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Fractal Fract., Volume 7, Issue 5 (May 2023) – 72 articles

Cover Story (view full-size image): A Riemann–Liouville distributed-order space fractional operator was introduced to model diffusion phenomena, exhibiting multi-scale varying characteristics. The proposed two-dimensional fractional diffusion model was solved by an alternating direction implicit scheme. Following this, we proved the stability and convergence of the numerical method. An extrapolated technique was implemented to improve the convergence order of the implicit method. Furthermore, a fast Bi-CGSTAB algorithm was developed to reduce computational costs. Finally, two numerical examples were provided to verify the effectiveness of the proposed numerical methods. This research may offer more insights into the application of fractional calculus and numerical techniques for solving fractional systems. View this paper
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4 pages, 218 KiB  
Editorial
Editorial for the Special Issue “Operators of Fractional Calculus and Their Multidisciplinary Applications”
by Hari Mohan Srivastava
Fractal Fract. 2023, 7(5), 415; https://doi.org/10.3390/fractalfract7050415 - 22 May 2023
Cited by 3 | Viewed by 1128
Abstract
This Special Issue of the MDPI journal, Fractal and Fractional, on the subject area of “Operators of Fractional Calculus and Their Multidisciplinary Applications” consists of 19 peer-reviewed papers, including some invited feature articles, originating from all over the world [...] Full article
9 pages, 281 KiB  
Brief Report
A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem
by Danila Cherkashin and Yana Teplitskaya
Fractal Fract. 2023, 7(5), 414; https://doi.org/10.3390/fractalfract7050414 - 20 May 2023
Cited by 1 | Viewed by 950
Abstract
We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that SA is connected, where A is a given compact subset of a given complete metric space X; a solution is called [...] Read more.
We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that SA is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties). Full article
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16 pages, 323 KiB  
Article
Analysis of Cauchy Problems and Diffusion Equations Associated with the Hilfer–Prabhakar Fractional Derivative via Kharrat–Toma Transform
by Ved Prakash Dubey, Jagdev Singh, Sarvesh Dubey and Devendra Kumar
Fractal Fract. 2023, 7(5), 413; https://doi.org/10.3390/fractalfract7050413 - 20 May 2023
Cited by 8 | Viewed by 822
Abstract
In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP [...] Read more.
In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat–Toma transform. The solutions of Cauchy problems and the diffusion equations modeled with the HP fractional derivative are computed in the form of the generalized Mittag–Leffler function. Full article
5 pages, 225 KiB  
Editorial
Special Issue: Nonlinear Dynamics in Complex Systems via Fractals and Fractional Calculus
by Viorel-Puiu Paun
Fractal Fract. 2023, 7(5), 412; https://doi.org/10.3390/fractalfract7050412 - 20 May 2023
Viewed by 759
Abstract
Advances in our knowledge of nonlinear dynamical networks, systems and processes (as well as their unified repercussions) currently allow us to study many typical complex phenomena taking place in nature, from the nanoscale to the extra-galactic scale, in an comprehensive manner [...] Full article
23 pages, 362 KiB  
Article
Some New Applications of the q-Analogous of Differential and Integral Operators for New Subclasses of q-Starlike and q-Convex Functions
by Suha B. Al-Shaikh, Ahmad A. Abubaker, Khaled Matarneh and Mohammad Faisal Khan
Fractal Fract. 2023, 7(5), 411; https://doi.org/10.3390/fractalfract7050411 - 19 May 2023
Viewed by 807
Abstract
In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of [...] Read more.
