Subjects -> MATHEMATICS (Total: 1013 journals)
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    - MATHEMATICS (714 journals)
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    - PROBABILITIES AND MATH STATISTICS (113 journals)

MATHEMATICS (714 journals)            First | 1 2 3 4 | Last

Showing 201 - 400 of 538 Journals sorted alphabetically
Educação Matemática Debate     Open Access  
Edumatica : Jurnal Pendidikan Matematika     Open Access  
EduMatSains     Open Access  
Electronic Journal of Differential Equations     Open Access  
Electronic Journal of Graph Theory and Applications     Open Access   (Followers: 3)
Em Teia : Revista de Educação Matemática e Tecnológica Iberoamericana     Open Access  
Emergent Scientist     Open Access  
Energy for Sustainable Development     Hybrid Journal   (Followers: 13)
Enseñanza de las Ciencias : Revista de Investigación y Experiencias Didácticas     Open Access  
Entropy     Open Access   (Followers: 5)
ESAIM: Control Optimisation and Calculus of Variations     Open Access   (Followers: 2)
Euclid     Open Access  
European Journal of Applied Mathematics     Hybrid Journal  
European Journal of Combinatorics     Full-text available via subscription   (Followers: 3)
European Journal of Mathematics     Hybrid Journal   (Followers: 1)
European Scientific Journal     Open Access   (Followers: 1)
Examples and Counterexamples     Open Access  
Experimental Mathematics     Hybrid Journal   (Followers: 5)
Expositiones Mathematicae     Hybrid Journal   (Followers: 2)
Extracta Mathematicae     Open Access  
Facta Universitatis, Series : Mathematics and Informatics     Open Access  
Finite Fields and Their Applications     Full-text available via subscription   (Followers: 5)
Fixed Point Theory and Applications     Open Access  
Formalized Mathematics     Open Access  
Forum of Mathematics, Pi     Open Access   (Followers: 1)
Forum of Mathematics, Sigma     Open Access   (Followers: 1)
Foundations and Trends® in Econometrics     Full-text available via subscription   (Followers: 6)
Foundations and Trends® in Networking     Full-text available via subscription   (Followers: 1)
Foundations and Trends® in Stochastic Systems     Full-text available via subscription   (Followers: 1)
Foundations and Trends® in Theoretical Computer Science     Full-text available via subscription   (Followers: 1)
Foundations of Computational Mathematics     Hybrid Journal  
Fractal and Fractional     Open Access  
Fractals     Hybrid Journal   (Followers: 1)
Frontiers of Mathematics in China     Hybrid Journal  
Fuel Cells Bulletin     Full-text available via subscription   (Followers: 9)
Functional Analysis and Other Mathematics     Hybrid Journal   (Followers: 4)
Fundamental Journal of Mathematics and Applications     Open Access  
Funktsional'nyi Analiz i ego Prilozheniya     Full-text available via subscription  
Fuzzy Optimization and Decision Making     Hybrid Journal   (Followers: 8)
Game Theory     Open Access   (Followers: 2)
Games     Open Access   (Followers: 4)
Games and Economic Behavior     Hybrid Journal   (Followers: 26)
Gamm - Mitteilungen     Hybrid Journal  
GANIT : Journal of Bangladesh Mathematical Society     Open Access  
GEM - International Journal on Geomathematics     Hybrid Journal   (Followers: 1)
General Mathematics     Open Access  
Glasgow Mathematical Journal     Full-text available via subscription  
Global Journal of Mathematical Sciences     Full-text available via subscription  
Graphs and Combinatorics     Hybrid Journal   (Followers: 4)
Grey Systems : Theory and Application     Hybrid Journal  
Groups, Complexity, Cryptology     Open Access   (Followers: 2)
GSTF Journal of Mathematics, Statistics and Operations Research     Open Access   (Followers: 1)
Historia Mathematica     Full-text available via subscription  
Historical Methods: A Journal of Quantitative and Interdisciplinary History     Hybrid Journal   (Followers: 28)
IMA Journal of Applied Mathematics     Hybrid Journal  
IMA Journal of Numerical Analysis - advance access     Hybrid Journal  
ImmunoInformatics     Open Access   (Followers: 1)
Indagationes Mathematicae     Open Access  
Indian Journal of Pure and Applied Mathematics     Hybrid Journal   (Followers: 4)
Indonesian Journal of Combinatorics     Open Access  
Indonesian Journal of Science and Mathematics Education     Open Access   (Followers: 2)
Infinite Dimensional Analysis, Quantum Probability and Related Topics     Hybrid Journal   (Followers: 1)
Infinity Jurnal Matematika dan Aplikasinya     Open Access   (Followers: 4)
Information and Inference     Free  
InfoTekJar : Jurnal Nasional Informatika dan Teknologi Jaringan     Open Access  
InfraMatics     Open Access  
Insight - Non-Destructive Testing and Condition Monitoring     Full-text available via subscription   (Followers: 114)
International Electronic Journal of Algebra     Open Access  
International Journal for Numerical Methods in Engineering     Hybrid Journal   (Followers: 37)
International Journal for Numerical Methods in Fluids     Hybrid Journal   (Followers: 20)
International Journal of Advanced Mathematical Sciences     Open Access  
International Journal of Advanced Mechatronic Systems     Hybrid Journal   (Followers: 2)
International Journal of Advanced Research in Mathematics     Open Access  
International Journal of Advances in Engineering Sciences and Applied Mathematics     Hybrid Journal   (Followers: 10)
International Journal of Algebra and Computation     Hybrid Journal   (Followers: 1)
International Journal of Algebra and Statistics     Open Access   (Followers: 3)
International Journal of Applied and Computational Mathematics     Hybrid Journal  
International Journal of Applied Mathematical Research     Open Access   (Followers: 1)
International Journal of Applied Mathematics and Computer Science     Open Access   (Followers: 7)
International Journal of Applied Mechanics     Hybrid Journal   (Followers: 8)
International Journal of Applied Nonlinear Science     Hybrid Journal  
International Journal of Autonomic Computing     Hybrid Journal   (Followers: 1)
International Journal of Bifurcation and Chaos     Hybrid Journal   (Followers: 4)
International Journal of Biomathematics     Hybrid Journal   (Followers: 2)
International Journal of Computational Complexity and Intelligent Algorithms     Hybrid Journal  
International Journal of Computational Economics and Econometrics     Hybrid Journal   (Followers: 6)
International Journal of Computational Geometry and Applications     Hybrid Journal   (Followers: 2)
International Journal of Computational Intelligence and Applications     Hybrid Journal   (Followers: 2)
International Journal of Computational Methods     Hybrid Journal   (Followers: 4)
International Journal of Computer Processing Of Languages     Hybrid Journal   (Followers: 1)
International Journal of Control, Automation and Systems     Hybrid Journal   (Followers: 15)
International Journal of Dynamical Systems and Differential Equations     Hybrid Journal   (Followers: 1)
International Journal of Economics and Accounting     Hybrid Journal   (Followers: 1)
International Journal of Foundations of Computer Science     Hybrid Journal   (Followers: 3)
International Journal of Fuzzy Computation and Modelling     Hybrid Journal   (Followers: 2)
International Journal of Image and Graphics     Hybrid Journal   (Followers: 5)
International Journal of Industrial Electronics and Drives     Hybrid Journal   (Followers: 3)
International Journal of Low-Carbon Technologies     Open Access   (Followers: 1)
International Journal of Mathematical Education in Science and Technology     Hybrid Journal   (Followers: 9)
International Journal of Mathematical Modelling & Computations     Open Access   (Followers: 3)
International Journal of Mathematical Modelling and Numerical Optimisation     Hybrid Journal   (Followers: 5)
International Journal of Mathematical Sciences and Computing     Open Access  
International Journal of Mathematics     Hybrid Journal   (Followers: 4)
International Journal of Mathematics & Computation     Full-text available via subscription  
International Journal of Mathematics and Mathematical Sciences     Open Access   (Followers: 4)
International Journal of Mathematics in Operational Research     Hybrid Journal   (Followers: 2)
International Journal of Metaheuristics     Hybrid Journal   (Followers: 1)
International Journal of Modelling in Operations Management     Hybrid Journal   (Followers: 2)
International Journal of Modern Nonlinear Theory and Application     Open Access   (Followers: 1)
International Journal of Number Theory     Hybrid Journal   (Followers: 2)
International Journal of Partial Differential Equations     Open Access   (Followers: 2)
International Journal of Polymer Science     Open Access   (Followers: 25)
International Journal of Pure Mathematical Sciences     Open Access  
International Journal of Reliability, Quality and Safety Engineering     Hybrid Journal   (Followers: 14)
International Journal of Research in Undergraduate Mathematics Education     Hybrid Journal   (Followers: 5)
International Journal of Sediment Research     Full-text available via subscription   (Followers: 2)
International Journal of Shape Modeling     Hybrid Journal   (Followers: 1)
International Journal of Theoretical and Mathematical Physics     Open Access   (Followers: 13)
International Journal of Trends in Mathematics Education Research     Open Access   (Followers: 5)
International Journal of Ultra Wideband Communications and Systems     Hybrid Journal  
International Journal of Wavelets, Multiresolution and Information Processing     Hybrid Journal  
International Journal on Artificial Intelligence Tools     Hybrid Journal   (Followers: 9)
International Mathematics Research Notices     Hybrid Journal   (Followers: 1)
Internet Mathematics     Hybrid Journal   (Followers: 1)
Inventiones mathematicae     Hybrid Journal   (Followers: 3)
Inverse Problems in Science and Engineering     Hybrid Journal   (Followers: 3)
Investigations in Mathematics Learning     Hybrid Journal  
Iranian Journal of Optimization     Open Access   (Followers: 2)
Israel Journal of Mathematics     Hybrid Journal  
Ithaca : Viaggio nella Scienza     Open Access  
ITM Web of Conferences     Open Access  
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya     Full-text available via subscription  
Jahresbericht der Deutschen Mathematiker-Vereinigung     Hybrid Journal  
Japan Journal of Industrial and Applied Mathematics     Hybrid Journal  
Japanese Journal of Mathematics     Hybrid Journal  
JIPM (Jurnal Ilmiah Pendidikan Matematika)     Open Access  
JMPM : Jurnal Matematika dan Pendidikan Matematika     Open Access  
JOHME : Journal of Holistic Mathematics Education     Open Access   (Followers: 3)
Johnson Matthey Technology Review     Open Access  
Jornal Internacional de Estudos em Educação Matemática     Open Access  
Journal d'Analyse Mathématique     Hybrid Journal   (Followers: 2)
Journal de Mathématiques Pures et Appliquées     Full-text available via subscription   (Followers: 3)
Journal for Research in Mathematics Education     Full-text available via subscription   (Followers: 29)
Journal für Mathematik-Didaktik     Hybrid Journal  
Journal of Advanced Mathematics and Applications     Full-text available via subscription   (Followers: 1)
Journal of Algebra     Full-text available via subscription   (Followers: 3)
Journal of Algebra and Its Applications     Hybrid Journal   (Followers: 3)
Journal of Algebraic Combinatorics     Hybrid Journal   (Followers: 3)
Journal of Algorithms & Computational Technology     Open Access  
Journal of Applied Mathematics     Open Access   (Followers: 3)
Journal of Applied Mathematics and Computation     Open Access   (Followers: 2)
Journal of Applied Mathematics and Computing     Hybrid Journal  
Journal of Applied Mathematics, Statistics and Informatics     Open Access   (Followers: 1)
Journal of Artificial Intelligence and Data Mining     Open Access   (Followers: 10)
Journal of Classification     Hybrid Journal   (Followers: 5)
Journal of Combinatorial Designs     Hybrid Journal   (Followers: 4)
Journal of Combinatorial Optimization     Hybrid Journal   (Followers: 7)
Journal of Combinatorial Theory, Series A     Full-text available via subscription   (Followers: 5)
Journal of Combinatorial Theory, Series B     Full-text available via subscription   (Followers: 3)
Journal of Complex Analysis     Open Access   (Followers: 2)
Journal of Complex Networks     Hybrid Journal   (Followers: 1)
Journal of Complexity     Hybrid Journal   (Followers: 6)
Journal of Computational and Applied Mathematics     Hybrid Journal   (Followers: 6)
Journal of Computational Biology     Hybrid Journal   (Followers: 9)
Journal of Computational Mathematics and Data Science     Open Access  
Journal of Computational Multiphase Flows     Open Access   (Followers: 1)
Journal of Computational Physics     Hybrid Journal   (Followers: 60)
Journal of Computational Physics : X     Open Access   (Followers: 1)
Journal of Computer Engineering, System and Science (CESS)     Open Access  
Journal of Contemporary Mathematical Analysis     Hybrid Journal  
Journal of Cryptology     Hybrid Journal   (Followers: 5)
Journal of Difference Equations and Applications     Hybrid Journal  
Journal of Differential Equations     Full-text available via subscription   (Followers: 1)
Journal of Discrete Mathematics     Open Access   (Followers: 1)
Journal of Dynamics and Differential Equations     Hybrid Journal  
Journal of Engineering Mathematics     Hybrid Journal   (Followers: 2)
Journal of Evolution Equations     Hybrid Journal  
Journal of Experimental Algorithmics     Full-text available via subscription  
Journal of Flood Risk Management     Hybrid Journal   (Followers: 14)
Journal of Function Spaces     Open Access  
Journal of Functional Analysis     Full-text available via subscription   (Followers: 3)
Journal of Geochemical Exploration     Hybrid Journal   (Followers: 4)
Journal of Geological Research     Open Access   (Followers: 1)
Journal of Geovisualization and Spatial Analysis     Hybrid Journal  
Journal of Global Optimization     Hybrid Journal   (Followers: 6)
Journal of Global Research in Mathematical Archives     Open Access  
Journal of Homotopy and Related Structures     Hybrid Journal  
Journal of Honai Math     Open Access  
Journal of Humanistic Mathematics     Open Access   (Followers: 1)
Journal of Hyperbolic Differential Equations     Hybrid Journal  
Journal of Indian Council of Philosophical Research     Hybrid Journal  
Journal of Industrial Mathematics     Open Access   (Followers: 2)
Journal of Inequalities and Applications     Open Access  
Journal of Infrared, Millimeter and Terahertz Waves     Hybrid Journal   (Followers: 3)
Journal of Integrable Systems     Open Access  
Journal of Knot Theory and Its Ramifications     Hybrid Journal   (Followers: 2)
Journal of Liquid Chromatography & Related Technologies     Hybrid Journal   (Followers: 7)
Journal of Logical and Algebraic Methods in Programming     Hybrid Journal   (Followers: 1)
Journal of Manufacturing Systems     Full-text available via subscription   (Followers: 3)
Journal of Mathematical Analysis and Applications     Full-text available via subscription   (Followers: 3)

