the necessary and sufficient conditions along with certain other useful properties of these newly defined subclasses have been calculated by using q-difference operator. Full article
(This article belongs to the Special Issue New Trends in Geometric Function Theory)
Article
Continuum Damage Dynamic Model Combined with Transient Elastic Equation and Heat Conduction Equation to Solve RPV Stress
Fractal Fract. 2022, 6(4), 215; https://doi.org/10.3390/fractalfract6040215 - 11 Apr 2022
Cited by 2 | Viewed by 1133
Abstract
The development of the world cannot be separated from energy: the energy crisis has become a major challenge in this era, and nuclear energy has been applied to many fields. This paper mainly studies the stress change of reaction pressure vessels (RPV). We [...] Read more.
The development of the world cannot be separated from energy: the energy crisis has become a major challenge in this era, and nuclear energy has been applied to many fields. This paper mainly studies the stress change of reaction pressure vessels (RPV). We established several different physical models to solve the same mechanical problem. Numerical methods range from 1D to 3D; the 1D model is mainly based on the mechanical equilibrium equations established by the internal pressure of RPV, the hoop stress, and the axial stress. We found that the hoop stress is twice the axial stress; this model is a rough estimate. For 2D RPV mechanical simulation, we proposed a new method, which combined the continuum damage dynamic model with the transient cross-section finite element method (CDDM-TCFEM). The advantage is that the temperature and shear strain can be linked by the damage factor effect on the elastic model and Poission ratio. The results show that with the increase of temperature (damage factor μ^,d^), the Young’s modulus decreases point by point, and the Poisson’s ratio increases with the increase of temperature (damage factor μ^,Et). The advantage of the CDDM-TCFEM is that the calculation efficiency is high. However, it is unable to obtain the overall mechanical cloud map. In order to solve this problem, we established the axisymmetric finite element model, and the results show that the stress value at both ends of RPV is significantly greater than that in the middle of the container. Meanwhile, the shape changes of 2D and 3D RPV are calculated and visualized. Finally, a 3D thermal–mechanical coupling model is established, and the cloud map of strain and displacement are also visualized. We found that the stress of the vessel wall near the nozzle decreases gradually from the inside surface to the outside, and the hoop stress is slightly larger than the axial stress. The main contribution of this paper is to establish a CDDM-TCFEM model considering the influence of temperature on elastic modulus and Poission ratio. It can dynamically describe the stress change of RPV; we have given the fitting formula of the internal temperature and pressure of RPV changing with time. We also establish a 3D coupling model and use the adaptive mesh to discretize the pipe. The numerical discrete theory of FDM-FEM is given, and the numerical results are visualized well. In addition, we have given error estimation for h-type and p-type adaptive meshes. So, our research can provide mechanical theoretical support for nuclear energy safety applications and RPV design. Full article
(This article belongs to the Topic Fractional Calculus: Theory and Applications)
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Article
Image Enhancement Based on Rough Set and Fractional Order Differentiator
Fractal Fract. 2022, 6(4), 214; https://doi.org/10.3390/fractalfract6040214 - 11 Apr 2022
Cited by 11 | Viewed by 1022
Abstract
In the paper, an image enhancement algorithm based on a rough set and fractional order differentiator is proposed. By combining the rough set theory with a Gaussian mixture model, a new image segmentation algorithm with higher immunity is obtained. This image segmentation algorithm [...] Read more.
In the paper, an image enhancement algorithm based on a rough set and fractional order differentiator is proposed. By combining the rough set theory with a Gaussian mixture model, a new image segmentation algorithm with higher immunity is obtained. This image segmentation algorithm can obtain more image layers with concentrating information and preserve more image details than traditional algorithms. After preprocessing, the segmentation layers will be enhanced by a new adaptive fractional order differential mask in the Fourier domain. Experimental results and numerical analysis have verified the effectiveness of the proposed algorithm. Full article
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Article
Design and Application of an Interval Estimator for Nonlinear Discrete-Time SEIR Epidemic Models
Fractal Fract. 2022, 6(4), 213; https://doi.org/10.3390/fractalfract6040213 - 09 Apr 2022
Cited by 1 | Viewed by 834
Abstract
This paper designs an interval estimator for a fourth-order nonlinear susceptible-exposed-infected-recovered (SEIR) model with disturbances using noisy counts of susceptible people provided by Public Health Services (PHS). Infectious diseases are considered the main cause of deaths among the top ten worldwide, as per [...] Read more.