In the geometric function theory of complex analysis, the investigation of the geometric properties of analytic functions using q-analogues of differential and integral operators is an important area of study, offering powerful tools for applications in numerical analysis and the solution of differential equations. Many topics, including complex analysis, hypergeometric series, and particle physics, have been generalized in q-calculus. In this study, first of all, we define the q-analogues of a differential operator (DRλ,qm,n) by using the basic idea of q-calculus and the definition of convolution. Additionally, using the newly constructed operator (DRλ,qm,n), we establish the q-analogues of two new integral operators (Fλ,γ1,γ2,γlm,n,q and Gλ,γ1,γ2,γlm,n,q), and by employing these operators, new subclasses of the q-starlike and q-convex functions are defined. Sufficient conditions for the functions (f) that belong to the newly defined classes are investigated. Additionally, certain subordination findings for the differential operator (DRλ,qm,n) and novel geometric characteristics of the q-analogues of the integral operators in these classes are also obtained. Our results are generalizations of results that were previously proven in the literature. Full article
(This article belongs to the Special Issue Fractional Calculus and Hypergeometric Functions in Complex Analysis)
24 pages, 468 KiB  
Article
Generalized ρ-Almost Periodic Sequences and Applications
by Marko Kostić, Belkacem Chaouchi, Wei-Shih Du and Daniel Velinov
Fractal Fract. 2023, 7(5), 410; https://doi.org/10.3390/fractalfract7050410 - 18 May 2023
Cited by 1 | Viewed by 748
Abstract
In this paper, we analyze the Bohr ρ-almost periodic type sequences and the generalized ρ-almost periodic type sequences of the form F:I×XY, where IZn, X and Y are [...] Read more.
In this paper, we analyze the Bohr ρ-almost periodic type sequences and the generalized ρ-almost periodic type sequences of the form F:I×XY, where IZn, X and Y are complex Banach spaces and ρ is a general binary relation on Y. We provide many structural results, observations and open problems about the introduced classes of ρ-almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given. Full article
(This article belongs to the Special Issue Abstract Fractional Differential Inclusions)
20 pages, 840 KiB  
Article
Control Design for Fractional Order Leader and Follower Systems with Mixed Time Delays: A Resilience-Based Approach
by Asad Khan, Azmat Ullah Khan Niazi, Waseem Abbasi, Airish Jamil and Jaleel Ahsan Malik
Fractal Fract. 2023, 7(5), 409; https://doi.org/10.3390/fractalfract7050409 - 18 May 2023
Cited by 2 | Viewed by 828
Abstract
In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into [...] Read more.
In this article, we consider the problem of resilient base containment control for fractional-order multi-agent systems (FOMASs) with mixed time delays using a reliable and simple approach, where the communication topology among followers is a weighted digraph. A disturbance term is introduced into the delayed and non-delayed controller part to make it more practical. Our method involves proposing algebraic criteria by utilizing non-delayed and delayed protocols, applying the Razumikhin technique and graph theory respectively. The presented method can well overcome the difficulty resulting from fractional calculus, time delays and fractional derivatives. To demonstrate the validity and effectiveness of our findings, we provide an example at the end of our study. Full article
(This article belongs to the Special Issue Robust and Adaptive Control of Fractional-Order Systems)
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14 pages, 1630 KiB  
Article
P-Bifurcation Analysis for a Fractional Damping Stochastic Nonlinear Equation with Gaussian White Noise
by Yujie Tang, Yun Peng, Guitian He, Wenjie Liang and Weiting Zhang
Fractal Fract. 2023, 7(5), 408; https://doi.org/10.3390/fractalfract7050408 - 18 May 2023
Viewed by 716
Abstract
This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furthermore, the expressions of drift and diffusion coefficients [...] Read more.
This work aims to address the P-bifurcation of a stochastic nonlinear system with fractional damping driven by Gaussian white noise. Based on a stochastic averaging method, a fractional damping stochastic nonlinear equation has been studied. Furthermore, the expressions of drift and diffusion coefficients of the Fokker–Planck equation (FPKE) have been obtained. The probability density function (PDF), the steady solution of FPKE, has also been derived. Then, PDFs of two fractional damping Morse oscillators have been obtained. One can note that the analytical results coincide with the results of numerical simulation. Importantly, stochastic P-bifurcation of a fractional damping stochastic nonlinear Morse oscillator has been further addressed and analyzed. Full article
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25 pages, 1447 KiB  
Article
On a Novel Dynamics of a SIVR Model Using a Laplace Adomian Decomposition Based on a Vaccination Strategy
by Prasantha Bharathi Dhandapani, Víctor Leiva, Carlos Martin-Barreiro and Maheswari Rangasamy
Fractal Fract. 2023, 7(5), 407; https://doi.org/10.3390/fractalfract7050407 - 18 May 2023
Cited by 2 | Viewed by 1227
Abstract
In this paper, we introduce a SIVR model using the Laplace Adomian decomposition. This model focuses on a new trend in mathematical epidemiology dedicated to studying the characteristics of vaccination of infected communities. We analyze the epidemiological parameters using equilibrium stability and numerical [...] Read more.