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Fractal and Fractional
Number of Followers: 0  

  This is an Open Access Journal Open Access journal
ISSN (Online) 2504-3110
Published by MDPI Homepage  [84 journals]
  • Fractal Fract, Vol. 6, Pages 344: Simultaneous Identification of
           Volatility and Mean-Reverting Parameter for European Option under
           Fractional CKLS Model

    • Authors: Jiajia Zhao, Zuoliang Xu
      First page: 344
      Abstract: In this paper, we reconstruct the time-dependent volatility function of the underlying asset and the mean-reverting parameter γ of the interest rate for European options under the fractional Chan–Karolyi–Longstaff–Sanders (CKLS) stochastic interest rate model. Tikhonov regularization is used to solve the ill-posedness of the inverse problem. The existence and stability of the solution of the regularization problem are given. We employ the alternating direction method of multipliers (ADMM) to iteratively optimize the volatility function and the parameter γ. Finally, numerical simulations and the empirical analysis are presented to illustrate the efficiency of the proposed method.
      Citation: Fractal and Fractional
      PubDate: 2022-06-21
      DOI: 10.3390/fractalfract6070344
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 345: Particle Swarm Optimization Fractional
           Slope Entropy: A New Time Series Complexity Indicator for Bearing Fault
           Diagnosis

    • Authors: Yuxing Li, Lingxia Mu, Peiyuan Gao
      First page: 345
      Abstract: Slope entropy (SlEn) is a time series complexity indicator proposed in recent years, which has shown excellent performance in the fields of medical and hydroacoustics. In order to improve the ability of SlEn to distinguish different types of signals and solve the problem of two threshold parameters selection, a new time series complexity indicator on the basis of SlEn is proposed by introducing fractional calculus and combining particle swarm optimization (PSO), named PSO fractional SlEn (PSO-FrSlEn). Then we apply PSO-FrSlEn to the field of fault diagnosis and propose a single feature extraction method and a double feature extraction method for rolling bearing fault based on PSO-FrSlEn. The experimental results illustrated that only PSO-FrSlEn can classify 10 kinds of bearing signals with 100% classification accuracy by using double features, which is at least 4% higher than the classification accuracies of the other four fractional entropies.
      Citation: Fractal and Fractional
      PubDate: 2022-06-21
      DOI: 10.3390/fractalfract6070345
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 346: On the Modeling of COVID-19 Transmission
           Dynamics with Two Strains: Insight through Caputo Fractional Derivative

    • Authors: Fatmawati, Endang Yuliani, Cicik Alfiniyah, Maureen L. Juga, Chidozie W. Chukwu
      First page: 346
      Abstract: The infection dynamics of COVID-19 is difficult to contain due to the mutation nature of the SARS-CoV-2 virus. This has been a public health concern globally with the impact of the pandemic on the world’s economy and mode of living. In the present work, we formulate and examine a fractional model of COVID-19 considering the two variants of concern on the disease transmission pathways, namely SARS-CoV-2 and D614G on our model formulation. The existence and uniqueness of our model solutions were analyzed using the fixed point theory. Mathematical analyses were presented, and the model’s basic reproduction numbers R01 and R02 were determined. The model has three equilibria: the disease-free equilibrium, that endemic for strain 1, and that endemic for strain 2. The locally asymptotic stability of the equilibria was established based on the R01 and R02 values. Caputo fractional operator was used to simulate the model to study the dynamics of the model solution. Results from numerical simulations envisaged that an increase in the transmission parameters of strain 1 leads to an increase in the number of infected individuals. On the other hand, an increase in the strain 2 transmission rate gives rise to more infection. Furthermore, it was established that there is an increased number of infections with a negative impact of strain 1 on strain 2 dynamics and vice versa.
      Citation: Fractal and Fractional
      PubDate: 2022-06-21
      DOI: 10.3390/fractalfract6070346
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 347: Effects of Relative Density and Grading
           on the Particle Breakage and Fractal Dimension of Granular Materials

    • Authors: Gui Yang, Zhuanzhuan Chen, Yifei Sun, Yang Jiang
      First page: 347
      Abstract: Particle breakage was reported to have great influence on the mechanical property of granular materials. However, limited studies were conducted to quantify the detailed effects of relative density and initial grading on the particle breakage behaviour of granular materials under different confining pressures. In this study, a series of monotonic drained triaxial tests were performed on isotropically consolidated granular materials with four different initial gradings and relative densities. It is observed that particle breakage increases as the confining pressure or relative density increases, whereas it decreases with the increasing coefficient of uniformity. Due to particle breakage, the grading curves of granular materials after triaxial tests can be simulated by a power-law function with fractal dimension. As the confining pressure increases, the fractal dimension approaches the limit of granular materials, i.e., 2.6. A unique normalized relation between the particle breakage extent and confining pressure by considering relative density and grading index was found.
      Citation: Fractal and Fractional
      PubDate: 2022-06-22
      DOI: 10.3390/fractalfract6070347
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 348: Novel Fractional Swarming with Key Term
           Separation for Input Nonlinear Control Autoregressive Systems

    • Authors: Faisal Altaf, Ching-Lung Chang, Naveed Ishtiaq Chaudhary, Khalid Mehmood Cheema, Muhammad Asif Zahoor Raja, Chi-Min Shu, Ahmad H. Milyani
      First page: 348
      Abstract: In recent decades, fractional order calculus has become an important mathematical tool for effectively solving complex problems through better modeling with the introduction of fractional differential/integral operators; fractional order swarming heuristics are also introduced and applied for better performance in different optimization tasks. This study investigates the nonlinear system identification problem of the input nonlinear control autoregressive (IN-CAR) model through the novel implementation of fractional order particle swarm optimization (FO-PSO) heuristics; further, the key term separation technique (KTST) is introduced in the FO-PSO to solve the over-parameterization issue involved in the parameter estimation of the IN-CAR model. The proposed KTST-based FO-PSO, i.e., KTST-FOPSO accurately estimates the parameters of an unknown IN-CAR system with robust performance in cases of different noise scenarios. The performance of the KTST-FOPSO is investigated exhaustively for different fractional orders as well as in comparison with the standard counterpart. The results of statistical indices through Monte Carlo simulations endorse the reliability and stability of the KTST-FOPSO for IN-CAR identification.
      Citation: Fractal and Fractional
      PubDate: 2022-06-22
      DOI: 10.3390/fractalfract6070348
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 349: Local Discontinuous Galerkin Method
           Coupled with Nonuniform Time Discretizations for Solving the
           Time-Fractional Allen-Cahn Equation

    • Authors: Zhen Wang, Luhan Sun, Jianxiong Cao
      First page: 349
      Abstract: This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α∈(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t→0+ albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results.
      Citation: Fractal and Fractional
      PubDate: 2022-06-22
      DOI: 10.3390/fractalfract6070349
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 350: Asymptotic and Finite-Time
           Synchronization of Fractional-Order Memristor-Based Inertial Neural
           Networks with Time-Varying Delay

    • Authors: Yeguo Sun, Yihong Liu, Lei Liu
      First page: 350
      Abstract: This paper emphasized on studying the asymptotic synchronization and finite synchronization of fractional-order memristor-based inertial neural networks with time-varying latency. The fractional-order memristor-based inertial neural network model is offered as a more general and flexible alternative to the integer-order inertial neural network. By utilizing the properties of fractional calculus, two lemmas on asymptotic stability and finite-time stability are provided. Based on the two lemmas and the constructed Lyapunov functionals, some updated and valid criteria have been developed to achieve asymptotic and finite-time synchronization of the addressed systems. Finally, the effectiveness of the proposed method is demonstrated by a number of examples and simulations.
      Citation: Fractal and Fractional
      PubDate: 2022-06-23
      DOI: 10.3390/fractalfract6070350
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 351: Permeability Prediction of Saturated
           Geomaterials with Revised Pore–Solid Fractal Model and Critical Path
           Analysis

    • Authors: Lei Kou, Wuxue Li, Jujie Wu
      First page: 351
      Abstract: The revised pore–solid fractal (PSF) model is presented by using the largest inscribed circle-based geometries of squares or cubes to replace the original pore or solid subregions as the new pore or solid phase in porous media. The revised PSF model changes the discrete lacunar pore and solid phases in the original PSF model to integrated. Permeability is an intrinsic property of geomaterials and has broad applications in exploring fluid flow and species transport. Based on the revised PSF model and critical path analysis, a fractal model for predicting the permeability of saturated geomaterials is proposed. The permeability prediction model is verified by comparison with the existing predicted model and the laboratory testing. The results show that the predicted permeabilities match the measured values very well. This work provides a theoretical framework for the revised PSF model and its application in predicting the permeability of geomaterials.
      Citation: Fractal and Fractional
      PubDate: 2022-06-23
      DOI: 10.3390/fractalfract6070351
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 352: Radially Symmetric Solution for
           Fractional Laplacian Systems with Different Negative Powers

    • Authors: Haiyong Xu, Bashir Ahmad, Guotao Wang, Lihong Zhang
      First page: 352
      Abstract: By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: (−Δ)α2u(x)+u−γ(x)+v−q(x)=0,x∈RN, (−Δ)β2v(x)+v−σ(x)+u−p(x)=0,x∈RN, u(x)≳ x a,v(x)≳ x bas x →∞, where α,β∈(0,2), and a,b>0 are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers.
      Citation: Fractal and Fractional
      PubDate: 2022-06-23
      DOI: 10.3390/fractalfract6070352
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 353: Fractal Description of Rock Fracture
           Networks Based on the Space Syntax Metric

    • Authors: Sui, Wang, Wu, Zhang, Yu, Ma, Sun
      First page: 353
      Abstract: Fractal characteristics and the fractal dimension are widely used in the description and characterization of rock fracture networks. They are important tools for coal mining, oil and gas transportation, and other engineering problems. However, due to the complexity of rock fracture networks and the difficulty in directly applying the limit definition of the fractal dimension, the definition and application of the fractal dimension have become hot topics in related projects. In this paper, the traditional fractal calculation methods were reviewed. Using the traditional fractal theory and the head/tail breaks method, a new fractal dimension quantization model was established as a simple method of fractal calculation. This simple method of fractal calculation was used to calculate the fractal dimensions of three rock fracture networks. Through comparison with the box-counting dimension calculation results, it was verified that the model could calculate the fractal dimension of the fracture length of rock fracture networks, as well as quantify it accurately and effectively. In addition, we found a number of similarities between rock fracture networks and urban road traffic networks in GIS. The application of the space syntax metric to rock fracture networks prevents controversy with respect to the definition of the axis and it showed a good effect. Using the space syntax metric as a parameter can better reflect the space relationship of rock fractures than length. Through the calculation of the fractal dimension of the connection value and control value, it was found that the trend of the length fractal dimension was the same as that of the control value, whereas the fractal dimension of the connection value was the opposite. This further verifies the applicability of the space syntax metric in rock fracture networks.
      Citation: Fractal and Fractional
      PubDate: 2022-06-23
      DOI: 10.3390/fractalfract6070353
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 354: Multivalent Functions and Differential
           Operator Extended by the Quantum Calculus

