Fractals最新文献

{"title":"A COMPUTATIONAL ALGORITHM FOR THE NUMERICAL SOLUTION OF TWO-DIMENSIONAL TELEGRAPH INTERFACE PROBLEMS","authors":"KHAWAJA SHAMSUL HAQ,MUHAMMAD ASIF,MUHAMMAD FAHEEM,MUHAMMAD AHSAN,SALAH BOULAARAS,QASEM AL MDALLAL","doi":"10.1142/s0218348x25402819","DOIUrl":"https://doi.org/10.1142/s0218348x25402819","url":null,"abstract":"This paper introduces the hyperbolic two-dimensional telegraph interface model with regular interfaces, employing a hybrid numerical technique that combines Haar wavelets and the finite difference method for both linear and nonlinear problems. Spatial derivatives are approximated using truncated Haar wavelet series, while time derivatives are managed via the finite difference method. For linear problems, the Gauss elimination technique is used to solve the resulting algebraic system, whereas Newton’s quasi-linearization technique addresses nonlinearity in nonlinear problems. Performance metrics, the computation of maximum absolute errors, root mean square errors, and computational convergence rates is carried out for various configurations of collocation points demonstrating the method’s stability and accuracy, even with sharp transitions or discontinuities. Numerical experiments and analysis confirm the method’s broad applicability and precision.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149090371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Investigation on properties of continuous functions under variable-order fractional integral and its application in joint fire allocation modeling","authors":"Wang Jun,Tao Yuben,Zhang Chengzhuo,Zou Danqing,Li Xuan,Feng Ketao","doi":"10.1142/s0218348x26501513","DOIUrl":"https://doi.org/10.1142/s0218348x26501513","url":null,"abstract":"As a natural extension of constant-order fractional operators, variable-order fractional calculus can characterize the spatiotemporal heterogeneity of systems in a more refined manner, yet its fundamental mathematical theory remains to be fully established. This paper is devoted to the regularity preservation problem of continuous functions under the variable-order Riemann-Liouville integral. It is rigorously proved that if the original function is continuous and of bounded variation, and the order function is continuously differentiable and satisfies an appropriate positive lower bound condition, then the function obtained after the variable-order integration remains continuous and of bounded variation. The proof, centered on the Jordan decomposition theorem, the Lebesgue dominated convergence theorem, and the integration-byparts estimation of total variation, provides a complete and rigorous argument. Furthermore, two typical functions of bounded variation are selected for numerical experiments. Under the action of different variable-order functions, the variable-order integrals are computed and the total variations are estimated. Numerical results demonstrate that the total variations of the integrated functions all remain finite, and the profiles exhibit a distinct smoothing effect, which intuitively validates the correctness of the theoretical conclusions. Meanwhile, partial conclusions are applied to military modeling to verify the effectiveness of the research findings. The work of this paper provides a rigorous mathematical foundation for the property preservation of variable-order fractional calculus on function spaces, and offers a reliable analytical framework for subsequent studies on topics such as Hölder continuity and fractal dimension evolution.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Which ETFs Protect Against U.S. Equity Crashes? Regime-Dependent Hedging and Safe-Haven Evidence from DCC-GJR-GARCH","authors":"Fernando Henrique Antunes de Araujo,Fabio Sandro dos Santos,Kerolly Kedma Felix do Nascimento","doi":"10.1142/s0218348x26501525","DOIUrl":"https://doi.org/10.1142/s0218348x26501525","url":null,"abstract":"Which liquid ETFs actually protect investors when U.S. equities crash? This paper answers that question using daily observations from 2012-03-02 to 2026-02-27 for SPY and eight cross-asset ETFs spanning gold, oil, broad commodities, the U.S. dollar, Treasuries, real estate, high-yield credit, and investment-grade credit. Pairwise DCC-GJR-GARCH models recover time-varying correlations, and nested downside regressions identify behavior in the lower 10%, 5%, and 1% tails of SPY returns. To prevent an incremental correlation decline from being misclassified as a safe haven, inference is based on HC3-robust Wald tests of the cumulative tail-state correlations. In the full sample, the U.S. dollar and U.S. Treasuries qualify as hedges, while the U.S. dollar and U.S. Treasuries reach statistically negative correlation in at least one downside state. Protection is sharply regime dependent. During COVID-19, effective safe-haven behavior is limited to the U.S. dollar and U.S. Treasuries; during the Russia-Ukraine regime it is observed for the U.S. dollar; and in the tariff window the hedge set narrows to gold and U.S. Treasuries. Across specifications, dollar exposure appears most frequently, while sovereign duration produces the deepest negative dependence in the full sample, COVID-19, and the tariff window. Gold remains episodic, and cyclical ETFs generally retain positive equity dependence. The findings show that hedge and safe-haven labels are state-contingent properties of the cross-asset correlation structure rather than permanent attributes of individual ETFs.