In this paper, we introduce a SIVR model using the Laplace Adomian decomposition. This model focuses on a new trend in mathematical epidemiology dedicated to studying the characteristics of vaccination of infected communities. We analyze the epidemiological parameters using equilibrium stability and numerical analysis techniques. New mathematical strategies are also applied to establish our epidemic model, which is a pandemic model as well. In addition, we mathematically establish the chance for the next wave of any pandemic disease and show that a consistent vaccination strategy could control it. Our proposal is the first model introducing a vaccination strategy to actively infected cases. We are sure this work will serve as the basis for future research on COVID-19 and pandemic diseases since our study also considers the vaccinated population. Full article
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7 pages, 293 KiB  
Editorial
Advances in Boundary Value Problems for Fractional Differential Equations
by Rodica Luca
Fractal Fract. 2023, 7(5), 406; https://doi.org/10.3390/fractalfract7050406 - 17 May 2023
Cited by 2 | Viewed by 971
Abstract
Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image [...] Read more.
Fractional-order differential and integral operators and fractional differential equations have extensive applications in the mathematical modelling of real-world phenomena which occur in scientific and engineering disciplines such as physics, chemistry, biophysics, biology, medical sciences, financial economics, ecology, bioengineering, control theory, signal and image processing, aerodynamics, transport dynamics, thermodynamics, viscoelasticity, hydrology, statistical mechanics, electromagnetics, astrophysics, cosmology, and rheology [...] Full article
32 pages, 422 KiB  
Article
New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator
by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh
Fractal Fract. 2023, 7(5), 405; https://doi.org/10.3390/fractalfract7050405 - 16 May 2023
Cited by 3 | Viewed by 1259
Abstract
In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, [...] Read more.
In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the (m,tgs)-type s-preinvex function, Godunova–Levin (s,m)-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings. Full article
(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)
18 pages, 349 KiB  
Article
On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator
by Moldir Muratbekova, Bakhtiyar Kadirkulov, Maira Koshanova and Batirkhan Turmetov
Fractal Fract. 2023, 7(5), 404; https://doi.org/10.3390/fractalfract7050404 - 16 May 2023
Cited by 1 | Viewed by 803
Abstract
The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of [...] Read more.
The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Full article
(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)
7 pages, 545 KiB  
Brief Report
Application of Fractional Calculus to Establish Equations of State for Solid Metals
by Vladimir Kulish, Navid Aslfattahi and Michal Schmirler
Fractal Fract. 2023, 7(5), 403; https://doi.org/10.3390/fractalfract7050403 - 16 May 2023
Viewed by 981
Abstract
Fractional differ-integral operators are used to obtain the equation of state for a substance that can be seen as fractal. Two equations of state have been obtained, the first of which depends on two parameters that characterize the fractal dimension of the material [...] Read more.
Fractional differ-integral operators are used to obtain the equation of state for a substance that can be seen as fractal. Two equations of state have been obtained, the first of which depends on two parameters that characterize the fractal dimension of the material and the thermal energy of the particles, respectively. The second equation involves three parameters, and expressions for the Helmholtz free energy and the bulk modulus have also been obtained for this equation. The model presented in this study has been validated using experimental data available in literature, and fractional exponent have been determined for various metals. Full article
(This article belongs to the Section Engineering)
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21 pages, 1509 KiB  
Article
Fractional-Order Accumulative Generation with Discrete Convolution Transformation
by Tao Chen
Fractal Fract. 2023, 7(5), 402; https://doi.org/10.3390/fractalfract7050402 - 16 May 2023
Cited by 1 | Viewed by 728
Abstract
A new fractional accumulation technique based on discrete sequence convolution transform was developed. The accumulation system, whose unit impulse response is the accumulation convolution sequence, was constructed; then, the order was extended to fractional orders. The fractional accumulative convolution grey forecasting model GM [...] Read more.