    • Authors: Samir B. Hadid, Rabha W. Ibrahim, Shaher Momani
      First page: 354
      Abstract: We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of the proposed functions are geometrically investigated. Our approach was influenced by the theory of differential subordination. As a result, we called attention to a few well-known corollaries of our main conclusions.
      Citation: Fractal and Fractional
      PubDate: 2022-06-24
      DOI: 10.3390/fractalfract6070354
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 355: Fractional-Order Interval Observer for
           Multiagent Nonlinear Systems

    • Authors: Haoran Zhang, Jun Huang, Siyuan He
      First page: 355
      Abstract: A framework of distributed interval observers is introduced for fractional-order multiagent systems in the presence of nonlinearity. First, a frame was designed to construct the upper and lower bounds of the system state. By using monotone system theory, the positivity of the error dynamics could be ensured, which implies that the bounds could trap the original state. Second, a sufficient condition was applied to guarantee the boundedness of distributed interval observers. Then, an extension of Lyapunov function in the fractional calculus field was the basis of the sufficient condition. An algorithm associated with the procedure of the observer design is also provided. Lastly, a numerical simulation is used to demonstrate the effectiveness of the distributed interval observer.
      Citation: Fractal and Fractional
      PubDate: 2022-06-25
      DOI: 10.3390/fractalfract6070355
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 356: Minimization Problems for Functionals
           Depending on Generalized Proportional Fractional Derivatives

    • Authors: Ricardo Almeida
      First page: 356
      Abstract: In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz’s variational problem and when in presence of time delays are also considered.
      Citation: Fractal and Fractional
      PubDate: 2022-06-25
      DOI: 10.3390/fractalfract6070356
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 357: Thermophysical Study of Oldroyd-B Hybrid
           Nanofluid with Sinusoidal Conditions and Permeability: A Prabhakar
           Fractional Approach

    • Authors: Juan Zhang, Ali Raza, Umair Khan, Qasim Ali, Aurang Zaib, Wajaree Weera, Ahmed M. Galal
      First page: 357
      Abstract: The functional implications of substances, such as retardation and relaxation, can be studied for magnetized diffusion coefficient based on the relative increase throughout magnetization is a well-known realization. In this context, we have explored the Oldroyd-B hybrid nanofluid flowing through a pored oscillating plate along with an inclined applied magnetics effect. The slipping effect and sinusoidal heating conditions are also supposed to be under consideration. An innovative and current classification of fractional derivatives, i.e., Prabhakar fractional derivative and Laplace transform, are implemented for the result of transformed leading equations. The graphical representation is also described to understand the physical implementation of all effecting parameters. In order to justify and physically examine the considered problem, some limiting cases, the rate of heat and mass transfer, and friction factors are also analyzed. As a result, we have concluded that the thermal enhancement can be improved more progressively with the interaction of silver-water-based nanofluid suspension compared to copper-nanoparticles mixed nanofluid. Furthermore, It has examined the impact of both parameters, i.e., time relaxation and retardation is opposite of the momentum field.
      Citation: Fractal and Fractional
      PubDate: 2022-06-26
      DOI: 10.3390/fractalfract6070357
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 358: Solution of the Ill-Posed Cauchy Problem
           for Systems of Elliptic Type of the First Order

    • Authors: Davron Aslonqulovich Juraev, Ali Shokri, Daniela Marian
      First page: 358
      Abstract: We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space Rm. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of the solutions.
      Citation: Fractal and Fractional
      PubDate: 2022-06-26
      DOI: 10.3390/fractalfract6070358
      Issue No: Vol. 6, No. 7 (2022)
       
  • Fractal Fract, Vol. 6, Pages 282: A New Method of Quantifying the
           Complexity of Fractal Networks

    • Authors: Matej Babič, Dragan Marinković, Miha Kovačič, Branko Šter, Michele Calì
      First page: 282
      Abstract: There is a large body of research devoted to identifying the complexity of structures in networks. In the context of network theory, a complex network is a graph with nontrivial topological features—features that do not occur in simple networks, such as lattices or random graphs, but often occur in graphs modeling real systems. The study of complex networks is a young and active area of scientific research inspired largely by the empirical study of real-world networks, such as computer networks and logistic transport networks. Transport is of great importance for the economic and cultural cooperation of any country with other countries, the strengthening and development of the economic management system, and in solving social and economic problems. Provision of the territory with a well-developed transport system is one of the factors for attracting population and production, serving as an important advantage for locating productive forces and providing an integration effect. In this paper, we introduce a new method for quantifying the complexity of a network based on presenting the nodes of the network in Cartesian coordinates, converting to polar coordinates, and calculating the fractal dimension using the ReScaled ranged (R/S) method. Our results suggest that this approach can be used to determine complexity for any type of network that has fixed nodes, and it presents an application of this method in the public transport system.
      Citation: Fractal and Fractional
      PubDate: 2022-05-24
      DOI: 10.3390/fractalfract6060282
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 283: Oscillators Based on Fractional-Order
           Memory Elements

    • Authors: Ivo Petráš
      First page: 283
      Abstract: This paper deals with the new oscillator structures that contain new elements, so-called memory elements, known as memristor, meminductor, and memcapacitor. Such circuits can exhibit oscillations as well as chaotic behavior. New mathematical models of fractional-order elements and whole oscillator circuits are proposed as well. An illustrative example to demonstrate the oscillations and the chaotic behavior through the numerical solution of the fractional-order circuit model is provided.
      Citation: Fractal and Fractional
      PubDate: 2022-05-24
      DOI: 10.3390/fractalfract6060283
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 284: Editorial for Special Issue “New
           Advancements in Pure and Applied Mathematics via Fractals and Fractional
           Calculus”

    • Authors: Asifa Tassaddiq, Muhammad Yaseen
      First page: 284
      Abstract: Fractional calculus has reshaped science and technology since its first appearance in a letter received to Gottfried Wilhelm Leibniz from Guil-laume de l’Hôpital in the year 1695 [...]
      Citation: Fractal and Fractional
      PubDate: 2022-05-25
      DOI: 10.3390/fractalfract6060284
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 285: Existence and Stability Results for a
           Tripled System of the Caputo Type with Multi-Point and Integral Boundary
           Conditions

    • Authors: Murugesan Manigandan, Muthaiah Subramanian, Thangaraj Nandha Gopal, Bundit Unyong
      First page: 285
      Abstract: In this paper, we introduce and investigate the existence and stability of a tripled system of sequential fractional differential equations (SFDEs) with multi-point and integral boundary conditions. The existence and uniqueness of the solutions are established by the principle of Banach’s contraction and the alternative of Leray–Schauder. The stability of the Hyer–Ulam solutions are investigated. A few examples are provided to identify the major results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060285
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 286: A New Homotopy Transformation Method for
           Solving the Fuzzy Fractional Black–Scholes European Option Pricing
           Equations under the Concept of Granular Differentiability

    • Authors: Jianke Zhang, Yueyue Wang, Sumei Zhang
      First page: 286
      Abstract: The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating in the actual financial market option pricing, such as risk-free interest rate, stock price, option underlying price, and security price volatility may be inaccurate in the real world. Therefore, it is of great practical significance to study the fractional fuzzy option pricing model. In this paper, we proposed a reliable approximation method, the Elzaki transform homotopy perturbation method (ETHPM) based on granular differentiability, to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Firstly, the fuzzy function is converted to a real number function based on the horizontal membership function (HMF). Secondly, the specific steps of the ETHPM are given to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Finally, some examples demonstrate that the new approach is simple, efficient, and accurate. In addition, the fuzzy approximation solutions have been visualized at the end of this paper.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060286
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 287: A Bi-Geometric Fractional Model for the
           Treatment of Cancer Using Radiotherapy

    • Authors: Mohammad Momenzadeh, Olivia Ada Obi, Evren Hincal
      First page: 287
      Abstract: Our study is based on the modification of a well-known predator-prey equation, or the Lotka–Volterra competition model. That is, a system of differential equations was established for the population of healthy and cancerous cells within the tumor tissue of a patient struggling with cancer. Besides, fractional differentiation remedies the situation by obtaining a meticulous model with more flexible parameters. Furthermore, a specific type of non-Newtonian calculus, bi-geometric calculus, can describe the model in terms of proportions and implies the alternative aspect of a dynamic system. Moreover, fractional operators in bi-geometric calculus are formulated in terms of Hadamard fractional operators. In this article, the development of fractional operators in non-Newtonian calculus was investigated. The model was extended in these criteria, and the existence and uniqueness of the model were considered and guaranteed in the first step by applying the Arzelà–Ascoli. The bi-geometric analogue of the numerical method provided a suitable tool to solve the model approximately. In the end, the visual graphs were obtained by using the MATLAB program.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060287
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 288: Modification of the Optimal Auxiliary
           Function Method for Solving Fractional Order KdV Equations

    • Authors: Hakeem Ullah, Mehreen Fiza, Ilyas Khan, Nawa Alshammari, Nawaf N. Hamadneh, Saeed Islam
      First page: 288
      Abstract: In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. A rapidly convergent series solution is obtained from the FOAFM and is validated by comparing with other results. The analysis proves that our method is simplified and applicable, contains less computational work, and has fast convergence. The beauty of this method is that there is no need to assume a small parameter such as in the perturbation method. The effectiveness and accuracy of the method is proven by numerical and graphical results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060288
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 289: Finite Time Stability of Fractional
           Order Systems of Neutral Type

    • Authors: Abdellatif Ben Makhlouf, Dumitru Baleanu
      First page: 289
      Abstract: This work deals with a new finite time stability (FTS) of neutral fractional order systems with time delay (NFOTSs). In light of this, FTSs of NFOTSs are demonstrated in the literature using the Gronwall inequality. The innovative aspect of our proposed study is the application of fixed point theory to show the FTS of NFOTSs. Finally, using two examples, the theoretical contributions are confirmed and substantiated.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060289
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 290: On a Nonlinear Fractional Langevin
           Equation of Two Fractional Orders with a Multiplicative Noise

    • Authors: McSylvester Ejighikeme Omaba, Eze R. Nwaeze
      First page: 290
      Abstract: We consider a stochastic nonlinear fractional Langevin equation of two fractional orders Dβ(Dα+γ)ψ(t)=λϑ(t,ψ(t))w˙(t),0<t≤1. Given some suitable conditions on the above parameters, we prove the existence and uniqueness of the mild solution to the initial value problem for the stochastic nonlinear fractional Langevin equation using Banach fixed-point theorem (Contraction mapping theorem). The upper bound estimate for the second moment of the mild solution is given, which shows exponential growth in time t at a precise rate of 3c1expc3t2(α+β)−1+c4t2α−1 on the parameters α>1 and α+β>1 for some positive constants c1,c3 and c4.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060290
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 291: The Sharp Bounds of Hankel Determinants
           for the Families of Three-Leaf-Type Analytic Functions

    • Authors: Muhammad Arif, Omar Mohammed Barukab, Sher Afzal Khan, Muhammad Abbas
      First page: 291
      Abstract: The theory of univalent functions has shown strong significance in the field of mathematics. It is such a vast and fully applied topic that its applications exist in nearly every field of applied sciences such as nonlinear integrable system theory, fluid dynamics, modern mathematical physics, the theory of partial differential equations, engineering, and electronics. In our present investigation, two subfamilies of starlike and bounded turning functions associated with a three-leaf-shaped domain were considered. These classes are denoted by BT3l and S3l*, respectively. For the class BT3l, we study various coefficient type problems such as the first four initial coefficients, the Fekete–Szegö and Zalcman type inequalities and the third-order Hankel determinant. Furthermore, the existing third-order Hankel determinant bounds for the second class will be improved here. All the results that we are going to prove are sharp.
      Citation: Fractal and Fractional
      PubDate: 2022-05-26
      DOI: 10.3390/fractalfract6060291
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 292: Tychonoff Solutions of the
           Time-Fractional Heat Equation

    • Authors: Giacomo Ascione
      First page: 292
      Abstract: In the literature, one can find several applications of the time-fractional heat equation, particularly in the context of time-changed stochastic processes. Stochastic representation results for such an equation can be used to provide a Monte Carlo simulation method, upon proving that the solution is actually unique. In the classical case, however, this is not true if we do not consider any additional assumption, showing, thus, that the Monte Carlo simulation method identifies only a particular solution. In this paper, we consider the problem of the uniqueness of the solutions of the time-fractional heat equation with initial data. Precisely, under suitable assumptions about the regularity of the initial datum, we prove that such an equation admits an infinity of classical solutions. The proof mimics the construction of the Tychonoff solutions of the classical heat equation. As a consequence, one has to add some addtional conditions to the time-fractional Cauchy problem to ensure the uniqueness of the solution.
      Citation: Fractal and Fractional
      PubDate: 2022-05-27
      DOI: 10.3390/fractalfract6060292
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 293: Fractional View Analysis of
           Cahn–Allen Equations by New Iterative Transform Method