This paper designs an interval estimator for a fourth-order nonlinear susceptible-exposed-infected-recovered (SEIR) model with disturbances using noisy counts of susceptible people provided by Public Health Services (PHS). Infectious diseases are considered the main cause of deaths among the top ten worldwide, as per the World Health Organization (WHO). Therefore, tracking and estimating the evolution of these diseases are important to make intervention strategies. We study a real case in which some uncertain variables such as model disturbances, uncertain input and output measurement noise are not exactly available but belong to an interval. Moreover, the uncertain transmission bound rate from the susceptible towards the exposed stage is not available for measurement. We designed an interval estimator using an observability matrix that generates a tight interval vector for the actual states of the SEIR model in a guaranteed way without computing the observer gain. As the developed approach is not dependent on observer gain, our method provides more freedom. The convergence of the width to a known value in finite time is investigated for the estimated state vector to prove the stability of the estimation error, significantly improving the accuracy for the proposed approach. Finally, simulation results demonstrate the satisfying performance of the proposed algorithm. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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Article
New Riemann–Liouville Fractional-Order Inclusions for Convex Functions via Interval-Valued Settings Associated with Pseudo-Order Relations
Fractal Fract. 2022, 6(4), 212; https://doi.org/10.3390/fractalfract6040212 - 09 Apr 2022
Cited by 8 | Viewed by 849
Abstract
In this study, we focus on the newly introduced concept of LR-convex interval-valued functions to establish new variants of the Hermite–Hadamard (H-H) type and Pachpatte type inequalities for Riemann–Liouville fractional integrals. By presenting some numerical examples, we also verify the correctness of the [...] Read more.
In this study, we focus on the newly introduced concept of LR-convex interval-valued functions to establish new variants of the Hermite–Hadamard (H-H) type and Pachpatte type inequalities for Riemann–Liouville fractional integrals. By presenting some numerical examples, we also verify the correctness of the results that we have derived in this paper. Because the results, which are related to the differintegral of the ϱ1+ϱ22 type, are novel in the context of the LR-convex interval-valued functions, we believe that this will be a useful contribution for motivating future research in this area. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
Article
Certain Hybrid Matrix Polynomials Related to the Laguerre-Sheffer Family
Fractal Fract. 2022, 6(4), 211; https://doi.org/10.3390/fractalfract6040211 - 09 Apr 2022
Cited by 1 | Viewed by 729
Abstract
The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials [...] Read more.
The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific members of the matrix polynomial family are shown. Full article
Article
On a Class of Partial Differential Equations and Their Solution via Local Fractional Integrals and Derivatives
Fractal Fract. 2022, 6(4), 210; https://doi.org/10.3390/fractalfract6040210 - 08 Apr 2022
Cited by 2 | Viewed by 896
Abstract
This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave [...] Read more.
This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
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Article
Further Generalization and Approximation of Fractional-Order Filters and Their Inverse Functions of the Second-Order Limiting Form
Fractal Fract. 2022, 6(4), 209; https://doi.org/10.3390/fractalfract6040209 - 08 Apr 2022
Cited by 3 | Viewed by 971
Abstract
This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts [...] Read more.
This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts the real part of the roots of the numerator and denominator polynomials of the proposed rational approximant to negative values is formulated. Consequently, stable inverse filter characteristics can also be achieved using the suggested method. Accuracy of the proposed low-pass, high-pass, band-pass, and band-stop filters for various combinations of design parameters is evaluated using the absolute relative magnitude/phase error metrics. Current feedback operational amplifier-based circuit simulations validate the efficacy of the four types of designed filters and their inverse functions. Experimental results for the frequency and time-domain performances of the proposed fractional-order band-pass filter and its inverse counterpart are also presented. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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Article
Weighted Fractional Calculus: A General Class of Operators
Fractal Fract. 2022, 6(4), 208; https://doi.org/10.3390/fractalfract6040208 - 07 Apr 2022
Cited by 7 | Viewed by 935
Abstract
We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the [...] Read more.