A new fractional accumulation technique based on discrete sequence convolution transform was developed. The accumulation system, whose unit impulse response is the accumulation convolution sequence, was constructed; then, the order was extended to fractional orders. The fractional accumulative convolution grey forecasting model GMr*(1,1) was established on the sequence convolution. From the viewpoint of sequence convolution, we can better understand the mechanism of accumulative generation. Real cases were used to verify the validity and effectiveness of the fractional accumulative convolution method. Full article
(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)
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19 pages, 5570 KiB  
Article
Analysis and Experimental Verification of the Sealing Performance of PEM Fuel Cell Based on Fractal Theory
by Bao Lv, Kai Han, Yongzhen Wang and Xiaolong Li
Fractal Fract. 2023, 7(5), 401; https://doi.org/10.3390/fractalfract7050401 - 15 May 2023
Cited by 3 | Viewed by 1160
Abstract
Aiming to accurately predict the leakage rate of the sealing interface, this work proposes a two-dimensional finite element model of a proton exchange membrane fuel cell, which includes the microscopic surface morphology and the asperity contact process of the components. First of all, [...] Read more.
Aiming to accurately predict the leakage rate of the sealing interface, this work proposes a two-dimensional finite element model of a proton exchange membrane fuel cell, which includes the microscopic surface morphology and the asperity contact process of the components. First of all, we constructed the surface morphology of the seal by the two-dimensional W-M (Weierstrass–Mandelbrot) fractal function and explored the influence of fractal dimension (D) and scale parameter (G) on the surface profile. Furthermore, the finite element method and Poiseuille fluid theory were adopted to obtain the deformation variables of the asperity under different clamping pressures and leakage rates. Moreover, we quantitatively analyzed the impact of surface roughness on the clamping pressure and leakage rate. It was found that both the surface amplitude and surface roughness are positively correlated with G and negatively correlated with D. Surface morphology is proportional to D but has no relationship with G. Additionally, the deformation asperity decreases exponentially with growing clamping pressure, and the leakage rate is consistent with the experimental values at a clamping pressure of 0.54 MPa. With the same leakage rate, when the seal surface roughness value is less than 1 μm, a doubled roughness value leads to an increase of 31% in the clamping pressure. In contrast, when the surface roughness of the seal is greater than 1 μm, a doubled roughness value induces an increase of 50% in the corresponding clamping pressure. Full article
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21 pages, 776 KiB  
Article
Qualitative and Quantitative Analysis of Fractional Dynamics of Infectious Diseases with Control Measures
by Sultan Alyobi and Rashid Jan
Fractal Fract. 2023, 7(5), 400; https://doi.org/10.3390/fractalfract7050400 - 15 May 2023
Cited by 5 | Viewed by 1005
Abstract
Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems may be less equipped to handle them. It is [...] Read more.
Infectious diseases can have a significant economic impact, both in terms of healthcare costs and lost productivity. This can be particularly significant in developing countries, where infectious diseases are more prevalent, and healthcare systems may be less equipped to handle them. It is recognized that the hepatitis B virus (HBV) infection remains a critical global public health issue. In this study, we develop a comprehensive model for HBV infection that includes vaccination and hospitalization through a fractional framework. It has been shown that the solutions of the recommended system of HBV infection are positive and bounded. We examine the steady states of the model and determine the basic reproduction number; denoted by R0. The qualitative and quantitative behavior of the model is demonstrated using mathematical skills and numerical techniques. It has been proved that the infection-free steady state of the system is locally asymptotically stable if R0<1 and unstable otherwise. Furthermore, the Ulam–Hyers stability (UHS) of the recommended fractional models is investigated and the significant conditions are provided. We present an iterative technique to visualize the dynamical behavior of the system. We perform different simulations to illustrate the effect of different input factors on the solution pathways of the system of HBV infection to conceptualize the role of parameters in the control and prevention of the infection. Full article
(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)
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17 pages, 380 KiB  
Article
Note on the Generalized Branching Random Walk on the Galton–Watson Tree
by Najmeddine Attia, Rim Amami and Rimah Amami
Fractal Fract. 2023, 7(5), 399; https://doi.org/10.3390/fractalfract7050399 - 14 May 2023
Viewed by 1009
Abstract
Let T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of T along which the sequence [...] Read more.
