The Aigerim-Ratbay-Shyngys equation is used to model ultrashort optical soliton propagation in nonlinear media, Alfvén soliton waves in magnetized plasmas, soliton dynamics in anharmonic lattices, and traffic soliton phenomena. The Akbota-Bayan-Myrzakulov equation describes nonlinear spin dynamics, geometric motion of curves and surfaces, and optical soliton propagation in telecommunications. In this article, we have investigated the fresh and broad spectral soliton solutions for the above models and their corresponding wave profiles as well by using the enhanced modified simple equation method. It is illustrated that the waveform features play an important role in the change of the associated free parameters. The soliton solutions are obtained by equating the exponents of the highest-order linear and nonlinear terms in the governing models. As a result, a variety of exact solutions for both models are obtained, including dark solitons, bright solitons, and singular profiles. The Lyapunov stability analysis demonstrates that both systems are dynamically stable. The solutions obtained in this study are further general and widely applicable and contain some unique forms that have not been reported in previous studies.
In general, it is very hard to establish the Variational Principle (VP) for the Nonlinear Partial Differential Equations (NPDEs) in mathematics and physics. In this paper, a fractal nonlinear Rangwala–Rao equation is proposed with the fractal derivative and its fractal VP is explored for the first time. With the aid of the Semi-Inverse Method (SIM), a novel Trial Lagrange Function (TLF) is introduced and then updated step by step to construct the fractal VP. The developed fractal VP is also confirmed by computing the functional stationary conditions. Especially, the fractal VP becomes the VP of the classic Rangwala–Rao equation when the fractal order [Formula: see text]. The obtained fractal VP can offer some new insights into the fractal variational and numerical methods in physics.
In this study, we solve a class of fractional problems using the Natural Ervenila (NE) integral transform. For nonlinear cases, approximate solutions are obtained by combining the NE integral transform with other iterative methods. The fractional operator is considered in the Caputo and constant proportional Caputo (CPC) senses. We demonstrate the efficiency of the proposed integral transform on several interesting models. Fractional electrical circuits, fractional honeybee, and fractional population dynamics models are solved as test examples.
The temperature-dependent effective thermal conductivity (ETC) is a vital parameter for characterizing the thermophysical properties of porous rocks. However, the thermo-mechanical coupling and complex multiscale pore-structures of porous rocks are challenging to the prediction of ETC. In this work, a novel pore-scale model based on fractal geometry theory is proposed to predict the effect of temperature on the ETC of porous rocks. The proposed ETC model relates temperature and the pore-structural parameters and physical properties of porous rocks, including the initial porosity (phi(0)), the initial maximum pore diameter (lambda(max,0)), the initial fractal dimension (D-f,D-0) and the fractal dimension for tortuosity (D-T,D-T) as well as the thermal expansion coefficient (alpha(T)) of pore volume. The proposed ETC model is validated against experimental data of four distinct types of saturated rocks over a temperature range from 273 K to 873 K. The model predictions are in good agreement with all experimental results, with an error of less than 4%. In addition, the present model demonstrates superior performance in comparison with classical models, accurately predicting the ETC of various types of saturated rocks with lower errors ( <2.7%). A detailed analysis is conducted to elucidate the effects of pore-structural parameters and physical properties on dimensionless ETC and the fractal dimensions for pore area and tortuosity. It is found that the thermal expansion increases porosity and pore fractal dimension but decreases tortuosity fractal dimension with rising temperature, leading to complex variations in ETC. When k(s)/k(f) < 1, the ETC increases significantly with increasing temperature; conversely, when k(s)/k(f) > 1, the ETC decreases moderately. When k(s)/k(f) = 1, even slight pore structural changes can induce a redistribution of heat flow paths, resulting in the ETC being most sensitive to pore structural variations near this critical point. These quantitative relationships provide essential constraints for fractal-based thermal conductivity models, enabling more accurate predictions of rock behavior under varying thermal regimes in geothermal and petroleum engineering applications.
Fractals, with their intrinsic self-similarity and structural elegance, have become powerful tools for modeling complex graphical structures. Among these, self-similar graphs are pointed out for their recursive node connectivity and hierarchical organization governed by fractal iterative principles. This study focuses on the Sierpinski Rhombus graph, a quintessential example of fractal structures, and introduces a computational approach for exploring its multivariate independence polynomials across iterative levels n = 1, 2, 3. Through a systematic algorithm, we construct the multivariate independence polynomials through independent sets, their cardinalities, and their inverses using SAGE mathematical software, thereby unveiling deeper algebraic patterns embedded within the graph's structure. Beyond their combinatorial significance, these multivariate independence polynomial descriptors serve as meaningful and interpretable feature for machine learning models and bridge the gap between structural complexity and data-driven modeling of graph network with intelligent predictive analysis. The synergy of fractal-based graph polynomials with machine learning algorithm in this work holds immense potential, thereby positioning this formulated approach as a promising foundation for advancements in complexity-aware learning and predictive modeling of fractal graphs in graph-theoretic intelligence.