    • Authors: Liaqat Ali, Rasool Shah, Wajaree Weera
      First page: 293
      Abstract: In this article, the new iterative transform method is applied to evaluate the time-fractional Cahn–Allen model solution. In this technique, Elzaki transformation is a mixture of the new iteration technique. Two problems are studied to demonstrate and confirm the accuracy of the proposed technique. The current technique’s mathematical analysis showed that the method is simple to understand and reliable. These solutions indicate that the proposed technique is advantageous and simple to apply in science and engineering problems.
      Citation: Fractal and Fractional
      PubDate: 2022-05-27
      DOI: 10.3390/fractalfract6060293
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 294: A New Fifth-Order Finite Difference
           Compact Reconstruction Unequal-Sized WENO Scheme for Fractional
           Differential Equations

    • Authors: Yan Zhang, Jun Zhu
      First page: 294
      Abstract: This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain nonoscillatory properties near discontinuities. The fractional Laplacian operator of order β(0<β<1) is split into the integral part and the first derivative term. Using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new finite difference CRUS-WENO scheme is presented to discretize the first derivative term of the fractional equation. This new CRUS-WENO scheme has the advantages of a narrower large stencil and high spectral resolution. In addition, the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Some numerical examples are given to show the effectiveness and feasibility of this new CRUS-WENO scheme in solving fractional equations containing the fractional Laplacian operator.
      Citation: Fractal and Fractional
      PubDate: 2022-05-27
      DOI: 10.3390/fractalfract6060294
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 295: Cluster Oscillation of a
           Fractional-Order Duffing System with Slow Variable Parameter Excitation

    • Authors: Xianghong Li, Yanli Wang, Yongjun Shen
      First page: 295
      Abstract: The complicated dynamic behavior of a fractional-order Duffing system with slow variable parameter excitation is investigated. The stability and bifurcation behavior of the fast subsystem are analyzed by using the dynamic theory of fractional-order systems. The pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation are discussed in detail, and it was found that Hopf bifurcation only happens while the fractional order is bigger than 1. On the other hand, the influence of the amplitude of parametric excitation on cluster oscillation models is discussed. The results show that amplitude regulates cluster oscillation models with different bifurcation types. The point–point cluster oscillation only relates to pitchfork bifurcation. The point–cycle cluster oscillation includes pitchfork bifurcation and Hopf bifurcation. The point–cycle–cycle cluster oscillation involves three kinds of bifurcation, i.e., the pitchfork bifurcation, Hopf bifurcation and limit cycle bifurcation. The larger the amplitude, the more bifurcation types are involved. The research results of cluster oscillation and its generation mechanism will provide valuable theoretical basis for mechanical manufacturing and engineering practice.
      Citation: Fractal and Fractional
      PubDate: 2022-05-28
      DOI: 10.3390/fractalfract6060295
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 296: Properties of q-Differential Equations
           of Higher Order and Visualization of Fractal Using q-Bernoulli Polynomials
           

    • Authors: Cheon-Seoung Ryoo, Jung-Yoog Kang
      First page: 296
      Abstract: We introduce several q-differential equations of higher order which are related to q-Bernoulli polynomials and obtain a symmetric property of q-differential equations of higher order in this paper. By giving q-varying variations, we identify the shape of the approximate roots of q-Bernoulli polynomials, a solution of q-differential equations of higher order, and find several conjectures associated with them. Furthermore, based on q-Bernoulli polynomials, we create a Mandelbrot set and a Julia set to find a variety of related figures.
      Citation: Fractal and Fractional
      PubDate: 2022-05-28
      DOI: 10.3390/fractalfract6060296
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 297: Dirichlet Averages of Generalized
           Mittag-Leffler Type Function

    • Authors: Dinesh Kumar, Jeta Ram, Junesang Choi
      First page: 297
      Abstract: Since Gösta Magus Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903 and studied its features in five subsequent notes, passing the first half of the 20th century during which the majority of scientists remained almost unaware of the function, the Mittag-Leffler function and its various extensions (referred to as Mittag-Leffler type functions) have been researched and applied to a wide range of problems in physics, biology, chemistry, and engineering. In the context of fractional calculus, Mittag-Leffler type functions have been widely studied. Since Carlson established the notion of Dirichlet average and its different variations, these averages have been explored and used in a variety of fields. This paper aims to investigate the Dirichlet and modified Dirichlet averages of the R-function (an extended Mittag-Leffler type function), which are provided in terms of Riemann-Liouville integrals and hypergeometric functions of several variables. Principal findings in this article are (possibly) applicable. This article concludes by addressing an open problem.
      Citation: Fractal and Fractional
      PubDate: 2022-05-28
      DOI: 10.3390/fractalfract6060297
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 298: Finite-Time Projective Synchronization
           and Parameter Identification of Fractional-Order Complex Networks with
           Unknown External Disturbances

    • Authors: Shuguo Wang, Song Zheng, Linxiang Cui
      First page: 298
      Abstract: This paper is devoted to exploring the finite-time projective synchronization (FTPS) of fractional-order complex dynamical networks (FOCDNs) with unknown parameters and external disturbances. Based on the stability theory of fractional-order differential systems, synchronization criteria between drive-response networks were obtained and both the uncertain parameters and external disturbances were identified or conquered simultaneously. Moreover, the upper limit of the settling-time function was obtained. Finally, a numerical example was given to verify the effectiveness of the results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-29
      DOI: 10.3390/fractalfract6060298
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 299: On a System of Riemann–Liouville
           Fractional Boundary Value Problems with ϱ-Laplacian Operators and
           Positive Parameters

    • Authors: Johnny Henderson, Rodica Luca, Alexandru Tudorache
      First page: 299
      Abstract: In this paper, we study the existence and nonexistence of positive solutions of a system of Riemann–Liouville fractional differential equations with ϱ-Laplacian operators, supplemented with coupled nonlocal boundary conditions containing Riemann–Stieltjes integrals, fractional derivatives of various orders, and positive parameters. We apply the Schauder fixed point theorem in the proof of the existence result.
      Citation: Fractal and Fractional
      PubDate: 2022-05-29
      DOI: 10.3390/fractalfract6060299
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 300: Series with Binomial-like Coefficients
           for the Investigation of Fractal Structures Associated with the Riemann
           Zeta Function

    • Authors: Igoris Belovas, Martynas Sabaliauskas, Lukas Kuzma
      First page: 300
      Abstract: The paper continues the study of efficient algorithms for the computation of zeta functions over the complex plane. We aim to apply the modifications of algorithms to the investigation of underlying fractal structures associated with the Riemann zeta function. We discuss the computational complexity and numerical aspects of the implemented algorithms based on series with binomial-like coefficients.
      Citation: Fractal and Fractional
      PubDate: 2022-05-29
      DOI: 10.3390/fractalfract6060300
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 301: Fractional Integral Inequalities of
           Hermite–Hadamard Type for (h,g;m)-Convex Functions with Extended
           Mittag-Leffler Function

    • Authors: Maja Andrić
      First page: 301
      Abstract: Several fractional integral inequalities of the Hermite–Hadamard type are presented for the class of (h,g;m)-convex functions. Applied fractional integral operators contain extended generalized Mittag-Leffler functions as their kernel, thus enabling new fractional integral inequalities that extend and generalize the known results. As an application, the upper bounds of fractional integral operators for (h,g;m)-convex functions are given.
      Citation: Fractal and Fractional
      PubDate: 2022-05-29
      DOI: 10.3390/fractalfract6060301
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 302: A Visually Secure Image Encryption Based
           on the Fractional Lorenz System and Compressive Sensing

    • Authors: Hua Ren, Shaozhang Niu, Jiajun Chen, Ming Li, Zhen Yue
      First page: 302
      Abstract: Recently, generating visually secure cipher images by compressive sensing (CS) techniques has drawn much attention among researchers. However, most of these algorithms generate cipher images based on direct bit substitution and the underlying relationship between the hidden and modified data is not considered, which reduces the visual security of cipher images. In addition, performing CS on plain images directly is inefficient, and CS decryption quality is not high enough. Thus, we design a novel cryptosystem by introducing vector quantization (VQ) into CS-based encryption based on a 3D fractional Lorenz chaotic system. In our work, CS compresses only the sparser error matrix generated from the plain and VQ images in the secret generation phase, which improves CS compression performance and the quality of decrypted images. In addition, a smooth function is used in the embedding phase to find the underlying relationship and determine relatively suitable modifiable values for the carrier image. All the secret streams are produced by updating the initial values and control parameters from the fractional chaotic system, and then utilized in CS, diffusion, and embedding. Simulation results demonstrate the effectiveness of the proposed method.
      Citation: Fractal and Fractional
      PubDate: 2022-05-29
      DOI: 10.3390/fractalfract6060302
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 303: Stationary Wong–Zakai
           Approximation of Fractional Brownian Motion and Stochastic Differential
           Equations with Noise Perturbations

    • Authors: Lauri Viitasaari, Caibin Zeng
      First page: 303
      Abstract: In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H∈(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions.
      Citation: Fractal and Fractional
      PubDate: 2022-05-30
      DOI: 10.3390/fractalfract6060303
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 304: An Efficient Numerical Simulation for
           the Fractional COVID-19 Model Using the GRK4M Together with the Fractional
           FDM

    • Authors: Yasser Ibrahim, Mohamed Khader, Ahmed Megahed, Fawzy Abd El-Salam, Mohamed Adel
      First page: 304
      Abstract: In this research, we studied a mathematical model formulated with six fractional differential equations to characterize a COVID-19 outbreak. For the past two years, the disease transmission has been increasing all over the world. We included the considerations of people with infections who were both asymptomatic and symptomatic as well as the fact that an individual who has been exposed is either quarantined or moved to one of the diseased classes, with the chance that a susceptible individual could also migrate to the quarantined class. The suggested model is solved numerically by implementing the generalized Runge–Kutta method of the fourth order (GRK4M). We discuss the stability analysis of the GRK4M as a general study. The acquired findings are compared with those obtained using the fractional finite difference method (FDM), where we used the Grünwald–Letnikov approach to discretize the fractional differentiation operator. The FDM is mostly reliant on correctly converting the suggested model into a system of algebraic equations. By applying the proposed methods, the numerical results reveal that these methods are straightforward to apply and computationally very effective at presenting a numerical simulation of the behavior of all components of the model under study.
      Citation: Fractal and Fractional
      PubDate: 2022-05-31
      DOI: 10.3390/fractalfract6060304
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 305: Contribution of Using Hadamard
           Fractional Integral Operator via Mellin Integral Transform for Solving
           Certain Fractional Kinetic Matrix Equations

    • Authors: Mohamed Abdalla, Mohamed Akel
      First page: 305
      Abstract: Recently, the importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in electrodynamics, control systems, economic, physics, geophysics and hydrodynamics. Among the many fractional differential equations are kinetic equations. Fractional-order kinetic Equations (FOKEs) are a unifying tool for the description of load vector behavior in disorderly media. In this article, we employ the Hadamard fractional integral operator via Mellin integral transform to establish the generalization of some fractional-order kinetic equations including extended (k,τ)-Gauss hypergeometric matrix functions. Solutions to certain fractional-order kinetic matrix Equations (FOKMEs) involving extended (k,τ)-Gauss hypergeometric matrix functions are also introduced. Moreover, several special cases of our main results are archived.
      Citation: Fractal and Fractional
      PubDate: 2022-05-31
      DOI: 10.3390/fractalfract6060305
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 306: Hidden and Coexisting Attractors in a
           Novel 4D Hyperchaotic System with No Equilibrium Point

    • Authors: Chengwei Dong, Jiahui Wang
      First page: 306
      Abstract: The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-31
      DOI: 10.3390/fractalfract6060306
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 307: Existence of Solutions and Relative
           Controllability of a Stochastic System with Nonpermutable Matrix
           Coefficients

    • Authors: Kinda Abuasbeh, Nazim I. Mahmudov, Muath Awadalla
      First page: 307
      Abstract: In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for the relative approximate controllability of time-delayed linear stochastic dynamical systems. In addition, by applying the Banach fixed point theorem, we establish some sufficient relative approximate controllability conditions for semilinear time-delayed stochastic differential systems. Finally, concrete examples are given to illustrate the main results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-31
      DOI: 10.3390/fractalfract6060307
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 308: On the Solvability of Some Boundary
           Value Problems for the Nonlocal Poisson Equation with Boundary Operators
           of Fractional Order