Let T be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of T along which the sequence SnX(t)/SnX˜(t) has a given set of limit points, where SnX(t) and SnX˜(t) are two branching random walks defined on T. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence s=(sn), we consider Eα,s=tT:SnX(t)αSnX˜(t)snasn+. We will give a sufficient condition on (sn) so that Eα,s has a maximal Hausdorff and packing dimension. Full article
16 pages, 2308 KiB  
Article
High-Dimensional Chaotic Lorenz System: Numerical Treatment Using Changhee Polynomials of the Appell Type
by Mohamed Adel, Mohamed M. Khader and Salman Algelany
Fractal Fract. 2023, 7(5), 398; https://doi.org/10.3390/fractalfract7050398 - 13 May 2023
Cited by 8 | Viewed by 909
Abstract
Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested approximation technique to achieve this goal. [...] Read more.
Presenting and simulating the numerical treatment of the nine-dimensional fractional chaotic Lorenz system is the goal of this work. The spectral collocation method (SCM), which makes use of Changhee polynomials of the Appell type, is the suggested approximation technique to achieve this goal. A rough formula for the Caputo fractional derivative is first derived, and it is used to build the numerical strategy for the suggested model’s solution. This procedure creates a system of algebraic equations from the model that was provided. We validate the effectiveness and precision of the provided approach by evaluating the residual error function (REF). We compare the results obtained with the fourth-order Runge–Kutta technique and other existing published work. The outcomes demonstrate that the technique used is a simple and effective tool for simulating such models. Full article
(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)
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15 pages, 412 KiB  
Article
A Sufficient and Necessary Condition for the Power-Exponential Function 1+1xαx to Be a Bernstein Function and Related nth Derivatives
by Jian Cao, Bai-Ni Guo, Wei-Shih Du and Feng Qi
Fractal Fract. 2023, 7(5), 397; https://doi.org/10.3390/fractalfract7050397 - 13 May 2023
Viewed by 1071
Abstract
In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions [...] Read more.
In the paper, the authors find a sufficient and necessary condition for the power-exponential function 1+1xαx to be a Bernstein function, derive closed-form formulas for the nth derivatives of the power-exponential functions 1+1xαx and (1+x)α/x, and present a closed-form formula of the partial Bell polynomials Bn,k(H0(x),H1(x),,Hnk(x)) for nk0, where Hk(x)=0eu1ueuuk1exudu for k0 are completely monotonic on (0,). Full article
(This article belongs to the Special Issue Fractional Calculus and Nonlinear Analysis: Theory and Applications)
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18 pages, 340 KiB  
Article
Boundary Value Problem for Impulsive Delay Fractional Differential Equations with Several Generalized Proportional Caputo Fractional Derivatives
by Ravi P. Agarwal and Snezhana Hristova
Fractal Fract. 2023, 7(5), 396; https://doi.org/10.3390/fractalfract7050396 - 12 May 2023
Cited by 1 | Viewed by 1091
Abstract
A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in [...] Read more.
A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results. Full article
(This article belongs to the Special Issue Feature Papers for the 'General Mathematics, Analysis' Section)
16 pages, 1764 KiB  
Article
A New Class of Generalized Fractal and Fractal-Fractional Derivatives with Non-Singular Kernels
by Khalid Hattaf
Fractal Fract. 2023, 7(5), 395; https://doi.org/10.3390/fractalfract7050395 - 12 May 2023
Cited by 16 | Viewed by 2230
Abstract
The present paper introduces a new class of generalized differential and integral operators. This class includes and generalizes a large number of definitions of fractal-fractional derivatives and integral operators used to model the complex dynamics of many natural and physical phenomena found in [...] Read more.