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149022461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling of Fractal Curves by an Interpolatory Quinary Refinement Scheme","authors":"Pakeeza Ashraf,Md. Nur Alam,Asbat Habib,Amna Kalsoom,Abdul Ghaffar,Md. Shohel Sayeed","doi":"10.1142/s0218348x26501604","DOIUrl":"https://doi.org/10.1142/s0218348x26501604","url":null,"abstract":"This study introduces a 4-point interpolatory quinary refinement scheme with two shape parameters and investigates the scheme’s convergence and smoothness. The scheme’s fundamental properties, such as approximation order, polynomial reproduction, and polynomial generation, are also examined. To the best of our knowledge, this is the first quinary interpolatory scheme shown to generate fractal curves with endpoint interpolation, and its versatility is demonstrated by its ability to generate fractal curves within a specific range of the shape parameters. A modified 4-point quinary scheme for endpoint interpolation is also presented to enhance the scheme’s modeling capability. This research offers valuable insights into the behavior and potential applications of the scheme in engineering and industry.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"From Tara to Mvog: A theory of deformable connected geometric evolution","authors":"Abdon Atangana","doi":"10.1142/s0218348x26501586","DOIUrl":"https://doi.org/10.1142/s0218348x26501586","url":null,"abstract":"This paper introduces an axiomatic theory of Mvogs, rooted, generation-indexed geometric structures that evolve through the reproduction, placement, and attachment of three-dimensional cells. Every Mvog originates from a distinguished ancestral cell, called Tara, whose descendants are generated through a prescribed birth law. Reproductively active descendants may subsequently assume the role of Tara relative to their own offspring, thereby producing nested genealogical families. The evolution is governed by a Growth Operator defined as the composition of birth, spatial placement, and geometric connection. A Mvog is surface-connected when every pair of cells can be joined by a finite chain of nonempty direct boundary interfaces. When the cellular support is not surface-connected, the Mvog is classified as branch-mediated if the inclusion of prescribed three-dimensional branches produces a connected augmented structure, and as completely disconnected otherwise. The framework is developed within the Trinition geometry induced by the positive-definite metric tensor [Formula: see text], where [Formula: see text] controls anisotropic deformation while preserving the intrinsic genealogical and attachment structures. The principal results establish recursive existence and uniqueness relative to prescribed admissible deterministic data, preservation of surface connectedness under explicit attachment hypotheses, population.volume growth laws, evolution of the total interface measure, and finite termination under a uniform lower-volume condition. The [Formula: see text]-connectivity ratio [Formula: see text] and the interface measure [Formula: see text] are introduced to quantify global cellular connectedness and the geometric magnitude of positive-area attachments. Detailed comparisons with iterated function systems, cell complexes, body packings, contact graphs, branching processes, cellular automata, stochastic aggregation, continuum percolation, and geometric-flow methods clarify that these established frameworks retain only particular projections or specializations of the complete Mvog state. Numerical constructions involving completely disconnected populations, branch-mediated organizations, strongly surface-connected assemblies, double-helical and anthropomorphic forms, and Mvogs supported by Cesáro–Koch and Sierpiński geometries illustrate the scope of the theory. The resulting framework provides a mathematical foundation for studying geometry generated through reproduction and attachment rather than through contraction, lattice updating, or continuous front propagation.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractal-Informed Risk Modeling in Energy Systems: From Extreme Events to Resilience-Oriented Control","authors":"Usman Nursusanto,Mohamad Fani Sulaima,Hamidreza Namazi","doi":"10.1142/s0218348x26300096","DOIUrl":"https://doi.org/10.1142/s0218348x26300096","url":null,"abstract":"Extreme events, renewable-generation variability, and high-impact low-probability disturbances challenge energy-system risk assessment under nonstationary operating conditions. This paper critically reviews fractal and multifractal methods relevant to energy-system risk analysis and proposes a Fractal-Informed Risk Modeling (FIRM) architecture linking data acquisition, fractal feature extraction, probabilistic risk inference, AI-based prediction, and resilience-oriented control. Fractal descriptors, including the Hurst exponent, fractal dimension, and multifractal-spectrum measures, are treated as complementary indicators rather than deterministic predictors of failure. Their operational value depends on estimation uncertainty, window selection, threshold calibration, and validation against strong conventional baselines. FIRM is therefore presented as a conceptual reference architecture rather than an implementation-ready or experimentally validated model. The framework specifies inter-layer interfaces, uncertainty propagation, and the distinction between literature-demonstrated applications and proposed extensions. A validation roadmap is also provided, covering candidate datasets, baseline models, extreme-event scenarios, predictive and calibration metrics, and resilience indicators. Overall, this study identifies where fractal information may add value to established risk methods and defines the methodological safeguards required before such information can support operational decision-making.