This paper examines hourly wind speed dynamics at 12 INMET stations in Piau & iacute;, Brazil, asking whether the local multifractal organization of the wind field moves closer to or farther from a random benchmark across meteorological seasons. The observational base is reorganized into a rolling multifractal framework with windows of 720 h observations and steps of 168 h. For each full sample and rolling window, the singularity-spectrum peak is translated into a stochastic inefficiency index equal to its absolute distance from 0.5, so lower values indicate greater stochastic efficiency. The analysis combines this index with mean wind speed and with the empirical third moment of speed, E(v3), proportional to mean wind power density under constant air density. Full-sample results identify PIRIPIRI, GILBUES, and TERESINA as closest to the random benchmark, whereas PARNAIBA is the most resource-rich station and the most structurally distant from that benchmark. Rolling results reveal marked seasonality: mean stochastic inefficiency is highest in autumn (0.2820) and lowest in winter (0.2488). Fixed-effects and quantile regressions show that winter and spring reduce stochastic inefficiency relative to summer, whereas autumn is statistically indistinguishable from the summer benchmark in most specifications. Overall, GILBUES, CANTO DO BURITI, OEIRAS, and CARACOL provide the most balanced combination of wind-resource magnitude and stochastic efficiency in the active 12-station subset.
This study introduces the concept of generalized Hukuhara fractal differentiability for fuzzy-valued functions and explores the necessary conditions for differentiability. The Atangana-Baleanu-Caputo fuzzy fractal-fractional derivative is developed, along with the corresponding integral operators. Theorems on existence and uniqueness of solutions are established, and a computational scheme is proposed for numerically solving fuzzy fractal-fractional differential equations. The effectiveness of the proposed method is demonstrated through the exponential population growth model, with numerical solutions compared against actual data using Mean Absolute Percentage Error. Comparative results highlight the improved accuracy of the fuzzy fractal-fractional approach over classical ordinary differential equations and fractional differential models. The proposed framework supports integration with machine learning by capturing memory, non-locality, and structural complexity, enabling advanced feature extraction, compact data representation, and improved model interpretability.
Wind turbines operate under highly variable aerodynamic, structural, and environmental conditions, producing complex, nonlinear, and nonstationary signals. Traditional linear and spectral analysis approaches often fail to capture long-range dependencies, multiscale fluctuations, and intermittency that characterize wind speed, vibration, acoustic emission, and power output signals. Fractal theory — encompassing monofractal and multifractal scaling, Hurst exponent estimation, fractal dimension analysis, and complexity measures — has emerged as a powerful framework for analyzing these irregular signals. This review synthesizes the significant body of research applying fractal methods to wind turbine monitoring, forecasting, and control. We examine fractal-based approaches in wind resource characterization, turbine performance analysis, structural health monitoring, fault detection, and power output modeling. The review highlights the strengths and limitations of existing methods and identifies open research directions for integrating fractal features with machine learning, digital twins, and real-time control systems.
Bilateral contraction, a recently introduced hybrid form of contraction, has opened new avenues in fixed point (FP) theory. In this paper, we build upon this concept and the structure of b-metric spaces (b-MS) to propose a novel contraction type, referred to as the Hybrid L-fuzzy contraction. This new approach unifies and extends key elements from the Jaggi and Dass-Gupta contraction principles within the framework of L-fuzzy set theory in b-metric spaces. Our aim is to broaden the theoretical foundation and enhance the applicability of these contractions. To illustrate the effectiveness of the proposed method, a detailed example is presented. Furthermore, we demonstrate its relevance through an application to a fractional differential equation (FDE). The existence of a solution is validated using a modern machine learning (ML) technique, specifically Physics-Informed Neural Networks (PINNs), which integrate the underlying differential equation directly into the learning process. This fusion of analytical theory and computational modeling highlights the potential of ML in supporting and extending traditional mathematical analysis.
This work presents a bibliographic analysis of structural dynamic modeling and applications of dynamic vibration absorbers in energy generation systems. First, the paper introduces the electrical energy generation systems and provides an overview of the bibliographic review content. Topics, such as horizontal axis wind turbines (HAWTs), buoy-type wave energy converters (WECs), bladeless wind turbines (BWTs), and fractional-order modeling, are considered in the bibliographic analysis. Then, a general overview of the dynamic vibration absorbers is addressed, including issues such as the operation modes and optimal design of the DVAs. A comprehensive review of the research on the structural dynamic modeling of each electrical energy generation system is presented. Likewise, applications of damping systems used as antivibration systems in HAWTs and enhancement systems for power generation by buoy-type WECs are reviewed. Subsequently, investigations on dynamic vibration absorbers modeling by fractional order derivatives are discussed. Moreover, the advantages and disadvantages of using BWTs compared to HAWTs are shown. Finally, the applications of the coupled with or decoupled from damping devices HAWTs’ fractional order modeling are provided.