We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the classical Riemann–Liouville fractional calculus, and use them to prove many fundamental properties of these operators. As examples, we consider special cases such as tempered, Hadamard-type, and Erdélyi–Kober operators. We also define appropriate modifications of the Laplace transform and convolution operations, and solve some ordinary differential equations in the setting of these general classes of operators. Full article
Article
Numerical Solutions of Variable-Coefficient Fractional-in-Space KdV Equation with the Caputo Fractional Derivative
Fractal Fract. 2022, 6(4), 207; https://doi.org/10.3390/fractalfract6040207 - 07 Apr 2022
Cited by 8 | Viewed by 984
Abstract
In this paper, numerical solutions of the variable-coefficient Korteweg-De Vries (vcKdV) equation with space described by the Caputo fractional derivative operator is developed. The propagation and interaction of vcKdV equation in different cases, such as breather soliton and periodic suppression soliton, are numerically [...] Read more.
In this paper, numerical solutions of the variable-coefficient Korteweg-De Vries (vcKdV) equation with space described by the Caputo fractional derivative operator is developed. The propagation and interaction of vcKdV equation in different cases, such as breather soliton and periodic suppression soliton, are numerically simulated. Especially, the Fourier spectral method is used to solve the fractional-in-space vcKdV equation with breather soliton. From numerical simulations and compared with other methods, it can be easily seen that our method has low computational complexity and higher precision. Full article
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Article
Numerical Analysis of Local Discontinuous Galerkin Method for the Time-Fractional Fourth-Order Equation with Initial Singularity
Fractal Fract. 2022, 6(4), 206; https://doi.org/10.3390/fractalfract6040206 - 07 Apr 2022
Cited by 2 | Viewed by 1141
Abstract
In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time t=0. The Caputo time-fractional derivative with [...] Read more.
In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time t=0. The Caputo time-fractional derivative with derivative order α(0,1) is discretized by the well-known L1 formula on nonuniform meshes; for the spatial derivative, the local discontinuous Galerkin (LDG) finite element method is used. Based on the discrete fractional Gronwall’s inequality, we prove the stability of the proposed scheme and the optimal error estimate for the solution, i.e., (2α)-order accurate in time and (k+1)-order accurate in space, when piece-wise polynomials of degree at most k are used. Moreover, a second-order and nonuniform time-stepping scheme is developed for the fractional model. The scheme uses the L2-1σ formula for the time fractional derivative and the LDG method for the space approximation. The stability and temporal optimal second-order convergence of the scheme are also shown. Finally, some numerical experiments are presented to confirm the theoretical results. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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Article
Macro- and Micromechanical Assessment of the Influence of Non-Plastic Fines and Stress Anisotropy on the Dynamic Shear Modulus of Binary Mixtures
Fractal Fract. 2022, 6(4), 205; https://doi.org/10.3390/fractalfract6040205 - 06 Apr 2022
Cited by 2 | Viewed by 1196
Abstract
Resonant column tests were carried out on Hostun sand mixed with 5%, 10% and 20% non-plastic fines (defined as grains smaller than 0.075 mm) in order to quantify the combined influence of the void ratio (e), anisotropic stress state (defined as [...] Read more.
Resonant column tests were carried out on Hostun sand mixed with 5%, 10% and 20% non-plastic fines (defined as grains smaller than 0.075 mm) in order to quantify the combined influence of the void ratio (e), anisotropic stress state (defined as σv/σh) and fines content (fc) on the maximum small-strain shear modulus Gmax. A significant reduction in the Gmax with increasing fc was observed. Using the empirical model forwarded by Roesler, the influence of e and σv/σh on Gmax was captured, although the model was unable to capture the influence of varying fines content using a single equation. From the micro-CT images, a qualitative observation of the initial skeletal structure of the ‘fines-in-sand’ grains was performed and the equivalent granular void ratio e* was determined. The e was henceforth replaced by e* in Roesler’s equation in order to capture the variation in fc. The new modification was quantified in terms of the mean square error R2. Furthermore, the Gmax of Hostun sand–fine mixtures was predicted with good accuracy by replacing e with e*. Additionally, a micromechanical interpretation based on the experimental observation was developed. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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Article
Design and Experimental Results of an Adaptive Fractional-Order Controller for a Quadrotor
Fractal Fract. 2022, 6(4), 204; https://doi.org/10.3390/fractalfract6040204 - 06 Apr 2022
Cited by 4 | Viewed by 1004
Abstract
The use of multi-copter systems started to grow over the last years in various applications. The designed solutions require high stability and maneuverability. To fulfill these specifications, a robust control strategy must be designed and integrated. Focusing on this challenge, this research proposes [...] Read more.