    • Authors: Kairat Usmanov, Batirkhan Turmetov, Kulzina Nazarova
      First page: 308
      Abstract: In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given.
      Citation: Fractal and Fractional
      PubDate: 2022-05-31
      DOI: 10.3390/fractalfract6060308
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 309: Hermite–Hadamard-Type Inequalities
           for h-Convex Functions Involving New Fractional Integral Operators with
           Exponential Kernel

    • Authors: Yaoqun Wu
      First page: 309
      Abstract: In this paper, we use two new fractional integral operators with exponential kernel about the midpoint of the interval to construct some Hermite–Hadamard type fractional integral inequalities for h-convex functions. Taking two integral identities about the first and second derivatives of the function as auxiliary functions, the main results are obtained by using the properties of h-convexity and the module. In order to illustrate the application of the results, we propose four examples and plot function images to intuitively present the meaning of the inequalities in the main results, and we verify the correctness of the conclusion. This study further expands the generalization of Hermite–Hadamard-type inequalities and provides some research references for the study of Hermite–Hadamard-type inequalities.
      Citation: Fractal and Fractional
      PubDate: 2022-06-01
      DOI: 10.3390/fractalfract6060309
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 310: Asymptotic Autonomy of Attractors for
           Stochastic Fractional Nonclassical Diffusion Equations Driven by a
           Wong–Zakai Approximation Process on Rn

    • Authors: Hong Li, Fuzhi Li
      First page: 310
      Abstract: In this paper, we consider the backward asymptotically autonomous dynamical behavior for fractional non-autonomous nonclassical diffusion equations driven by a Wong–Zakai approximations process in Hs(Rn) with s∈(0,1). We first prove the existence and backward time-dependent uniform compactness of tempered pullback random attractors when the growth rate of nonlinearities have a subcritical range. We then show that, under the Wong–Zakai approximations process, the components of the random attractors of a non-autonomous dynamical system in time can converge to those of the random attractor of the limiting autonomous dynamical system in Hs(Rn).
      Citation: Fractal and Fractional
      PubDate: 2022-06-01
      DOI: 10.3390/fractalfract6060310
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 311: Galerkin Approximation for Stochastic
           Volterra Integral Equations with Doubly Singular Kernels

    • Authors: Yuyuan Li, Wanqing Song, Yanan Jiang, Aleksey Kudreyko
      First page: 311
      Abstract: This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0<α1+β1<1, 0<α2+β2<12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.
      Citation: Fractal and Fractional
      PubDate: 2022-06-01
      DOI: 10.3390/fractalfract6060311
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 312: Regularization for a Sideways Problem of
           the Non-Homogeneous Fractional Diffusion Equation

    • Authors: Yonggang Chen, Yu Qiao, Xiangtuan Xiong
      First page: 312
      Abstract: In this article, we investigate a sideways problem of the non-homogeneous time-fractional diffusion equation, which is highly ill-posed. Such a model is obtained from the classical non-homogeneous sideways heat equation by replacing the first-order time derivative by the Caputo fractional derivative. We achieve the result of conditional stability under an a priori assumption. Two regularization strategies, based on the truncation of high frequency components, are constructed for solving the inverse problem in the presence of noisy data, and the corresponding error estimates are proved.
      Citation: Fractal and Fractional
      PubDate: 2022-06-02
      DOI: 10.3390/fractalfract6060312
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 313: Even-Order Neutral Delay Differential
           Equations with Noncanonical Operator: New Oscillation Criteria

    • Authors: Osama Moaaz, Barakah Almarri, Fahd Masood, Doaa Atta
      First page: 313
      Abstract: The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and reflection of developments in the study of second-order equations. We also derive criteria for improving conditions that exclude the decreasing positive solutions of the considered equation.
      Citation: Fractal and Fractional
      PubDate: 2022-06-02
      DOI: 10.3390/fractalfract6060313
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 314: A Higher-Order Numerical Scheme for
           Two-Dimensional Nonlinear Fractional Volterra Integral Equations with
           Uniform Accuracy

    • Authors: Zi-Qiang Wang, Qin Liu, Jun-Ying Cao
      First page: 314
      Abstract: In this paper, based on the modified block-by-block method, we propose a higher-order numerical scheme for two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. This approach involves discretizing the domain into a large number of subdomains and using biquadratic Lagrangian interpolation on each subdomain. The convergence of the high-order numerical scheme is rigorously established. We prove that the numerical solution converges to the exact solution with the optimal convergence order O(hx4−α+hy4−β) for 0<α,β<1. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method.
      Citation: Fractal and Fractional
      PubDate: 2022-06-02
      DOI: 10.3390/fractalfract6060314
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 315: Existence of Positive Solutions for a
           Singular Second-Order Changing-Sign Differential Equation on Time Scales

    • Authors: Hui Tian, Xinguang Zhang, Yonghong Wu, Benchawan Wiwatanapataphee
      First page: 315
      Abstract: In this paper, we focus on the existence of positive solutions for a boundary value problem of the changing-sign differential equation on time scales. By constructing a translation transformation and combining with the properties of the solution of the nonhomogeneous boundary value problem, we transfer the changing-sign problem to a positone problem, then by means of the known fixed-point theorem, several sufficient conditions for the existence of positive solutions are established for the case in which the nonlinear term of the equation may change sign.
      Citation: Fractal and Fractional
      PubDate: 2022-06-03
      DOI: 10.3390/fractalfract6060315
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 316: Bazilevič Functions of Complex
           Order with Respect to Symmetric Points

    • Authors: Daniel Breaz, Kadhavoor R. Karthikeyan, Gangadharan Murugusundaramoorthy
      First page: 316
      Abstract: In this paper, we familiarize a class of multivalent functions with respect to symmetric points related to the differential operator and discuss the impact of Janowski functions on conic regions. Inclusion results, the subordination property, and coefficient inequalities are obtained. Further, the applications of our results that are extensions of those given in earlier works are presented as corollaries.
      Citation: Fractal and Fractional
      PubDate: 2022-06-05
      DOI: 10.3390/fractalfract6060316
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 317: Effect of Heterogeneity on the Extension
           of Ubiquitiformal Cracks in Rock Materials

    • Authors: Beibei Yang, Xiaoshan Cao, Tielin Han, Panfeng Li, Junping Shi
      First page: 317
      Abstract: Fracture energy, as an important characteristic parameter of the fracture properties of materials, has been extensively studied by scholars. However, less research has been carried out on ubiquitiformal fracture energy and the main method used by scholars is the uniaxial tensile test. In this paper, based on previous research, the first Brazilian splitting test was used to study the ubiquitiformal crack extension of slate and granite, and the complexity and ubiquitiformal fracture energy of rock material were obtained. The heterogeneity of the material was then characterized by the Weibull statistical distribution, and the cohesive model is applied to the ABAQUS numerical software to simulate the effect of heterogeneity on the characteristics of ubiquitiformal cracks. The results demonstrate that the ubiquitiformal complexity of slate ranges from 1.54 to 1.60, and that of granite ranges from 1.58 to 1.62. The mean squared deviations of the slate and granite ubiquitiformal fracture energy are the smallest compared with the other fracture energies, which are 0.038 and 0.037, respectively. When the homogeneity of the heterogeneous model is less than 1.5, its heterogeneity has a greater influence on the Brazilian splitting strength, and the heterogeneity of the rock is obvious. However, when the homogeneity is greater than five, the effect on the Brazilian splitting strength is much less, and the Brazilian splitting strength tends to be the average strength. Therefore, it is particularly important to study the fracture problem of cracks from the nature of the material structure by combining the macroscopic and mesoscopic views through the ubiquitiform theory.
      Citation: Fractal and Fractional
      PubDate: 2022-06-05
      DOI: 10.3390/fractalfract6060317
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 318: Neutrosophic Double Controlled Metric
           Spaces and Related Results with Application

    • Authors: Fahim Uddin, Umar Ishtiaq, Aftab Hussain, Khalil Javed, Hamed Al Sulami, Khalil Ahmed
      First page: 318
      Abstract: In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result.
      Citation: Fractal and Fractional
      PubDate: 2022-06-06
      DOI: 10.3390/fractalfract6060318
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 319: Sensitivity of Uniformly Convergent
           Mapping Sequences in Non-Autonomous Discrete Dynamical Systems

    • Authors: Yongxi Jiang, Xiaofang Yang, Tianxiu Lu
      First page: 319
      Abstract: Let H be a compact metric space. The metric of H is denoted by d. And let (H,f1,∞) be a non-autonomous discrete system where f1,∞={fn}n=1∞ is a mapping sequence. This paper discusses infinite sensitivity, m-sensitivity, and m-cofinitely sensitivity of f1,∞. It is proved that, if fn(n∈N) are feebly open and uniformly converge to f:H→H, fi∘f=f∘fi for any i∈{1,2,…}, and ∑i=1∞D(fi,f)<∞, then (H,f) has the above sensitive property if and only if (H,f1,∞) has the same property where D(·,·) is the supremum metric.
      Citation: Fractal and Fractional
      PubDate: 2022-06-07
      DOI: 10.3390/fractalfract6060319
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 320: Correlating Morphology and Multifractal
           Spatial Patterns of the Leaf Surface Architecture of Anacardium
           occidentale L.

    • Authors: Glenda Quaresma Ramos, Robert Saraiva Matos, Abhijeet Das, Sanjeev Kumar, Ştefan Ţălu, Henrique Duarte da Fonseca Filho
      First page: 320
      Abstract: Plant leaf surfaces can contain interesting, reproducible spatial patterns that can be used for several industrial purposes. In this paper, the main goal was to analyze the surface microtexture of Amazon Anacardium occidentale L. using multifractal theory. AFM images were used to evaluate the multifractal spatial surface patterns of the adaxial and abaxial sides of the leaf. The 3D maps revealed that the abaxial side is dominated by stomach cells, while striated structures were observed on the adaxial side. The surface of the abaxial side is rougher than the adaxial side. The autocorrelation function calculations showed that the abaxial side has an isotropic surface compared to the adaxial side. Despite this, Minkowski functionals demonstrated that the morphological spatial patterns have robust statistical similarity. Both sides exhibit multifractal behavior, which was verified by the trend observed in the mass exponent and generalized dimension. However, the adaxial side exhibits stronger multifractality and increased vertical complexity compared to the abaxial side. Our findings show that the multifractal spatial patterns of the leaf surface depend on the rough dynamics of the topographic profile. The identification of the multifractal patterns of the structures present on the surface of plant leaves is useful for the fabrication of leaf-architecture-based materials.
      Citation: Fractal and Fractional
      PubDate: 2022-06-07
      DOI: 10.3390/fractalfract6060320
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 321: Study of Lagrange Points in the
           Earth–Moon System with Continuation Fractional Potential

    • Authors: Lata Kumari Bairwa, Ashok Kumar Pal, Reena Kumari, Sawsan Alhowaity, Elbaz I. Abouelmagd
      First page: 321
      Abstract: In this work, the restricted three-body system is studied in the framework of the continuation fractional potential with its application on the Earth–Moon system. With the help of a numerical technique, we obtained thirteen equilibrium points, such that nine of them are collinear while the remaining four are non-collinear points. We found that the collinear points near the smaller primary were shifted outward from the Moon, whereas the points near the bigger primary were shifted towards the Earth as the value of the continuation fractional parameter increased. We analyzed the zero-velocity curves and discussed the perturbation of the continuation fractional potential effect on the possible regions of the motion. We also discussed the linear stability of all the equilibrium points and found that out of thirteen only two were stable. Due to such a prevalence, the continuation fractional potential is a source of significant perturbation, which embodies the lack of sphericity of the body in the restricted three-body problem
      Citation: Fractal and Fractional
      PubDate: 2022-06-08
      DOI: 10.3390/fractalfract6060321
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 322: The Oscillatory Flow of Oldroyd-B Fluid
           with Magnetic Disturbance

    • Authors: Pujie Yue, Chunying Ming
      First page: 322
      Abstract: The magnetic field intensity will be nondeterminacy with the flow of charged particles thrown out by solar activities, the overlap of adjacent magnetic islands or non-axisymmetric magnetic interference in tokamaks and so on. The model of a generalized Oldroyd-B fluid with fractional derivative under oscillating pressure gradient and magnetic field with some disturbance will be considered in this paper. The disturbance is regarded as the background noise of the system, and the model is described by a fractional stochastic differential equation. Time and space are discretized by L1, L2 schemes based on piecewise linear interpolation and the central difference quotient method. We demonstrate the effects of the amplitude and period of the oscillating pressure gradient, magnetic parameter, fractional parameters and noise on the velocity field, and two special cases are given.
      Citation: Fractal and Fractional
      PubDate: 2022-06-08
      DOI: 10.3390/fractalfract6060322
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 323: Dynamical Behaviors of an Environmental
           Protection Expenses Model in Protected Areas with Two Delays