Hedonic and eudaimonic well-being are core components of happiness, as understood in positive psychology. This study develops a discrete-time, non-integer order framework using fractal-fractional (FF) differentials with both singular and non-singular kernels, integrated with a stochastic Levenberg-Marquardt backpropagation (LMB) neural network (NN) architecture. The algorithm effectively captures memory-dependent, nonlinear behavioral dynamics across a range of psychometric conditions. FF differential equations (FF-DEs), employing Mittag-Leffler kernels, enable chaotic and hyperchaotic behavior under both disturbed and undisturbed conditions. Simulations explore three cases: varying fractional order, fractal dimension, or both. Empirical validation is performed using Spanish happiness data from 2009-2023, with performance evaluated via & Rscr;2, MSE, AE, and regression analysis. The LMB-NN framework is trained on 78% of the dataset, with the remaining 22% split equally for validation and testing, achieving accuracy up to 5-7 decimal places. Error histograms, state transition metrics, and strong correlation validate the model's precision and robustness. Furthermore, emotional regulation and synchronization are examined, positioning this hybrid FF-LMB-NN framework as a powerful tool for psychotherapeutic modeling and personalized well-being analysis. This research integrates systems theory, control strategies, and psychological insight, offering valuable implications for psychotherapy and mental wellness.
This paper investigates the practical stability of nonlinear dynamical systems governed by the k-Caputo fractional derivative. Using a Lyapunov-based approach, we derive new practical Mittag-Leffler-based stability criteria that provide sufficient conditions ensuring that the trajectories of the considered systems converge to a bounded neighborhood of the equilibrium point. The developed theoretical framework is then applied to the challenging problem of simultaneous state and actuator-fault estimation for nonlinear fractional-order systems. In this setting, we construct an observer capable of jointly reconstructing the system states and the unknown actuator faults, while guaranteeing practical stability through Linear Matrix Inequality (LMI) conditions. The proposed methodology explicitly accounts for nonlinear dynamics through a Lipschitz condition and accommodates bounded actuator faults with bounded fractional derivatives. Numerical examples confirm the convergence of the state and fault-estimation errors and illustrate the practical Mittag-Leffler stability properties of the proposed approach. Overall, the results enrich the stability analysis of fractional-order systems and provide a feasible framework for fault diagnosis in complex nonlinear dynamical settings.
This paper explores the development of fresh mathematical inequalities operating under the umbrella of local fractional calculus. A central breakthrough of our research is the formulation of an innovative auxiliary fractal-fractional identity. Building upon this foundational equation, we deduce a series of new Milne-type bounds, tailored for mappings that possess generalized harmonically convex first-order local fractional derivatives. To expand the analytical reach of our study, we extract supplementary bounding constraints by deploying the generalized power mean and H & ouml;lder inequalities. Moreover, the theoretical propositions are strictly substantiated using a comprehensive numerical example; accompanying visual plots are supplied to verify the accuracy and reliability of the generated boundaries. Concluding the study, the tangible applicability of our newly constructed estimations is showcased through a specific implementation concerning distinct fractal averages.
This paper investigates the dynamics of Coronavirus Disease 2019 (COVID-19) by incorporating vaccination, partial immunity, and stochastic perturbations into a comprehensive mathematical framework. Three versions of the model are considered: a classical ordinary differential equation model, a fractal-fractional order model, and a stochastic differential equation model driven by Levy noise. The fractal-fractional order COVID- 19 (FFoCOVID-19) model is numerically solved using a fourth-order RungeKutta method with a power-law kernel to capture memory and hereditary effects in disease transmission. For the stochastic model, sufficient conditions for the existence and uniqueness of positive solutions, disease extinction, and persistence in the mean are established using Lyapunov functions and stochastic analysis techniques. The basic reproduction number, [Formula: see text] is derived and shown to serve as a threshold parameter governing the spread and control of the disease. Furthermore, a LevenbergMarquardt backpropagation neural network (L-MBNN) is employed to obtain numerical solutions of the COVID-19 model under three different initial-condition scenarios. The accuracy and efficiency of the proposed neural-network approach are evaluated through mean squared error (MSE) analysis, with Case 1 achieving training, testing, and validation MSE values of [Formula: see text], and [Formula: see text] respectively. Numerical simulations confirm the reliability of the proposed computational techniques and demonstrate that Levy noise can significantly influence disease dynamics, while the neural-network framework provides an effective and accurate tool for solving complex epidemiological models.