The use of multi-copter systems started to grow over the last years in various applications. The designed solutions require high stability and maneuverability. To fulfill these specifications, a robust control strategy must be designed and integrated. Focusing on this challenge, this research proposes an adaptive control design applied to a physical model of a quadrotor prototype. The proposed adaptive structure guarantees robustness, control flexibility, and stability to the whole process. The prototype components, structure, and laboratory testing equipment that are used to run the experiments are presented in this paper. The study is focused on the performance comparison of a classical PID controller and a fractional-order controller, which are both integrated into the adaptive scheme. Fractional-order controllers are preferred due to their recognized ability to increase the robustness of the overall closed-loop system. Furthermore, this work covers the design and the tuning method of this control approach. The research concludes with the actual results obtained for this comparative study that highlights the advantages of the fractional-order controller. Full article
(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)
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Article
On the Existence and Stability of a Neutral Stochastic Fractional Differential System
Fractal Fract. 2022, 6(4), 203; https://doi.org/10.3390/fractalfract6040203 - 06 Apr 2022
Cited by 7 | Viewed by 894
Abstract
The main purpose of this paper is to investigate the existence and Ulam-Hyers stability (U-Hs) of solutions of a nonlinear neutral stochastic fractional differential system. We prove the existence and uniqueness of solutions to the proposed system by using fixed point theorems and [...] Read more.
The main purpose of this paper is to investigate the existence and Ulam-Hyers stability (U-Hs) of solutions of a nonlinear neutral stochastic fractional differential system. We prove the existence and uniqueness of solutions to the proposed system by using fixed point theorems and the Banach contraction principle. Also, by using fundamental schemes of fractional calculus, we study the (U-Hs) to the solutions of our suggested system. Besides, we study an example, best describing our main result. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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Article
New Solutions of Nonlinear Dispersive Equation in Higher-Dimensional Space with Three Types of Local Derivatives
Fractal Fract. 2022, 6(4), 202; https://doi.org/10.3390/fractalfract6040202 - 04 Apr 2022
Cited by 2 | Viewed by 913
Abstract
The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study [...] Read more.
The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study of differential equations with local derivatives using the reduction technique. This higher-dimensional equation is considered with three types of local derivatives in the temporal sense. Different types of exact solutions in five cases are reported. Furthermore, with the help of the Maple package, the solutions found in this study are verified. Finally, several interesting 3D, 2D and density plots are demonstrated to visualize the nonlinear wave structures more efficiently. Full article
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Article
Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations
Fractal Fract. 2022, 6(4), 201; https://doi.org/10.3390/fractalfract6040201 - 02 Apr 2022
Cited by 2 | Viewed by 1089
Abstract
In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral [...] Read more.
In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results. Full article
(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)
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Article
A Numerical Approach to Solve the q-Fractional Boundary Value Problems
Fractal Fract. 2022, 6(4), 200; https://doi.org/10.3390/fractalfract6040200 - 02 Apr 2022
Viewed by 1015
Abstract
In this present paper, we study the difference method for solving a boundary value problem of the Caputo type q-fractional differential equation. This method is based on the numerical quadrature of the q-fractional derivative and the q-Taylor expansion of related [...] Read more.
In this present paper, we study the difference method for solving a boundary value problem of the Caputo type q-fractional differential equation. This method is based on the numerical quadrature of the q-fractional derivative and the q-Taylor expansion of related function. We first derive the truncation error boundness of O(xn2)-order and prove the existence and uniqueness of the numerical solution. Then, we prove the stability of the numerical solution and give the error estimation. Numerical experiments finally verify the validity of the theoretical analysis. Full article
Article
Applications for Unbounded Convergences in Banach Lattices
Fractal Fract. 2022, 6(4), 199; https://doi.org/10.3390/fractalfract6040199 - 01 Apr 2022
Cited by 2 | Viewed by 1021
Abstract
Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute [...] Read more.
Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices. Full article
Article
Influence of Groundwater Depth on Pile–Soil Mechanical Properties and Fractal Characteristics under Cyclic Loading
Fractal Fract. 2022, 6(4), 198; https://doi.org/10.3390/fractalfract6040198 - 01 Apr 2022
Cited by 55 | Viewed by 1679
Abstract
The analysis of the behavior of soil and foundations when the piles in offshore areas are subjected to long-term lateral loading (wind) is one of the major problems associated with the smooth operation of superstructure. The strength of the pile-soil system is influenced [...] Read more.
The analysis of the behavior of soil and foundations when the piles in offshore areas are subjected to long-term lateral loading (wind) is one of the major problems associated with the smooth operation of superstructure. The strength of the pile-soil system is influenced by variations in the water content of the soil. At present, there are no studies carried out analyzing the mechanical and deformational behavior of both the material of the laterally loaded piles and soil with groundwater level as a variable. In this paper, a series of 1-g model tests were conducted to explore the lateral behavior of both soil and monopile under unidirectional cyclic loading, based on the foundation of an offshore wind turbine near the island. The influence of underground water level and cyclic load magnitude on the performance of the pile–soil system was analyzed. To visualize the movements of soil particles during the experimental process, particle image velocimetry (PIV) was used to record the soil displacement field under various cyclic loading conditions. The relationship curves between pile top displacement and cyclic steps, as well as the relationship curves between cyclic stiffness and cyclic steps, were displayed. Combined with fractal theory, the fractal dimension of each curve was calculated to evaluate the sensitivity of the pile–soil interaction system. The results showed that cyclic loading conditions and groundwater depth are the main factors affecting the pile–soil interaction. The cyclic stiffness of the soil increased in all test groups as loading progressed; however, an increase in the cyclic load magnitude decreased the initial and cyclic stiffness. The initial and cyclic stiffness of dry soil was higher than that of saturated soil, but less than that of unsaturated soil. The ability of the unsaturated soil to limit the lateral displacement of the pile decreased as the depth of the groundwater level dropped. The greater the fluctuation of the pile top displacement, the larger the fractal dimension of each relationship curve, with a variation interval of roughly 1.24–1.38. The average increment of the cumulative pile top displacement between each cycle step following the cyclic loading was positively correlated with fractal dimension. Based on the PIV results, the changes in the pile–soil system were predominantly focused in the early stages of the experiment, and the short-term effects of lateral cyclic loading are greater than the long-term effects. In addition, this research was limited to a single soil layer. The pile–soil interaction under layered soil is investigated, and the results will be used in more complex ground conditions in the future. Full article
(This article belongs to the Special Issue Fractal and Fractional in Geomaterials)
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Article
Why Controlling the Asymptomatic Infection Is Important: A Modelling Study with Stability and Sensitivity Analysis
Fractal Fract. 2022, 6(4), 197; https://doi.org/10.3390/fractalfract6040197 - 31 Mar 2022
Cited by 6 | Viewed by 1174
Abstract
The large proportion of asymptomatic patients is the major cause leading to the COVID-19 pandemic which is still a significant threat to the whole world. A six-dimensional ODE system (SEIAQR epidemical model) is established to study the dynamics of COVID-19 spreading considering infection [...] Read more.
The large proportion of asymptomatic patients is the major cause leading to the COVID-19 pandemic which is still a significant threat to the whole world. A six-dimensional ODE system (SEIAQR epidemical model) is established to study the dynamics of COVID-19 spreading considering infection by exposed, infected, and asymptomatic cases. The basic reproduction number derived from the model is more comprehensive including the contribution from the exposed, infected, and asymptomatic patients. For this more complex six-dimensional ODE system, we investigate the global and local stability of disease-free equilibrium, as well as the endemic equilibrium, whereas most studies overlooked asymptomatic infection or some other virus transmission features. In the sensitivity analysis, the parameters related to the asymptomatic play a significant role not only in the basic reproduction number R0. It is also found that the asymptomatic infection greatly affected the endemic equilibrium. Either in completely eradicating the disease or achieving a more realistic goal to reduce the COVID-19 cases in an endemic equilibrium, the importance of controlling the asymptomatic infection should be emphasized. The three-dimensional phase diagrams demonstrate the convergence point of the COVID-19 spreading under different initial conditions. In particular, massive infections will occur as shown in the phase diagram quantitatively in the case R0>1. Moreover, two four-dimensional contour maps of Rt are given varying with different parameters, which can offer better intuitive instructions on the control of the pandemic by adjusting policy-related parameters. Full article
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