Fractal and Fractional

The fractional PDE-ODE coupled system with Neumann boundary condition is considered in this paper. We design a boundary state feedback controller using the Backstepping method to stabilize the considered system. According to operator semigroup theory, we obtain a unique solution of the investigated system. Based on this, by using the fractional Lyapunov scheme, we prove that the system is Mittag-Leffler stable. Full article

In this manuscript, we study a class of nabla fractional difference equations with summation boundary conditions that depend on a parameter. We construct the Green’s function related to the linear problem and we deduce some of its properties. First, we obtain an upper bound of the sum of it, and use this property to give an existence result for the considered problem based on the Leray–Shauder nonlinear alternative. Then, we establish some bounds on the parameter in which the Green’s function is positive, and by using Krasnoselski–Zabreiko fixed-point theorem, we deduce another existence result. Finally, we give some particular examples in order to demonstrate our primary findings. Full article

This work introduces a new family of multivariate hybrid special polynomials, motivated by their growing relevance in mathematical modeling, physics, and engineering. We explore their core properties, including recurrence relations and shift operators, within a unified structural framework. By employing the factorization method, we derive various governing equations such as differential, partial differential, and integrodifferential equations. Additionally, we establish a related fractional Volterra integral equation, which broadens the theoretical foundation and potential applications of these polynomials. To support the theoretical development, we carry out computational simulations to approximate their roots and visualize the distribution of their zeros, offering practical insights into their analytical behavior. Full article

In the field of deep learning, the design of optimization algorithms and neural network structures is crucial for improving model performance. Recent advances in medical image analysis have revealed that many pathological features exhibit fractal-like characteristics in their spatial distribution and morphological patterns. This observation has opened new possibilities for developing fractal-inspired deep learning approaches. In this study, we propose the following: (1) a novel Region-Module Adam (RMA) optimizer that incorporates fractal-inspired region-weighting to prioritize areas with higher fractal dimensionality, and (2) an ECA-Enhanced Shuffle MobileNet (ESM) architecture designed to capture multi-scale fractal patterns through its enhanced feature extraction modules. Our experiments demonstrate that this fractal-informed approach significantly improves classification accuracy compared to conventional methods. On gastrointestinal image datasets, the RMA algorithm achieved accuracies of 83.60%, 81.60%, and 87.30% with MobileNetV2, ShuffleNetV2, and ESM networks, respectively. For glaucoma fundus images, the corresponding accuracies reached 84.90%, 83.60%, and 92.73%. These results suggest that explicitly considering fractal properties in medical image analysis can lead to more effective diagnostic tools. Full article

This paper focuses on the problem, claimed in some works, of the non-existence of finite-time stable equilibria in nonlinear fractional differential equations. After dividing the equilibrium point into the initial equilibrium point and the finite-time equilibrium point, we provide sufficient conditions for the equilibrium point of a fractional stochastic differential equation. Then the finite-time stability of the equilibrium points of nonlinear fractional stochastic differential equations is presented. Finally, the correctness of the theoretical analysis is illustrated through an example. Full article

This paper focuses on the numerical modeling of drug diffusion in biological tissues using fractional time-dependent parabolic equations with non-local boundary conditions. The model includes a Caputo fractional derivative to capture the non-local effects and memory inherent in biological processes, such as drug absorption and transport. The theoretical framework of the problem is based on the work of Alhazzani, et al.,which demonstrates the solution’s goodness, existence, and uniqueness. Building on this foundation, we present a robust numerical method designed to deal with the complexity of fractional derivatives and non-local interactions at the boundaries of biological tissues. Numerical simulations reveal how fractal order and non-local boundary conditions affect the drug concentration distribution over time, providing valuable insights into drug delivery dynamics in biological systems. The results underscore the potential of fractal models to accurately represent diffusion processes in heterogeneous and complex biological environments. Full article

Community detection in complex networks plays a pivotal role in modern scientific research, including in social network analysis and protein structure analysis. Traditional community detection methods face challenges in integrating heterogeneous multi-source information, capturing global semantic relationships, and adapting to dynamic network evolution. This paper proposes a novel unsupervised multimodal community detection algorithm (UMM) based on fractal iteration. The core idea is to design a dual-channel encoder that comprehensively considers node semantic features and network topological structures. Initially, node representation vectors are derived from structural information (using feature vectors when available, or singular value decomposition to obtain feature vectors for nodes without attributes). Subsequently, a parameter-free graph convolutional encoder (PFGC) is developed based on fractal iteration principles to extract high-order semantic representations from structural encodings without requiring any training process. Furthermore, a semantic–structural dual-channel encoder (DC-SSE) is designed, which integrates semantic encodings—reduced in dimensionality via UMAP—with structural features extracted by PFGC to obtain the final node embeddings. These embeddings are then clustered using the K-means algorithm to achieve community partitioning. Experimental results demonstrate that the UMM outperforms existing methods on multiple real-world network datasets. Full article