The present paper introduces a new class of generalized differential and integral operators. This class includes and generalizes a large number of definitions of fractal-fractional derivatives and integral operators used to model the complex dynamics of many natural and physical phenomena found in diverse fields of science and engineering. Some properties of the newly introduced class are rigorously established. As applications of this new class, two illustrative examples are presented, one for a simple problem and the other for a nonlinear problem modeling the dynamical behavior of a chaotic system. Full article
(This article belongs to the Special Issue Operators of Fractional Integration and Their Applications)
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21 pages, 9915 KiB  
Article
Depth Image Enhancement Algorithm Based on Fractional Differentiation
by Tingsheng Huang, Xinjian Wang, Da Xie, Chunyang Wang and Xuelian Liu
Fractal Fract. 2023, 7(5), 394; https://doi.org/10.3390/fractalfract7050394 - 11 May 2023
Cited by 1 | Viewed by 953
Abstract
Depth image enhancement techniques can help to improve image quality and facilitate computer vision tasks. Traditional image-enhancement methods, which are typically based on integer-order calculus, cannot exploit the textural information of an image, and their enhancement effect is limited. To solve this problem, [...] Read more.
Depth image enhancement techniques can help to improve image quality and facilitate computer vision tasks. Traditional image-enhancement methods, which are typically based on integer-order calculus, cannot exploit the textural information of an image, and their enhancement effect is limited. To solve this problem, fractional differentiation has been introduced as an innovative image-processing tool. It enables the flexible use of local and non-local information by taking into account the continuous changes between orders, thereby improving the enhancement effect. In this study, a fractional differential is applied in depth image enhancement and used to establish a novel algorithm, named the fractional differential-inverse-distance-weighted depth image enhancement method. Experiments are performed to verify the effectiveness and universality of the algorithm, revealing that it can effectively solve edge and hole interference and significantly enhance textural details. The effects of the order of fractional differentiation and number of iterations on the enhancement performance are examined, and the optimal parameters are obtained. The process data of depth image enhancement associated with the optimal number of iterations and fractional order are expected to facilitate depth image enhancement in actual scenarios. Full article
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13 pages, 8503 KiB  
Article
Relationship between Remanence and Micromorphology of Nd-Fe-B Permanent Magnets Revealed by Fractal Theory and EBSD Data
by Qisong Sun, Minggang Zhu, Jiaming Bai and Qiang Wang
Fractal Fract. 2023, 7(5), 393; https://doi.org/10.3390/fractalfract7050393 - 09 May 2023
Viewed by 1173
Abstract
Remanence is an important parameter of magnetic property for Nd-Fe-B magnets, and high remanent magnetization is a prerequisite for high-performance magnets. In this paper, the surface morphology perpendicular to the texture orientation direction and parallel to the texture orientation direction is analyzed by [...] Read more.
Remanence is an important parameter of magnetic property for Nd-Fe-B magnets, and high remanent magnetization is a prerequisite for high-performance magnets. In this paper, the surface morphology perpendicular to the texture orientation direction and parallel to the texture orientation direction is analyzed by Nd-Fe-B permanent magnets with different compositions. For the first time, the relationship between the remanence of a magnet and the degree of texture orientation is explained in depth using the fractal dimension. The fractal dimension of surface morphology combined with the remanence equation yields the degree of texture orientation of the magnet, which is in agreement with the trend of the squareness factor of the demagnetization curves. Among the three samples, the Nd-Fe-B sample has the highest degree of texture orientation, the Pr-Nd-Fe-B sample has the lowest degree of texture orientation, and the Nd-Ce-Fe-B sample is in between the first two. The multiples of uniform (pole) density obtained by EBSD further prove the correctness of the degree of texture orientation calculated by the fractal dimension. The combination of EBSD morphology and fractal dimension to obtain novel insights into the correlation between remanence and the degree of texture orientation will contribute to the development of high-performance Nd-Fe-B with high remanence. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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17 pages, 387 KiB  
Article
Bernoulli-Type Spectral Numerical Scheme for Initial and Boundary Value Problems with Variable Order
by Zareen A. Khan, Sajjad Ahmad, Salman Zeb and Hussam Alrabaiah
Fractal Fract. 2023, 7(5), 392; https://doi.org/10.3390/fractalfract7050392 - 09 May 2023
Viewed by 920
Abstract
This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration [...] Read more.