    • Authors: Jun He, Ping Yang, Jinde Cao
      First page: 323
      Abstract: This paper investigates an environmental protection expenses model, which considers the relations between the visitors to the protected areas V, the quality of the environmental resource E, and the capital stock K. In this model, the total tourism income is used partly to increase the capital stock or as the environmental protection expenses. Two time delays are introduced into the number of visitors, since the visitors need time to respond the changes of the environment, and the environment will take time to respond to the input of money. Stability crossing curves in the plane of delays (τ1,τ2) are used to obtain the stable region of equilibrium. Numerical simulations represent the mutual transformation of the supercritical bifurcation and the subcritical bifurcation. Our model shows that under some parameter conditions, the share of tourism income η is related closely to the delay τ1, while the capital stock and the environmental quality can be maintained persistently if the delay τ1 is not too large.
      Citation: Fractal and Fractional
      PubDate: 2022-06-08
      DOI: 10.3390/fractalfract6060323
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 324: Generalized Fractional Integral
           Inequalities for p-Convex Fuzzy Interval-Valued Mappings

    • Authors: Muhammad Bilal Khan, Adriana Cătaș, Tareq Saeed
      First page: 324
      Abstract: The fuzzy order relation ≽ and fuzzy inclusion relation ⊇ are two different relations in fuzzy-interval calculus. Due to the importance of p-convexity, in this article we consider the introduced class of nonconvex fuzzy-interval-valued mappings known as p-convex fuzzy-interval-valued mappings (p-convex f-i-v-ms) through fuzzy order relation. With the support of a fuzzy generalized fractional operator, we establish a relationship between p-convex f-i-v-ms and Hermite–Hadamard (ℋ–ℋ) inequalities. Moreover, some related ℋ–ℋ inequalities are also derived by using fuzzy generalized fractional operators. Furthermore, we show that our conclusions cover a broad range of new and well-known inequalities for p-convex f-i-v-ms, as well as their variant forms as special instances. The theory proposed in this research is shown, with practical examples that demonstrate its usefulness. These findings and alternative methodologies may pave the way for future research in fuzzy optimization, modeling, and interval-valued mappings (i-v-m).
      Citation: Fractal and Fractional
      PubDate: 2022-06-09
      DOI: 10.3390/fractalfract6060324
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 325: Experimental Investigation of the
           Relationship between Surface Crack of Concrete Cover and Corrosion Degree
           of Steel Bar Using Fractal Theory

    • Authors: Weiwen Li, Meizhong Wu, Tiansheng Shi, Pengfei Yang, Zejie Pan, Wei Liu, Jun Liu, Xu Yang
      First page: 325
      Abstract: Conventionally, crack width is used to assess the corrosion level, whereas other important characteristics such as the variation in crack width at different locations on the surface are disregarded. These important characteristics of surface crack can be described comprehensively using the fractal theory to facilitate the assessment of the corrosion level. In this study, the relationship between steel corrosion and the fractal characterization of concrete surface cracking is investigated. Reinforced concrete prisms with steel bars of different diameters and with different corrosion rates were evaluated. High-resolution images of cracks on the surfaces of these specimens were captured and processed to obtain their fractal dimensions. Finally, a relationship between the fractal dimension, steel bar diameter, and the corrosion rate is established. The results show that the fractal dimension is associated closely with the corrosion rate and steel bar diameter. This study provides new ideas for evaluating corroded reinforced concrete structures.
      Citation: Fractal and Fractional
      PubDate: 2022-06-12
      DOI: 10.3390/fractalfract6060325
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 326: An Existence Study for a Multiplied
           System with p-Laplacian Involving φ-Hilfer Derivatives

    • Authors: Hamid Beddani, Moustafa Beddani, Carlo Cattani, Mountassir Hamdi Hamdi Cherif
      First page: 326
      Abstract: In this paper, we study the existence of solutions for a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the φ-Hilfer derivatives. The presented results are obtained by the fixed point theorems of Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such a problem is considered.
      Citation: Fractal and Fractional
      PubDate: 2022-06-12
      DOI: 10.3390/fractalfract6060326
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 327: The Multicomponent Higher-Order
           Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its
           N-Soliton Solution

    • Authors: Yong Zhang, Huanhe Dong, Yong Fang
      First page: 327
      Abstract: It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures.
      Citation: Fractal and Fractional
      PubDate: 2022-06-13
      DOI: 10.3390/fractalfract6060327
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 328: Fractal Properties of the Magnetic
           Polarity Scale in the Stochastic Hereditary αω-Dynamo Model

    • Authors: Gleb Vodinchar , Lyubov Feshchenko
      First page: 328
      Abstract: We study some fractal properties of the hereditary αω-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the α-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales.
      Citation: Fractal and Fractional
      PubDate: 2022-06-13
      DOI: 10.3390/fractalfract6060328
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 329: Investigation on Pore Structure and
           Permeability of Concrete–Rock Interfacial Transition Zones Based on
           Fractal Theory

    • Authors: Juan Yue, Jinchang Sheng, Huimin Wang, Yunjin Hu, Kailai Zhang, Yulong Luo, Qing Zhou, Meili Zhan
      First page: 329
      Abstract: The concrete–rock interfacial transition zone (ITZ) is generally considered the weak layer in hydraulic engineering, for it is more permeable than the intact concrete or rocks. The water permeability of the ITZ is a critical parameter concerned with structural safety and durability. However, the permeability and pore structure of the ITZ has not been investigated previously, and the mathematical model of ITZ permeability has not been established. This study performed multi-scale experiments on the concrete–rock ITZ with various rock types (limestone, granite, and sandstone). A series of quantitative and qualitative analysis techniques, including NMR, SEM-EDS, and XRD, characterize the ITZ pore structures. The controlled constant flow method was used to determine the permeability of the concrete, rock, and ITZ. The mathematical model of ITZ permeability was proposed using the fractal theory. The consistency between the experimental data and the proposed model indicates the reliability of this study. The results of the experiment show that ITZ permeability is between 4.08 × 10−18 m2 and 5.74 × 10−18 m2. The results of the experiment and the proposed model could determine ITZ permeability in hydraulic structure safety and durability analysis.
      Citation: Fractal and Fractional
      PubDate: 2022-06-13
      DOI: 10.3390/fractalfract6060329
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 330: Local and Global Existence and
           Uniqueness of Solution for Time-Fractional Fuzzy Navier–Stokes
           Equations

    • Authors: Kinda Abuasbeh, Ramsha Shafqat, Azmat Ullah Khan Niazi, Muath Awadalla
      First page: 330
      Abstract: Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order β∈(0,1). In Hγ,℘, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space.
      Citation: Fractal and Fractional
      PubDate: 2022-06-14
      DOI: 10.3390/fractalfract6060330
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 331: Backward Stochastic Differential
           Equations Driven by a Jump Markov Process with Continuous and
           Non-Necessary Continuous Generators

    • Authors: Khaoula Abdelhadi, Mhamed Eddahbi, Nabil Khelfallah, Anwar Almualim
      First page: 331
      Abstract: We deal with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs for short). We start by proving the existence and uniqueness of the solutions for this type of equation and present a comparison of the solutions in the case of Lipschitz conditions in the generator. With these tools in hand, we study the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies the linear growth condition. An existence result for BSDE with a left-continuous, increasing and bounded generator is also discussed. Finally, the general result is applied to solve one kind of quadratic BSDEJ.
      Citation: Fractal and Fractional
      PubDate: 2022-06-15
      DOI: 10.3390/fractalfract6060331
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 332: The Hausdorff Dimension and Capillary
           Imbibition

    • Authors: Didier Samayoa, Ernesto Pineda León, Lucero Damián Adame, Eduardo Reyes de Luna, Andriy Kryvko
      First page: 332
      Abstract: The time scaling exponent for the analytical expression of capillary rise ℓ∼tδ for several theoretical fractal curves is derived. It is established that the actual distance of fluid travel in self-avoiding fractals at the first stage of imbibition is in the Washburn regime, whereas at the second stage it is associated with the Hausdorff dimension dH. Mapping is converted from the Euclidean metric into the geodesic metric for linear fractals F governed by the geodesic dimension dg=dH/dℓ, where dℓ is the chemical dimension of F. The imbibition measured by the chemical distance ℓg is introduced. Approximate spatiotemporal maps of capillary rise activity are obtained. The standard differential equations proposed for the von Koch fractals are solved. Illustrative examples to discuss some physical implications are presented.
      Citation: Fractal and Fractional
      PubDate: 2022-06-16
      DOI: 10.3390/fractalfract6060332
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 333: Supervised Neural Network Procedures for
           the Novel Fractional Food Supply Model

    • Authors: Basma Souayeh, Zulqurnain Sabir, Muhammad Umar, Mir Waqas Alam
      First page: 333
      Abstract: This work presents the numerical performances of the fractional kind of food supply (FKFS) model. The fractional kinds of the derivatives have been used to acquire the accurate and realistic solutions of the FKFS model. The FKFSM system contains three types, special kind of the predator L(x), top-predator M(x) and prey populations N(x). The numerical solutions of three different cases of the FKFS model are provided through the stochastic procedures of the scaled conjugate gradient neural networks (SCGNNs). The data selection for the FKFS model is chosen as 82%, for training and 9% for both testing and authorization. The precision of the designed SCGNNs is provided through the achieved and Adam solutions. To rationality, competence, constancy, and correctness is approved by using the stochastic SCGNNs along with the simulations of the regression actions, mean square error, correlation performances, error histograms values and state transition measures.
      Citation: Fractal and Fractional
      PubDate: 2022-06-16
      DOI: 10.3390/fractalfract6060333
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 334: A Study of the Soliton Solutions with an
           Intrinsic Fractional Discrete Nonlinear Electrical Transmission Line

    • Authors: Hassan Almusawa, Adil Jhangeer
      First page: 334
      Abstract: This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end.
      Citation: Fractal and Fractional
      PubDate: 2022-06-16
      DOI: 10.3390/fractalfract6060334
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 335: Effects of Fly Ash Dosage on Shrinkage,
           Crack Resistance and Fractal Characteristics of Face Slab Concrete

    • Authors: Lei Wang, Zhiqiang Yu, Bo Liu, Feng Zhao, Shengwen Tang, Minmin Jin
      First page: 335
      Abstract: The crack resistance of face slab concretes to various shrinkages is crucial for the structural integrity and the normal operation of concrete-faced rockfill dams (CFRDs). In this work, the effects of fly ash with four dosages (i.e., 10%, 20%, 30% and 40%) on the drying shrinkage, autogenous shrinkage and the cracking resistance of face slab concrete were studied. Besides, the difference in shrinkage behavior due to fly ash addition was revealed from the viewpoint of the pore structure and fractal dimension of the pore surface (Ds). The findings demonstrate that (1) the incorporation of 10–40% fly ash could slightly reduce the drying shrinkage by about 2.2–13.5% before 14 days of hydration, and it could reduce the drying shrinkage at 180 days by about 5.1–23.2%. By contrast, the fly ash addition could markedly reduce the autogenous shrinkage at early, middle and long-term ages. (2) Increasing fly ash dosage from 0 to 40% considerably improves the crack resistance of concrete to plastic shrinkage. Nevertheless, the increase in fly ash dosage increases the drying-induced cracking risk under restrained conditions. (3) The pore structures of face slab concrete at 3 and 28 days become coarser with the increase in fly ash dosage up to 40%. At 180 days, the pore structures become more refined as the fly ash dosage increases to 30%; however, this refinement effect is not as appreciable as the fly ash dosage increases from 30% to 40%. (4) The Ds of face slab concrete is closely related with the concrete pore structures. The Ds of face slab concrete at a. late age increases from 2.902 to 2.946 with increasing of the fly ash dosage. The pore structure and Ds are closely correlated with the shrinkage of face slab concrete. (5) The fly ash dosage around 30% is optimal for face slab concretes in terms of lowering shrinkage and refining the pore structures, without compromising much mechanical property. However, the face slab concretes with a large fly ash dosage should be well cured under restrained and evaporation conditions at an initial hydration age.
      Citation: Fractal and Fractional
      PubDate: 2022-06-16
      DOI: 10.3390/fractalfract6060335
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 336: Application of Asymmetric Notched
           Semi-Circular Bending Specimen to Evaluate Mixed-Mode I-II Fracture
           Behaviors of Sandstone