With the rapid proliferation of real-time digital communication, particularly in multimedia applications, securing transmitted image data has become a vital concern. While chaotic systems have shown strong potential for cryptographic use, most existing approaches rely on low-dimensional, integer-order architectures, limiting their complexity and resistance to attacks. Advances in fractional calculus and memristive technologies offer new avenues for enhancing security through more complex and tunable dynamics. However, the practical deployment of high-dimensional fractional-order memristive chaotic systems in hardware remains underexplored. This study addresses this gap by presenting a secure image transmission system implemented on a field-programmable gate array (FPGA) using a universal high-dimensional memristive chaotic topology with arbitrary-order dynamics. The design leverages four- and five-dimensional hyperchaotic oscillators, analyzed through bifurcation diagrams and Lyapunov exponents. To enable efficient hardware realization, the chaotic dynamics are approximated using the explicit fractional-order Runge–Kutta (EFORK) method with the Caputo fractional derivative, implemented in VHDL. Deployed on the Xilinx Artix-7 AC701 platform, synchronized master–slave chaotic generators drive a multi-stage stream cipher. This encryption process supports both RGB and grayscale images. Evaluation shows strong cryptographic properties: correlation of $-6.1081\times {10}^{-5}$, entropy of 7.9991, NPCR of 99.9776%, UACI of 33.4154%, and a key space of $2^{1344}$, confirming high security and robustness. Full article

In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients $|c_2|$, $|c_3|$ for functions in these subclasses of bi-univalent functions. Additionally, by using the values of $a_2,a_3$ we determine the Fekete–Szegö inequality results. Moreover, a few new subclasses are deduced that have not been studied in relation to Liouville–Caputo fractional derivatives so far. The implications of the results are also emphasized. Our results are concrete examples of several earlier discoveries that are not only improved but also expanded upon. Full article

Research on reliability control for enhancing power systems under random loads holds significant and undeniable importance in maintaining system stability, performance, and safety. The primary challenge lies in determining the reliability index while optimizing system parameters. To effectively address this challenge, we developed a novel intelligent algorithm and conducted an optimal reliability assessment for a Negative Stiffness Device (NSD) seismic isolation structure incorporating fractional-order damping. This algorithm combines the Gaussian Radial Basis Function Neural Network (GRBFNN) with the Particle Swarm Optimization (PSO) algorithm. It takes the reliability function with unknown parameters as the objective function, while using the Backward Kolmogorov (BK) equation, which governs the reliability function and is accompanied by boundary and initial conditions, as the constraint condition. During the operation of this algorithm, the neural network is employed to solve the BK equation, thereby deriving the fitness function in each iteration of the PSO algorithm. Then the PSO algorithm is utilized to obtain the optimal parameters. The unique advantage of this algorithm is its ability to simultaneously achieve the optimization of implicit objectives and the solution of time-dependent BK equations.To evaluate the performance of the proposed algorithm, this study compared it with the algorithm combines the GRBFNN with Genetic Algorithm (GA-GRBFNN)across multiple dimensions, including performance and operational efficiency. The effectiveness of the proposed algorithm has been validated through numerical comparisons and Monte Carlo simulations. The control strategy presented in this paper provides a solid theoretical foundation for improving the reliability performance of mechanical engineering systems and demonstrates significant potential for practical applications. Full article

In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article

This study addresses the practical stability analysis of Riemann-Liouville fractional-order nonlinear systems with time delays. We first establish a rigorous formulation of initial conditions that aligns with the properties of Riemann-Liouville fractional derivatives. Subsequently, a generalized definition of practical stability is introduced, specifically tailored to accommodate the hybrid dynamics of fractional calculus and time-delay phenomena. By constructing appropriate Lyapunov-Krasovskii functionals and employing an enhanced Razumikhin-type technique, we derive sufficient conditions ensuring practical stability in the $L_p$-norm sense. The theoretical findings are validated through illustrative example for fractional order nonlinear systems with time delays. Full article

In this paper, we investigate the memory effects introduced by the time-fractional Schrödinger equation proposed by Naber on quantum entanglement and quantum dense coding under amplitude damping noise. Two formulations are analyzed: one with fractional operations applied to the imaginary unit and one without. Numerical results show that the formulation without fractional operations on the imaginary unit may be more suitable for describing non-Markovian (power-law) behavior in dissipative environments. This finding provides a more physically meaningful interpretation of the memory effects in time-fractional quantum dynamics and indirectly addresses fundamental concerns regarding the violation of unitarity and probability conservation in such frameworks. Our work offers a new perspective for the application of fractional quantum mechanics to realistic open quantum systems and shows promise in supporting the theoretical modeling of decoherence and information degradation. Full article

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

This study focuses on the analysis of soliton solutions within the framework of the time-fractional cubic–quintic nonlinear Schrödinger equation (TFCQ-NLSE), a powerful model with broad applications in complex physical phenomena such as fiber optic communications, nonlinear optics, optical signal processing, and laser–tissue interactions in medical science. The nonlinear effects exhibited by the model—such as self-focusing, self-phase modulation, and wave mixing—are influenced by the combined impact of the cubic and quintic nonlinear terms. To explore the dynamics of this model, we apply a robust analytical technique known as the sub-ODE method, which reveals a diverse range of soliton structures and offers deep insight into laser pulse interactions. The investigation yields a rich set of explicit soliton solutions, including hyperbolic, rational, singular, bright, Jacobian elliptic, Weierstrass elliptic, and periodic solutions. These waveforms have significant real-world relevance: bright solitons are employed in fiber optic communications for distortion-free long-distance data transmission, while both bright and dark solitons are used in nonlinear optics to study light behavior in media with intensity-dependent refractive indices. Solitons also contribute to advancements in quantum technologies, precision measurement, and fiber laser systems, where hyperbolic and periodic solitons facilitate stable, high-intensity pulse generation. Additionally, in nonlinear acoustics, solitons describe wave propagation in media where amplitude influences wave speed. Overall, this work highlights the theoretical depth and practical utility of soliton dynamics in fractional nonlinear systems. Full article

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel $\sigma$-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures. Full article