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometry-dependent chaos: A unified framework for anisotropic dynamical systems via Trinition deformation","authors":"Abdon Atangana,Areej A. Binsultan","doi":"10.1142/s0218348x26501598","DOIUrl":"https://doi.org/10.1142/s0218348x26501598","url":null,"abstract":"This work develops a Trinition-based framework for geometry-dependent chaotic dynamics. The Trinition construction is represented on a three-component primitive space, while its multiplication law generates both an anisotropic quadratic geometry and an additional interaction-accessible bivector sector, providing a precise mechanism for dimensional mutation. The associated deformation tensor depends on a parameter [Formula: see text], with [Formula: see text] recovering the Euclidean configuration and the endpoints [Formula: see text] and [Formula: see text] corresponding to singular geometric limits. When the inverse Trinition metric acts on a nonlinear vector field, equilibrium locations are preserved, whereas spectral properties, stability, symmetry, and bifurcation boundaries may vary with [Formula: see text]. Applied to the Lorenz system, the deformation breaks the classical symmetry between the two nontrivial equilibria and produces branch-dependent Hopf stability boundaries. The Rössler system similarly exhibits deformation-dependent modifications of its local stability and bifurcation structure. Numerical simulations further reveal pronounced [Formula: see text]-dependent changes in attractor morphology and long-time dynamics. The resulting framework provides a mathematically controlled mechanism through which algebraically induced anisotropic geometry can participate directly in nonlinear and chaotic evolution.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"105 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Using Self-Affine Fractal Functions and Dimensions to Characterize the Dynamic Phase Transition of Sardine Schools","authors":"Kunyuan Li,Xiongwei Zhang,Chenlong Xue,Meng Sun,Yin Li,Kefeng Liu,Xia Zhang,Yangjun Wang,Kui Yao","doi":"10.1142/s0218348x26501574","DOIUrl":"https://doi.org/10.1142/s0218348x26501574","url":null,"abstract":"This paper uses self-affine fractal functions to characterize the phase transition in dy- namic images of sardine schools and identifies critical points of phase transition through the rate of change in box dimension, generalizing the application of fractal theory in the study of phase transitions in ocean schools.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weak solutions for a generalized Katugampola fractional Langevin inclusion with boundary condition","authors":"Mohamed Adel,Lakhlifa Sadek,Mohamed Abbas El-Naggar,Mohamed Khalifa Saad,Hijaz Ahmad","doi":"10.1142/s0218348x26501537","DOIUrl":"https://doi.org/10.1142/s0218348x26501537","url":null,"abstract":"We investigate weak solutions of a fractional Langevin inclusion subject to a nonlocal boundary condition and involving the generalized Katugampola-Caputo derivative. The problem is posed in a real Banach space and the multivalued nonlinearity is treated through Pettis-integrable selections. We first derive a precise integral representation for the associated linear boundary problem. The derivation identifies an explicit non-resonance denominator and yields a contraction condition that guarantees uniqueness of the auxiliary problem. We then construct the induced multivalued integral operator and combine the De Blasi measure of weak noncompactness with a Mönch-type fixed-point principle to obtain sufficient conditions for the existence of weak solutions . The estimates retain the contributions of both the dissipative term and the nonlocal boundary term. To illustrate the applicability of the theoretical results, three examples are provided in which all assumptions are verified, including the summability requirements in [Formula: see text].","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractal-Aware Federated AI for Personalized Demand-Side Management: A Review of Household Heterogeneity, Privacy, and Flexibility Learning","authors":"Hamidreza Namazi,Mohamad Fani Sulaima,Norazryana Mat Dawi","doi":"10.1142/s0218348x26300084","DOIUrl":"https://doi.org/10.1142/s0218348x26300084","url":null,"abstract":"Federated artificial intelligence is increasingly viewed as a promising approach for privacy-preserving demand-side management because it enables distributed learning from household energy data without centralizing raw smart-meter records. However, existing federated energy models often treat household heterogeneity mainly as a non-IID statistical problem, while overlooking the multiscale behavioral structure of electricity demand. This paper proposes a fractal-aware federated AI framework for demand-side management, focusing on household heterogeneity, privacy, and personalized flexibility learning. This paper synthesizes literature on federated learning, smart-meter analytics, fractal and multifractal demand characterization, privacy-preserving energy management, and personalized demand response. The proposed framework argues that fractal indicators, such as persistence, irregularity, long-range dependence, and scale-dependent variability, can enrich federated models by capturing household-specific demand signatures. These indicators may support more accurate flexibility prediction, cluster-aware personalization, privacy-conscious consumer profiling, and adaptive incentive design. The paper contributes a conceptual foundation for future empirical research on federated DSM systems that are not only privacy-preserving, but also personalized, interpretable, behaviorally aware, and fair across heterogeneous households.","PeriodicalId":501262,"journal":{"name":"Fractals","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149021107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}