Fractal and Fractional最新文献

{"title":"Fractional Diffusion Equation under Singular and Non-Singular Kernel and Its Stability","authors":"Enrique C. Gabrick, Paulo R. Protachevicz, Ervin K. Lenzi, Elaheh Sayari, José Trobia, Marcelo K. Lenzi, Fernando S. Borges, Iberê L. Caldas, Antonio M. Batista","doi":"10.3390/fractalfract7110792","DOIUrl":"https://doi.org/10.3390/fractalfract7110792","url":null,"abstract":"The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method can be particularly employed for singular and non-singular kernels, such as the Riemann–Liouville, Caputo, Fabrizio–Caputo, and Atangana–Baleanu operators. Moreover, we obtained general inequalities that guarantee that the stability condition depends explicitly on the kernel. As an implementation of the method, we numerically solved the diffusion equation under the power-law and exponential kernels. For the power-law kernel, we solved by considering fractional time, space, and both operators. In another example, we considered the exponential kernel acting on the time derivative and compared the numerical results with the analytical ones. Our results showed that the numerical procedure developed in this work can be employed to solve fractional differential equations considering different kernels.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136102505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multi-Signal Multifractal Detrended Fluctuation Analysis for Uncertain Systems —Application to the Energy Consumption of Software Programs in Microcontrollers","authors":"Juan Carlos de la Torre, Pablo Pavón-Domínguez, Bernabé Dorronsoro, Pedro L. Galindo, Patricia Ruiz","doi":"10.3390/fractalfract7110794","DOIUrl":"https://doi.org/10.3390/fractalfract7110794","url":null,"abstract":"Uncertain systems are those wherein some variability is observed, meaning that different observations of the system will produce different measurements. Studying such systems demands the use of statistical methods over multiple measurements, which allows overcoming the uncertainty, based on the premise that a single measurement is not representative of the system’s behavior. In such cases, the current multifractal detrended fluctuation analysis (MFDFA) method cannot offer confident conclusions. This work presents multi-signal MFDFA (MS-MFDFA), a novel methodology for accurately characterizing uncertain systems using the MFDFA algorithm, which enables overcoming the uncertainty of the system by simultaneously considering a large set of signals. As a case study, we consider the problem of characterizing software (Sw) consumption. The difficulty of the problem mainly comes from the complexity of the interactions between Sw and hardware (Hw), as well as from the high uncertainty level of the consumption measurements, which are affected by concurrent Sw services, the Hw, and external factors such as ambient temperature. We apply MS-MFDFA to generate a signature of the Sw consumption profile, regardless of the execution time, the consumption levels, and uncertainty. Multiple consumption signals (or time series) are built from different Sw runs, obtaining a high frequency sampling of the instant input current for each of them while running the Sw. A benchmark of eight Sw programs for analysis is also proposed. Moreover, a fully functional application to automatically perform MS-MFDFA analysis has been made freely available. The results showed that the proposed methodology is a suitable approximation for the multifractal analysis of a large number of time series obtained from uncertain systems. Moreover, analysis of the multifractal properties showed that this approach was able to differentiate between the eight Sw programs studied, showing differences in the temporal scaling range where multifractal behavior is found.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"20 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136022920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications","authors":"Danish Ali, Shahbaz Ali, Darab Pompei-Cosmin, Turcu Antoniu, Abdullah A. Zaagan, Ali M. Mahnashi","doi":"10.3390/fractalfract7110790","DOIUrl":"https://doi.org/10.3390/fractalfract7110790","url":null,"abstract":"Fixed point theory is a branch of mathematics that studies solutions that remain unchanged under a given transformation or operator, and it has numerous applications in fields such as mathematics, economics, computer science, engineering, and physics. In the present article, we offer a quicker iteration technique, the D** iteration technique, for approximating fixed points in generalized α-nonexpansive mappings and nearly contracted mappings. In uniformly convex Banach spaces, we develop weak and strong convergence results for the D** iteration approach to the fixed points of generalized α-nonexpansive mappings. In order to demonstrate the effectiveness of our recommended iteration strategy, we provide comprehensive analytical, numerical, and graphical explanations. Here, we also demonstrate the stability consequences of the new iteration technique. We approximately solve a fractional Volterra–Fredholm integro-differential problem as an application of our major findings. Our findings amend and expand upon some previously published results.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136023301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Difference Equations and Julia Sets of Several Functions for Degenerate q-Sigmoid Polynomials","authors":"Jung-Yoog Kang, Cheon-Seoung Ryoo","doi":"10.3390/fractalfract7110791","DOIUrl":"https://doi.org/10.3390/fractalfract7110791","url":null,"abstract":"In this article, we construct a new type of degenerate q-sigmoid (DQS) polynomial for sigmoid functions containing quantum numbers and find several difference equations related to it. We check how each point moves by iteratively synthesizing a quartic degenerate q-sigmoid (DQS) polynomial that appears differently depending on q in the space of a complex structure. We also construct Julia sets associated with quartic DQS polynomials and find their features. Based on this, we make some conjectures.