    • Authors: Gang Ma, Jiangteng Li, Xiang Zhou, Lianying Zhang, Peitao Qiu, Yang Yu
      First page: 336
      Abstract: In this paper, to investigate mixed-mode I-II fracture behaviors, three different asymmetric notched semi-circular bending specimens (ANSCB) were designed by adjusting the angle and the distance between supporting rollers to conduct asymmetric three-point bending tests. Several aid technologies, including acoustic emission (AE), digital image correlation (DIC), crack propagation gauge (CPG), and scanning electron microscopy (SEM), was utilized to monitor and assess the fracture characteristic. Meanwhile, the fractal dimension of the fracture surface was assessed based on the reconstructed digital fracture surface. The results show that mixed-mode I-II ANSCB three-point bending fracture is a brittle failure with the characteristics of the main crack being rapidly transfixed and the bearing capacity decreasing sharply. Based on the DIC method, the whole fracture process consists of a nonlinear elastic stage, fracture process zone, crack initiation stage and crack propagation stage. The crack initiation is mainly caused by the tension-shear strain concentration at the pre-existing crack tip. At the microscale, the crack propagation path is always along the grain boundary where the resultant stress is weakest. According to the monitoring of the AE, it can be found that micro-tensile cracks are mainly responsible for the asymmetric three-point bending fracture. The data obtained by CPG suggest that the subcritical crack growth rate is positively correlated to the ultimate load. In addition, asymmetric loading leads to a coarser fracture surface, and thus a higher fractal dimension of the fracture surface. The current study can provide a better understanding of the mixed-mode I-II fracture behaviors of rock.
      Citation: Fractal and Fractional
      PubDate: 2022-06-17
      DOI: 10.3390/fractalfract6060336
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 337: Fractal Analysis on Pore Structure and
           Modeling of Hydration of Magnesium Phosphate Cement Paste

    • Authors: Yuxiang Peng, Shengwen Tang, Jiasheng Huang, Can Tang, Lei Wang, Yufei Liu
      First page: 337
      Abstract: Magnesium phosphate cement (MPC) paste is hardened by the acid–base reaction between magnesium oxide and phosphate. This work collects and evaluates the thermodynamic data at 25 ℃ and a pressure of 0.1 MPa and establishes the hydration reaction model of MPC pastes. The influence of the magnesium–phosphorus molar (M/P) ratio and water-to-binder (W/B) ratio on the hydration product is explored by the thermodynamic simulation. Following this, the initial and ultimate states of the hydration state of MPC pastes are visualized, and the porosity of different pastes as well as fractal analysis are presented. The result shows that a small M/P ratio is beneficial for the formation of main hydration products. The boric acid acts as a retarder, has a significant effect on lowering the pH of the paste, and slows down the formation of hydration products. After the porosity comparison, it can be concluded that the decreasing of M/P and W/B ratios helps reduce porosity. In addition, the fractal dimension Df of MPC pastes is positively proportional to the porosity, and small M/P ratios as well as small W/B ratios are beneficial for reducing the Df of MKPC pastes.
      Citation: Fractal and Fractional
      PubDate: 2022-06-17
      DOI: 10.3390/fractalfract6060337
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 338: Analytical and Numerical Solutions for a
           Kind of High-Dimensional Fractional Order Equation

    • Authors: Chang-Na Lu, Cun-Juan Hou, Ning Zhang
      First page: 338
      Abstract: In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the G′G-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method.
      Citation: Fractal and Fractional
      PubDate: 2022-06-17
      DOI: 10.3390/fractalfract6060338
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 339: A New Approach to Compare the Strong
           Convergence of the Milstein Scheme with the Approximate Coupling Method

    • Authors: Yousef Alnafisah
      First page: 339
      Abstract: Milstein and approximate coupling approaches are compared for the pathwise numerical solutions to stochastic differential equations (SDE) driven by Brownian motion. These methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the approximate coupling method. We use MATLAB to simulate these methods by applying them to a particular two-dimensional SDE. Then, we analyze the performance of both methods and the amount of time required to obtain the result. This comparison is essential in several areas, such as stochastic analysis, financial mathematics, and some biological applications.
      Citation: Fractal and Fractional
      PubDate: 2022-06-17
      DOI: 10.3390/fractalfract6060339
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 340: On the Global Well-Posedness of Rotating
           Magnetohydrodynamics Equations with Fractional Dissipation

    • Authors: Muhammad Zainul Abidin, Muhammad Marwan, Humaira Kalsoom, Omer Abdalrhman Omer
      First page: 340
      Abstract: This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation (−Δ)℘ for 12<℘≤1. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional rotating magnetohydrodynamics equations in a more generalized Besov spaces characterized by the time evolution semigroup related to the generalized linear Stokes–Coriolis operator.
      Citation: Fractal and Fractional
      PubDate: 2022-06-17
      DOI: 10.3390/fractalfract6060340
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 341: A Novel Modeling Method of
           Micro-Topography for Grinding Surface Based on Ubiquitiform Theory

    • Authors: Yue Liu, Qi An, Min Huang, Deyong Shang, Long Bai
      First page: 341
      Abstract: In order to simulate the grinding surface more accurately, a novel modeling method is proposed based on the ubiquitiform theory. Combined with the power spectral density (PSD) analysis of the measured surface, the anisotropic characteristics of the grinding surface are verified. Based on the isotropic fractal Weierstrass–Mandbrot (W-M) function, the expression of the anisotropic fractal surface is derived. Then, the lower bound of scale invariance δmin is introduced into the anisotropic fractal, and an anisotropic W-M function with ubiquitiformal properties is constructed. After that, the influence law of the δmin on the roughness parameters is discussed, and the δmin for modeling the grinding surface is determined to be 10−8 m. When δmin = 10−8 m, the maximum relative errors of Sa, Sq, Ssk, and Sku of the four surfaces are 5.98%, 6.06%, 5.77%, and 4.53%, respectively. In addition, the relative errors of roughness parameters under the fractal method and the ubiquitiformal method are compared. The comparison results show that the relative errors of Sa, Sq, Ssk, and Sku under the ubiquitiformal modeling method are 5.36%, 6.06%, 5.84%, and 4.53%, while the maximum relative errors under the fractal modeling method are 23.21%, 7.03%, 83.10%, and 7.25%. The comparison results verified the accuracy of the modeling method in this paper.
      Citation: Fractal and Fractional
      PubDate: 2022-06-19
      DOI: 10.3390/fractalfract6060341
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 342: Stationary Response of a Kind of
           Nonlinear Stochastic Systems with Variable Mass and Fractional Derivative
           Damping

    • Authors: Shuo Zhang, Lu Liu, Chunhua Wang
      First page: 342
      Abstract: Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelasticity and variable mass, stationary response of a kind of nonlinear stochastic systems with stochastic variable-mass and fractional derivative, damping is investigated in this paper. Firstly, an approximately equivalent system of the studied nonlinear stochastic system is presented according to the Taylor expansion technique. Then, based on stochastic averaging of energy envelope, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced, which gives an approximated analytical solution of stationary response. Finally, a nonlinear oscillator with variable mass and fractional derivative damping is proposed in numerical simulations. The approximated analytical solution is compared with Monte Carlo numerical solution, which could verify the effectiveness of the obtained results.
      Citation: Fractal and Fractional
      PubDate: 2022-06-20
      DOI: 10.3390/fractalfract6060342
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 343: Variable Step Hybrid Block Method for
           the Approximation of Kepler Problem

    • Authors: Joshua Sunday, Ali Shokri, Daniela Marian
      First page: 343
      Abstract: In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time.
      Citation: Fractal and Fractional
      PubDate: 2022-06-20
      DOI: 10.3390/fractalfract6060343
      Issue No: Vol. 6, No. 6 (2022)
       
  • Fractal Fract, Vol. 6, Pages 259: A New Parallelized Computation Method of
           HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation
           

    • Authors: Ren Liu, Xiaozhong Yang, Peng Lyu
      First page: 259
      Abstract: The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to O(τ2−α+h2) under strong regularity and O(τα+h2) under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE.
      Citation: Fractal and Fractional
      PubDate: 2022-05-07
      DOI: 10.3390/fractalfract6050259
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 260: Asymmetric Lévy Flights Are More
           Efficient in Random Search

    • Authors: Amin Padash, Trifce Sandev, Holger Kantz, Ralf Metzler, Aleksei V. Chechkin
      First page: 260
      Abstract: We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a δ-sink and an asymmetric space-fractional derivative operator with stable index α and asymmetry (skewness) parameter β. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with β≤0 (with a rightward bias) for short initial distances, while for β>0 (with a leftward bias) Lévy flights with α→1 are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for α=1 with β=0) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices α close to unity.
      Citation: Fractal and Fractional
      PubDate: 2022-05-08
      DOI: 10.3390/fractalfract6050260
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 261: Third Hankel Determinant for the
           Logarithmic Coefficients of Starlike Functions Associated with Sine
           Function

    • Authors: Bilal Khan, Ibtisam Aldawish, Serkan Araci, Muhammad Ghaffar Khan
      First page: 261
      Abstract: The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, we first study some well-known classes of starlike functions and then determine the third Hankel determinant bound for the logarithmic coefficients of certain subclasses of starlike functions that also involve the sine functions. We also obtain a number of coefficient estimates. Some of our results are shown to be sharp.
      Citation: Fractal and Fractional
      PubDate: 2022-05-09
      DOI: 10.3390/fractalfract6050261
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 262: Image Dehazing Based on Local and
           Non-Local Features

    • Authors: Qingliang Jiao, Ming Liu, Bu Ning, Fengfeng Zhao, Liquan Dong, Lingqin Kong, Mei Hui, Yuejin Zhao
      First page: 262
      Abstract: Image dehazing is a traditional task, yet it still presents arduous problems, especially in the removal of haze from the texture and edge information of an image. The state-of-the-art dehazing methods may result in the loss of some visual informative details and a decrease in visual quality. To improve dehazing quality, a novel dehazing model is proposed, based on a fractional derivative and data-driven regularization terms. In this model, the contrast constrained adaptive histogram equalization method is used as the data fidelity item; the fractional derivative is applied to avoid over-enhancement and noise amplification; and the proposed data-driven regularization terms are adopted to extract the local and non-local features of an image. Then, to solve the proposed model, half-quadratic splitting is used. Moreover, a dual-stream network based on Convolutional Neural Network (CNN) and Transformer is introduced to structure the data-driven regularization. Further, to estimate the atmospheric light, an atmospheric light model based on the fractional derivative and the atmospheric veil is proposed. Extensive experiments display the effectiveness of the proposed method, which surpasses the state-of-the-art methods for most synthetic and real-world images, quantitatively and qualitatively.
      Citation: Fractal and Fractional
      PubDate: 2022-05-09
      DOI: 10.3390/fractalfract6050262
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 263: Mixed Convection Flow over an Elastic,
           Porous Surface with Viscous Dissipation: A Robust Spectral Computational
           Approach

    • Authors: Lijun Zhang, Nafisa Tariq, Muhammad Mubashir Bhatti, Efstathios E. Michaelides
      First page: 263
      Abstract: A novel computational approach is developed to investigate the mixed convection, boundary layer flow over a nonlinear elastic (stretching or shrinking) surface. The viscous fluid is electrically conducting, incompressible, and propagating through a porous medium. The consequences of viscous dissipation, Joule heating, and heat sink/source of the volumetric rate of heat generation are also included in the energy balance equation. In order to formulate the mathematical modeling, a similarity analysis is performed. The numerical solution of nonlinear differential equations is accomplished through the use of a robust computational approach, which is identified as the Spectral Local Linearization Method (SLLM). The computational findings reported in this study show that, in addition to being simple to establish and numerically implement, the proposed method is very reliable in that it converges rapidly to achieve a specified goal and is more effective in resolving very complex models of nonlinear boundary value problems. In order to ensure the convergence of the proposed SLLM method, the Gauss–Seidel approach is used. The SLLM’s reliability and numerical stability can be optimized even more using Gauss–Seidel approach. The computational results for different emerging parameters are computed to show the behavior of velocity profile, skin friction coefficient, temperature profile, and Nusselt number. To evaluate the accuracy and the convergence of the obtained results, a comparison between the proposed approach and the bvp4c (built-in command in Matlab) method is presented. The Matlab software, which is used to generate machine time for executing the SLLM code, is also displayed in a table.
      Citation: Fractal and Fractional
      PubDate: 2022-05-10
      DOI: 10.3390/fractalfract6050263
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 264: High-Order Dissipation-Preserving
           Methods for Nonlinear Fractional Generalized Wave Equations

    • Authors: Yu Li, Wei Shan, Yanming Zhang
      First page: 264
      Abstract: In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretized by a fourth-order Riesz fractional difference operator in spatial discretization and the collocation methods in the temporal direction. Not only can the present method achieve fourth-order accuracy in the spatial direction and arbitrarily high-order accuracy in the temporal direction, but it also has long-time computing stability. Then, the unconditional discrete energy dissipation law of the present numerical schemes is proved. Finally, some numerical experiments are provided to certify the efficiency and the structure-preserving properties of the proposed schemes.
      Citation: Fractal and Fractional
      PubDate: 2022-05-10
      DOI: 10.3390/fractalfract6050264
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 265: Applications of Prabhakar-like
           Fractional Derivative for the Solution of Viscous Type Fluid with
           Newtonian Heating Effect