This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration and differentiation are developed. With the aid of these operational matrices, the considered problems are converted to algebraic-type equations, which can be easily solved using computational software. Various examples are solved by applying the method described above, and their graphical presentation and accuracy performance are provided. Full article
(This article belongs to the Special Issue Recent Advances in Time/Space-Fractional Evolution Equations)
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31 pages, 3141 KiB  
Review
Fractional-Order Control Techniques for Renewable Energy and Energy-Storage-Integrated Power Systems: A Review
by Masoud Alilou, Hatef Azami, Arman Oshnoei, Behnam Mohammadi-Ivatloo and Remus Teodorescu
Fractal Fract. 2023, 7(5), 391; https://doi.org/10.3390/fractalfract7050391 - 09 May 2023
Cited by 7 | Viewed by 2143
Abstract
The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and greenhouse emissions. The control system of renewable energy [...] Read more.
The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and greenhouse emissions. The control system of renewable energy units and energy storage systems has a high effect on their performance and absolutely on the efficiency of the total power network. Classical controllers are based on integer-order differentiation and integration, while the fractional-order controller has tremendous potential to change the order for better modeling and controlling the system. This paper presents a comprehensive review of the energy system of renewable energy units and energy storage devices. Various papers are evaluated, and their methods and results are presented. Moreover, the mathematical fundamentals of the fractional-order method are mentioned, and the various studies are categorized based on different parameters. Various definitions for fractional-order calculus are also explained using their mathematical formula. Different studies and numerical evaluations present appropriate efficiency and accuracy of the fractional-order techniques for estimating, controlling, and improving the performance of energy systems in various operational conditions so that the average error of the fractional-order methods is considerably lower than other ones. Full article
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13 pages, 294 KiB  
Article
Some Families of Differential Equations Associated with Multivariate Hermite Polynomials
by Badr Saad T. Alkahtani, Ibtehal Alazman and Shahid Ahmad Wani
Fractal Fract. 2023, 7(5), 390; https://doi.org/10.3390/fractalfract7050390 - 08 May 2023
Cited by 3 | Viewed by 832
Abstract
In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also [...] Read more.
In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered. Full article
15 pages, 501 KiB  
Article
An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process
by Samuel Megameno Nuugulu, Frednard Gideon and Kailash C. Patidar
Fractal Fract. 2023, 7(5), 389; https://doi.org/10.3390/fractalfract7050389 - 08 May 2023
Viewed by 1420
Abstract
After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an [...] Read more.
After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2). Full article
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18 pages, 5131 KiB  
Article
Evaluation of the Methods for Nonlinear Analysis of Heart Rate Variability
by Evgeniya Gospodinova, Penio Lebamovski, Galya Georgieva-Tsaneva and Mariya Negreva
Fractal Fract. 2023, 7(5), 388; https://doi.org/10.3390/fractalfract7050388 - 08 May 2023
Viewed by 1455
Abstract
The dynamics of cardiac signals can be studied using methods for nonlinear analysis of heart rate variability (HRV). The methods that are used in the article to investigate the fractal, multifractal and informational characteristics of the intervals between heartbeats (RR time intervals) are: [...] Read more.