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"29 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136023322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Properties for a Certain Subclass of Analytic Functions Associated with the Salagean q-Differential Operator","authors":"Abdel Moneim Y. Lashin, Abeer O. Badghaish, Fayzah A. Alshehri","doi":"10.3390/fractalfract7110793","DOIUrl":"https://doi.org/10.3390/fractalfract7110793","url":null,"abstract":"Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136022913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Development of an Efficient Variable Step-Size Gradient Method Utilizing Variable Fractional Derivatives","authors":"Luotang Ye, Yanmao Chen, Qixian Liu","doi":"10.3390/fractalfract7110789","DOIUrl":"https://doi.org/10.3390/fractalfract7110789","url":null,"abstract":"The fractional gradient method has garnered significant attention from researchers. The common view regarding fractional-order gradient methods is that they have a faster convergence rate compared to classical gradient methods. However, through conducting theoretical convergence analysis, we have revealed that the maximum convergence rate of the fractional-order gradient method is the same as that of the classical gradient method. This discovery implies that the superiority of fractional gradients may not reside in achieving fast convergence rates compared to the classical gradient method. Building upon this discovery, a novel variable fractional-type gradient method is proposed with an emphasis on automatically adjusting the step size. Theoretical analysis confirms the convergence of the proposed method. Numerical experiments demonstrate that the proposed method can converge to the extremum point both rapidly and accurately. Additionally, the Armijo criterion is introduced to ensure that the proposed gradient methods, along with various existing gradient methods, can select the optimal step size at each iteration. The results indicate that, despite the proposed method and existing gradient methods having the same theoretical maximum convergence speed, the introduced variable step size mechanism in the proposed method consistently demonstrates superior convergence stability and performance when applied to practical problems.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"178 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136023168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Global Nonexistence of a Solution for Wave Equations with Nonlinear Memory Term","authors":"Soufiane Bousserhane Reda, Amer Memou, Abdelhak Berkane, Ahmed Himadan, Abdelkader Moumen, Hicham Saber, Tariq Alraqad","doi":"10.3390/fractalfract7110788","DOIUrl":"https://doi.org/10.3390/fractalfract7110788","url":null,"abstract":"The paper is devoted to the problem of the local existence for a solution to a nonlinear wave equation, with the dissipation given by a nonlinear form with the presence of a nonlinear memory term. Moreover, the global nonexistence of a solution is established using the test function method. We combine the Fourier transform and fractional derivative calculus to achieve our goal.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"308 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136135003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Analytical Stochastic Solutions for the Stochastic Potential Yu–Toda–Sasa–Fukuyama Equation with Conformable Derivative Using Different Methods","authors":"Sahar Albosaily, Elsayed M. Elsayed, M. Daher Albalwi, Meshari Alesemi, Wael W. Mohammed","doi":"10.3390/fractalfract7110787","DOIUrl":"https://doi.org/10.3390/fractalfract7110787","url":null,"abstract":"We consider in this study the (3+1)-dimensional stochastic potential Yu–Toda–Sasa–Fukuyama with conformable derivative (SPYTSFE-CD) forced by white noise. For different kind of solutions of SPYTSFE-CD, including hyperbolic, rational, trigonometric and function, we use He’s semi-inverse and improved (G′/G)-expansion methods. Because it investigates solitons and nonlinear waves in dispersive media, plasma physics and fluid dynamics, the potential Yu–Toda–Sasa–Fukuyama theory may explain many intriguing scientific phenomena. We provide numerous 2D and 3D figures to address how the white noise destroys the pattern formation of the solutions and stabilizes the solutions of SPYTSFE-CD.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"4 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamic Modeling and Response Analysis of Dielectric Elastomer Incorporating Fractional Viscoelasticity and Gent Function","authors":"Qiaoyan Li, Zhongkui Sun","doi":"10.3390/fractalfract7110786","DOIUrl":"https://doi.org/10.3390/fractalfract7110786","url":null,"abstract":"Dielectric Elastomer (DE) has been recognized for its remarkable potential in actuation and sensing applications. However, the functionality of most DE materials is restricted by their high viscoelastic effects. Currently, there is a lack of dynamic models that consider both viscoelasticity and stiffening effects. To address this research gap, we propose a fractional-order model in this study. Specifically, the model comprehensively integrates both viscoelastic and stiffening effects under electromechanical coupling, utilizing the principle of virtual work. Further, the effects of the system parameters are analyzed. The results indicate that the fractional-order derivative influences the hysteresis behaviors during the transient state and affects the duration of the transient process. Furthermore, when the system’s energy surpasses a certain threshold, the steady-state response can transition between two distinct potential wells. Through the manipulation of electromechanical coupling parameters, bifurcation can be induced, and the occurrence of snap-through and snap-back behaviors can be controlled. These findings have significant implications for the design and optimization of DE materials in various applications.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"33 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family","authors":"Jia Mu, Zhiyuan Yuan, Yong Zhou","doi":"10.3390/fractalfract7110785","DOIUrl":"https://doi.org/10.3390/fractalfract7110785","url":null,"abstract":"Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"48 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}