    • Authors: Ali Raza, Umair Khan, Aurang Zaib, Emad E. Mahmoud, Wajaree Weera, Ibrahim S. Yahia, Ahmed M. Galal
      First page: 265
      Abstract: This article examines a natural convection viscous unsteady fluid flowing on an oscillating infinite inclined plate. The Newtonian heating effect, slip effect on the boundary wall, and constant mass diffusion conditions are also considered. In order to account for extended memory effects, the semi-analytical solution of transformed governed partial differential equations is attained with the help of a recent and more efficient fractional definition known as Prabhakar, like a thermal fractional derivative with Mittag-Leffler function. Fourier and Fick’s laws are also considered in the thermal profile and concentration field solution. The essentials’ preliminaries, fractional model, and execution approach are expansively addressed. The physical impacts of different parameters on all governed equations are plotted and compared graphically. Additionally, the heat transfer rate, mass diffusion rate, and skin friction are examined with different numerical techniques. Consequently, it is noted that the variation in fractional parameters results in decaying behavior for both thermal and momentum profiles while increasing with the passage of time. Furthermore, in comparing both numerical schemes and existing literature, the overlapping of both curves validates the attained solution of all governed equations.
      Citation: Fractal and Fractional
      PubDate: 2022-05-12
      DOI: 10.3390/fractalfract6050265
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 266: Existence and U-H Stability Results for
           Nonlinear Coupled Fractional Differential Equations with Boundary
           Conditions Involving Riemann–Liouville and
           Erdélyi–Kober Integrals

    • Authors: Muthaiah Subramanian, P. Duraisamy, C. Kamaleshwari, Bundit Unyong, R. Vadivel
      First page: 266
      Abstract: The purpose of this article is to discuss the existence, uniqueness, and Ulam–Hyers stability of solutions to a coupled system of fractional differential equations with Erdélyi–Kober and Riemann–Liouville integral boundary conditions. The Banach fixed point theorem is used to prove the uniqueness of solutions, while the Leray–Schauder alternative is used to prove the existence of solutions. Furthermore, we conclude that the solution to the discussed problem is Hyers–Ulam stable. The results are illustrated with examples.
      Citation: Fractal and Fractional
      PubDate: 2022-05-13
      DOI: 10.3390/fractalfract6050266
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 267: Hadamard-Type Fractional
           Integro-Differential Problem: A Note on Some Asymptotic Behavior of
           Solutions

    • Authors: Ahmad Mugbil, Nasser-Eddine Tatar
      First page: 267
      Abstract: As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our findings is provided.
      Citation: Fractal and Fractional
      PubDate: 2022-05-15
      DOI: 10.3390/fractalfract6050267
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 268: Left Riemann–Liouville Fractional
           Sobolev Space on Time Scales and Its Application to a Fractional Boundary
           Value Problem on Time Scales

    • Authors: Xing Hu, Yongkun Li
      First page: 268
      Abstract: First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained.
      Citation: Fractal and Fractional
      PubDate: 2022-05-15
      DOI: 10.3390/fractalfract6050268
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 269: On a Partial Fractional Hybrid Version
           of Generalized Sturm–Liouville–Langevin Equation

    • Authors: Zohreh Heydarpour, Javad Izadi, Reny George, Mehran Ghaderi, Shahram Rezapour
      First page: 269
      Abstract: As we know one of the most important equations which have many applications in various areas of physics, mathematics, and financial markets, is the Sturm–Liouville equation. In this paper, by using the α-ψ-contraction technique in fixed point theory and employing some functional inequalities, we study the existence of solutions of the partial fractional hybrid case of generalized Sturm–Liouville-Langevin equations under partial boundary value conditions. Towards the end, we present two examples with numerical and graphical simulation to illustrate our main results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-16
      DOI: 10.3390/fractalfract6050269
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 270: Understanding Dynamics and Bifurcation
           Control Mechanism for a Fractional-Order Delayed Duopoly Game Model in
           Insurance Market

    • Authors: Peiluan Li, Jinling Yan, Changjin Xu, Rong Gao, Ying Li
      First page: 270
      Abstract: Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies.
      Citation: Fractal and Fractional
      PubDate: 2022-05-17
      DOI: 10.3390/fractalfract6050270
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 271: A Special Family of m-Fold Symmetric
           Bi-Univalent Functions Satisfying Subordination Condition

    • Authors: Ibtisam Aldawish, Sondekola Rudra Swamy, Basem Aref Frasin
      First page: 271
      Abstract: In this paper, we introduce a special family Mσm(τ,ν,η,φ) of the function family σm of m-fold symmetric bi-univalent functions defined in the open unit disc D and obtain estimates of the first two Taylor–Maclaurin coefficients for functions in the special family. Further, the Fekete–Szegö functional for functions in this special family is also estimated. The results presented in this paper not only generalize and improve some recent works, but also give new results as special cases.
      Citation: Fractal and Fractional
      PubDate: 2022-05-17
      DOI: 10.3390/fractalfract6050271
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 272: Fixed Point Results for Generalized
           F-Contractions in b-Metric-like Spaces

    • Authors: Huaping Huang, Kastriot Zoto, Zoran D. Mitrović, Stojan Radenović
      First page: 272
      Abstract: The purpose of this paper is to introduce several generalized F-contractions in b-metric-like spaces and establish some fixed point theorems for such contractions. Moreover, some nontrivial examples are given to illustrate the superiority of our results. In addition, as an application, we find the existence and uniqueness of a solution to a class of integral equations in the context of b-metric-like spaces.
      Citation: Fractal and Fractional
      PubDate: 2022-05-17
      DOI: 10.3390/fractalfract6050272
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 273: A Survey on Recent Results on
           Lyapunov-Type Inequalities for Fractional Differential Equations

    • Authors: Sotiris K. Ntouyas, Bashir Ahmad, Jessada Tariboon
      First page: 273
      Abstract: This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, ψ-Caputo, ψ-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators.
      Citation: Fractal and Fractional
      PubDate: 2022-05-18
      DOI: 10.3390/fractalfract6050273
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 274: Application of the Explicit Euler Method
           for Numerical Analysis of a Nonlinear Fractional Oscillation Equation

    • Authors: Valentine Aleksandrovich Kim, Roman Ivanovich Parovik
      First page: 274
      Abstract: In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid.
      Citation: Fractal and Fractional
      PubDate: 2022-05-19
      DOI: 10.3390/fractalfract6050274
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 275: Enhancing the Accuracy of Solving
           Riccati Fractional Differential Equations

    • Authors: Antonela Toma, Flavius Dragoi, Octavian Postavaru
      First page: 275
      Abstract: In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if α=1.
      Citation: Fractal and Fractional
      PubDate: 2022-05-20
      DOI: 10.3390/fractalfract6050275
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 276: Ensemble FARIMA Prediction with Stable
           Infinite Variance Innovations for Supermarket Energy Consumption

    • Authors: Jing Wang, Yi Liu, Haiyan Wu, Shan Lu, Meng Zhou
      First page: 276
      Abstract: This paper concerns a fractional modeling and prediction method directly oriented toward an industrial time series with obvious non-Gaussian features. The hidden long-range dependence and the multifractal property are extracted to determine the fractional order. A fractional autoregressive integrated moving average model (FARIMA) is then proposed considering innovations with stable infinite variance. The existence and convergence of the model solutions are discussed in depth. Ensemble learning with an autoregressive moving average model (ARMA) is used to further improve upon accuracy and generalization. The proposed method is used to predict the energy consumption in a real cooling system, and superior prediction results are obtained.
      Citation: Fractal and Fractional
      PubDate: 2022-05-22
      DOI: 10.3390/fractalfract6050276
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 277: Seepage–Fractal Model of
           Embankment Soil and Its Application

    • Authors: Xiaoming Zhao, Binbin Yang, Shichong Yuan, Zhenzhou Shen, Di Feng
      First page: 277
      Abstract: Over time and across space, the hydraulic conductivity, fractal dimension, and porosity of embankment soil have strong randomness, which makes analyzing seepage fields difficult, affecting embankment risk analysis and early disaster warning. This strong randomness limits the application of fractal theory in embankment engineering and sometimes keeps it in the laboratory stage. Based on the capillary model of porous soil, an analytical formula of the fractal relationship between hydraulic conductivity and fractal dimension is derived herein. It is proposed that the influencing factors of hydraulic conductivity of embankment soil mainly include the capillary aperture, fractal dimension, and fluid viscosity coefficient. Based on random field theory and combined with the embankment parameters of Shijiu Lake, hydraulic conductivity is discretized, and then the soil fractal dimension is approximately solved to reveal the internal relationship between hydraulic gradient, fractal dimension, and hydraulic conductivity. The results show that an increased fractal dimension will reduce the connectivity of soil pores in a single direction, increase the hydraulic gradient, and reduce the hydraulic conductivity. A decreased fractal dimension will lead to consistency of seepage channels in the soil, increased hydraulic conductivity, and decreased hydraulic gradient.
      Citation: Fractal and Fractional
      PubDate: 2022-05-22
      DOI: 10.3390/fractalfract6050277
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 278: Financial Applications on Fractional
           Lévy Stochastic Processes

    • Authors: Reem Abdullah Aljethi, Adem Kılıçman
      First page: 278
      Abstract: In this present work, we perform a numerical analysis of the value of the European style options as well as a sensitivity analysis for the option price with respect to some parameters of the model when the underlying price process is driven by a fractional Lévy process. The option price is given by a deterministic representation by means of a real valued function satisfying some fractional PDE. The numerical scheme of the fractional PDE is obtained by means of a weighted and shifted Grunwald approximation.
      Citation: Fractal and Fractional
      PubDate: 2022-05-22
      DOI: 10.3390/fractalfract6050278
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 279: Stochastic Optimal Control Analysis of a
           Mathematical Model: Theory and Application to Non-Singular Kernels

    • Authors: Anwarud Din, Qura Tul Ain
      First page: 279
      Abstract: Some researchers believe fractional differential operators should not have a non-singular kernel, while others strongly believe that due to the complexity of nature, fractional differential operators can have either singular or non-singular kernels. This contradiction in thoughts has led to the publication of a few papers that are against differential operators with non-singular kernels, causing some negative impacts. Thus, publishers and some Editors-in-Chief are concerned about the future of fractional calculus, which has generally brought confusion among the vibrant and innovative young researchers who desire to apply fractional calculus within their respective fields. Thus, the present work aims to develop a model based on a stochastic process that could be utilized to portray the effect of arbitrary-order derivatives. A nonlinear perturbation is used to study the proposed stochastic model with the help of white noises. The required condition(s) for the existence of an ergodic stationary distribution is obtained via Lyapunov functional theory. The finding of the study indicated that the proposed noises have a remarkable impact on the dynamics of the system. To reduce the spread of a disease, we imposed some control measures on the stochastic model, and the optimal system was achieved. The models both with and without control were coded in MATLAB, and at the conclusion of the research, numerical solutions are provided.
      Citation: Fractal and Fractional
      PubDate: 2022-05-23
      DOI: 10.3390/fractalfract6050279
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 280: The Effect of Learning Rate on Fractal
           Image Coding Using Artificial Neural Networks

    • Authors: Rashad A. Al-Jawfi
      First page: 280
      Abstract: The amount by which the artificial neural network weights are updated during the training process is called the learning rate. More precisely, the learning rate is an adjustable parameter used in training neural networks in which small values, often in the interval [0, 1], are handled. The learning rate determines how quickly the model updates its weights to adapt to the problem. Smaller learning rates require more training periods due to small changes to the weights per refresh cycle, while larger learning rates lead to faster changes and require fewer training periods. In this paper, the effect of changing the learning rate value in the artificial neural network designed to solve the inverse problem of fractals was studied. Some results were obtained showing the impact of this change, whether when using large values of the learning rate or small values based on the type of fractal shape required to identify the recursive functions that generate it.
      Citation: Fractal and Fractional
      PubDate: 2022-05-23
      DOI: 10.3390/fractalfract6050280
      Issue No: Vol. 6, No. 5 (2022)
       
  • Fractal Fract, Vol. 6, Pages 281: Uniform Stability of a Class of
           Fractional-Order Fuzzy Complex-Valued Neural Networks in Infinite
           Dimensions

    • Authors: Xin Liu, Lili Chen, Yanfeng Zhao
      First page: 281
      Abstract: In this paper, the problem of the uniform stability for a class of fractional-order fuzzy impulsive complex-valued neural networks with mixed delays in infinite dimensions is discussed for the first time. By utilizing fixed-point theory, theory of differential inclusion and set-valued mappings, the uniqueness of the solution of the above complex-valued neural networks is derived. Subsequently, the criteria for uniform stability of the above complex-valued neural networks are established. In comparison with related results, we do not need to construct a complex Lyapunov function, reducing the computational complexity. Finally, an example is given to show the validity of the main results.
      Citation: Fractal and Fractional
      PubDate: 2022-05-23
      DOI: 10.3390/fractalfract6050281
      Issue No: Vol. 6, No. 5 (2022)
       
 
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