This paper presents a novel approach for solving coupled nonlinear variable-order time-fractional sine-Gordon equations (CNVOTF-SGEs) by employing two-dimensional wavelets. When the solu tion of a fractional differential equation is non-smooth, weakly regular, or singular, polynomialbased methods encounter significant challenges, as polynomials are not well-suited for approximating non smooth functions. In contrast, wavelet-based methods can approximate such functions with both accuracy and efficiency. Wavelet-based techniques often produce near-diagonal matrices, such as integral operational matrices, which significantly accelerate computations by reducing the overall computational time. Wavelets can be easily extended to 2D and 3D, which makes them highly ef fective for solving multidimensional fractional partial differential equations. The original equations are reduced to an algebraic system of equations (ASEs) by approximating the unknown functions with Jacobi and Chebyshev cardinal wavelets and by applying operational matrices together with collocation points. After solving the ASEs, we observed that the numerical results were better when using JWs compared to CCWs. Theoretical results on convergence and error estimation of the proposed method are also established. Several illustrative examples confirm the accuracy, efficiency, and applicability of the two-dimensional wavelet technique.
This study examined how cognitive load influences the joint dynamics of galvanic skin response (GSR) and facial electromyography (EMG) during graded mental effort. Thirty healthy adults completed four levels of mental-arithmetic tasks while GSR and facial EMG signals were recorded. Signal complexity was quantified using fractal dimension, sample entropy, and approximate entropy. Complexity measures were compared across task conditions, and cross-modal associations between corresponding GSR and EMG measures were evaluated. Across the three metrics, complexity generally increased with task difficulty in both modalities, although the observed changes were more pronounced for GSR than for EMG. Positive associations were also observed between corresponding GSR and EMG complexity measures across the experimental conditions, suggesting coordinated variation in autonomic and somatic responses during increasing cognitive demand. These findings indicate that complexity-based analysis may provide useful information about workload-related changes in electrodermal and facial muscular activity. However, further inferential testing, confidence-interval estimation, and validation in independent samples and real-world settings are required before these measures can be regarded as reliable markers of cognitive workload.
Multifractal cascades, as the generic constructors of multifractal measures, have been widely used in hydrometeorology and other areas of geophysics. In order to parametrize the cascade models, the inverse problem in this case consists in finding the generator parameters, given a realization of the multifractal measure. From the direct measurement of a field or time series (e.g. rainfall), its decomposition into breakdown coefficients is immediate, producing a microcanonical (deterministically normalized) multiplicative cascade over a certain range of scales. However, the canonical (stochastically normalized) construction at underlying scales may generate statistical properties that are not easy to reproduce. This work analyzes those properties for the case of a one-dimensional, beta–lognormal discrete multiplicative cascade.
The determination of exact Hausdorff centered measure of a fractal set is a difficult and important problem in fractal geometry. In this paper, we investigate the exact computation of the Hausdorff centered measure for the Cartesian product of the symmetric perfect set with itself, and an explicit calculating formula is given.
In this paper, we study the fractal characteristics of Horadam sequences and generalize a recent result on the Fibonacci sequence to Horadam sequences with positive discriminants. More precisely, for any integers [Formula: see text] with [Formula: see text], we prove that the Horadam sequence [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text] satisfies [Formula: see text] where [Formula: see text] denotes the box dimension. The main tool we use is the Binet formula, together with box-dimension estimates. The hypothesis [Formula: see text] is sharp.
In this paper, we study a Caputo–Fabrizio fractional-order climate model to capture the intrinsic nonlocal and fading-memory characteristics of coupled environmental processes. The framework incorporates five interacting state variables surface temperature [Formula: see text], atmospheric carbon concentration [Formula: see text], ice fraction [Formula: see text], atmospheric activity [Formula: see text], and oceanic carbon uptake [Formula: see text]. The non-singular exponential kernel embedded in the Caputo–Fabrizio operator ensures smooth temporal weighting and realistic climate inertia without introducing singular behavior. Theoretical analysis establishes existence, uniqueness, and dynamical stability of the proposed system within a fractional framework. For computational investigation, we perform numerical analysis using a fractional weighted discrete iterative scheme (FWDIS), which preserves the exponential memory structure while enabling stable and efficient time-marching of the nonlinear coupled model. Furthermore, an artificial neural network trained via the Levenberg–Marquardt optimization algorithm is employed to learn complex climate interactions and validate numerical accuracy. Strong agreement between analytical findings, discrete simulations, and intelligent predictions demonstrates that integrating fractional calculus with advanced numerical and learning strategies provides a robust and forward-looking framework for long-term climate forecasting and environmental risk assessment.