The dynamics of cardiac signals can be studied using methods for nonlinear analysis of heart rate variability (HRV). The methods that are used in the article to investigate the fractal, multifractal and informational characteristics of the intervals between heartbeats (RR time intervals) are: Rescaled Range, Detrended Fluctuation Analysis, Multifractal Detrended Fluctuation Analysis, Poincaré plot, Approximate Entropy and Sample Entropy. Two groups of people were studied: 25 healthy subjects (15 men, 10 women, mean age: 56.3 years) and 25 patients with arrhythmia (13 men, 12 women, mean age: 58.7 years). The results of the application of the methods for nonlinear analysis of HRV in the two groups of people studied are shown as mean ± std. The effectiveness of the methods was evaluated by t-test and the parameter Area Under the Curve (AUC) from the Receiver Operator Curve (ROC) characteristics. The studied 11 parameters have statistical significance (p < 0.05); therefore, they can be used to distinguish between healthy and unhealthy subjects. It was established by applying the ROC analysis that the parameters Hq=2(MFDFA), F(α)(MFDFA) and SD2(Poincaré plot) have a good diagnostic value; H(R/S), α1(DFA), SD1/SD2(Poincaré plot), ApEn and SampEn have a very good score; α2(DFA), αall(DFA) and SD1(Poincaré plot) have an excellent diagnostic score. In conclusion, the methods used for nonlinear analysis of HRV have been evaluated as effective, and with their help, new perspectives are opened in the diagnosis of cardiovascular diseases. Full article
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11 pages, 2949 KiB  
Article
Electromagnetic Scattering from Fractional Brownian Motion Surfaces via the Small Slope Approximation
by Antonio Iodice, Gerardo Di Martino, Alessio Di Simone, Daniele Riccio and Giuseppe Ruello
Fractal Fract. 2023, 7(5), 387; https://doi.org/10.3390/fractalfract7050387 - 08 May 2023
Cited by 3 | Viewed by 931
Abstract
Marine and terrestrial natural surfaces exhibit statistical scale invariance properties that are well modelled by fractional Brownian motion (fBm), two-dimensional random processes. Accordingly, for microwave remote sensing applications it is useful to evaluate the normalized radar cross section (NRCS) of fBm surfaces. This [...] Read more.
Marine and terrestrial natural surfaces exhibit statistical scale invariance properties that are well modelled by fractional Brownian motion (fBm), two-dimensional random processes. Accordingly, for microwave remote sensing applications it is useful to evaluate the normalized radar cross section (NRCS) of fBm surfaces. This task has been accomplished in the past by using either the Kirchhoff approximation (KA) or the small perturbation method (SPM). However, KA and SPM have rather limited ranges of application in terms of surface roughness and incidence angle: a wider range of application is achieved by the small slope approximation (SSA), more recently developed, but the latter has not been applied yet to fBm surfaces. In this paper, the first-order SSA (SSA-1) is applied to the evaluation of scattering from fBm surfaces obtaining an analytical formulation of their NRCS. It is then shown that the obtained SSA-1 expression reduces to the KA and SPM ones at near-specular and far-from-specular scattering directions, respectively. Finally, the results of the proposed method are compared to experimental data available in the literature. Full article
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22 pages, 583 KiB  
Article
The Continuous Bernoulli Distribution: Mathematical Characterization, Fractile Regression, Computational Simulations, and Applications
by Mustafa Ç. Korkmaz, Víctor Leiva and Carlos Martin-Barreiro
Fractal Fract. 2023, 7(5), 386; https://doi.org/10.3390/fractalfract7050386 - 06 May 2023
Cited by 5 | Viewed by 1822
Abstract
The continuous Bernoulli distribution is defined on the unit interval and has a unique property related to fractiles. A fractile is a position on a probability density function where the corresponding surface is a fixed proportion. This article presents the derivation of properties [...] Read more.
The continuous Bernoulli distribution is defined on the unit interval and has a unique property related to fractiles. A fractile is a position on a probability density function where the corresponding surface is a fixed proportion. This article presents the derivation of properties of the continuous Bernoulli distribution and formulates a fractile or quantile regression model for a unit response using the exponentiated continuous Bernoulli distribution. Monte Carlo simulation studies evaluate the performance of point and interval estimators for both the continuous Bernoulli distribution and the fractile regression model. Real-world datasets from science and education are analyzed to illustrate the modeling abilities of the continuous Bernoulli distribution and the exponentiated continuous Bernoulli quantile regression model. Full article
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