Fractal and Fractional

The fractional Langevin equation is used to analyze the erratic motion of a particle in a complex environment. Hydrodynamic backflows and restoring harmonic forces can work concurrently on the probe particle in addition to the constraints imposed by the complex environment, whose characteristics are represented in the friction force the particle encounters. The friction force, which is used for the modelling, corresponds to a damped oscillating function and it is expressed by a one-parameter Mittag-Leffler function. Analytical solutions for the relaxation functions are extracted, and through them observables like the position autocorrelation function, velocity autocorrelation function and mean square displacement are explicitly given. The competition between hydrodynamic fluctuations and frictional forces establishes a memory whose time scale is larger than the diffusional time scale, and during this time window the motion of the particle is superdiffusive. Full article

To further tap into the vibration isolation potential of vehicle mechatronic ISD (Inerter–Spring–Damper) suspension systems, a positive real synthesis design method for vehicle ISD suspensions is proposed based on the fractional-order bridge networks. Firstly, a quarter-car dynamic model of the mechatronic ISD suspension system is established. Subsequently, based on the impedance solution method for bridge networks, a basic bridge network structure and its corresponding series-parallel network configuration are constructed for comparative analysis. Then, with the aim of improving the overall performance of the vehicle suspension system, the particle swarm optimization algorithm is employed to determine the optimal parameters for the mechatronic ISD suspension. Finally, simulation results under random road excitation at a vehicle speed of 20 m/s indicate that, compared to the traditional passive suspension system, the fractional-order bridge network-based ISD suspension system achieves reductions of 3.9%, 27.5%, and 11.8% in the root mean square (RMS) values of vehicle body acceleration, suspension working space, and dynamic tire load, respectively. In contrast, the integer-order bridge network-based ISD suspension system only achieves reductions of 3.0%, 24.2%, and 4.5% in these respective performance metrics. The results confirm that, with an equivalent number of components, the fractional-order bridge network exhibits superior vibration isolation performance relative to series-parallel networks, thereby providing novel methodological guidance for the design of vehicle mechatronic ISD suspension. Full article

This work investigates the moderate deviation principle for a class of two-time-scale Caputo fractional stochastic differential equations. The driving noise of the slow variable is fractional Brownian motion with Hurst index $H\in ({\displaystyle \frac{1}{2}},1)$. The driving noise of the fast variable is standard Brownian motion. The fractional derivative operator of the slow variable is defined by Caputo, and the derivative of the fast variable is of the integer order. The proof process is mainly based on the weak convergence method of fractional Brownian motion variational representation. We first establish the moderate deviation principle by proving the weak convergence of the single-time-scale controlled version. Subsequently, we combine Khasminskii time discretization technology to extend the theoretical framework to two-time-scale systems. Finally, a concrete computational case is offered to demonstrate the applicability of the theoretical framework. Full article

This paper studies mixed impulsive boundary value problems involving tempered $\mathsf{\Psi }$-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered $\mathsf{\Psi }$-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results. Full article

In this study, we present a unified symmetry-conservation solution analysis of a well-posed resonant nonlinear Schrödinger (NLS)-type equation incorporating spatio-temporal dispersion and inter-modal dispersion. Working within the truncated M-fractional derivative framework, we first construct exact traveling-wave solution families via the Kudryashov expansion method, together with the corresponding parameter constraints and limiting cases. We then determine the admitted Lie point symmetries and establish the associated Lie algebra, including the commutator structure, adjoint representation, and an optimal system of one-dimensional subalgebras for classification. Using the conservation theorem, we derive conserved vectors associated with the fundamental invariances of the model; in the NLS setting and under suitable conditions, these quantities can be interpreted as generalized power (mass), momentum, and energy-type invariants. Overall, the results provide explicit wave profiles and structural invariants that enhance the interpretability of the model and offer benchmark expressions useful for further qualitative, numerical, and stability investigations in nonlinear dispersive wave dynamics. Full article

The Jurassic lacustrine oil shale in southwest China has become a primary production layer due to its high yield and substantial reserves. However, influenced by the lacustrine environment, the vertical profile of the lacustrine shale reservoir shows alternating deposits of shale and carbonate rock. This complex lithological combination results in significant heterogeneity in reservoir types, reservoir distribution, and internal structure. Currently, research on micro-pore structure and hydrocarbon storage mechanisms in lacustrine shales is insufficient, necessitating the elucidation of their micro-characteristics to support future exploration and development. This research focuses on the Da’anzhai Member of Jurassic Ziliujing Formation. Various techniques—including organic geochemical analysis, X-ray diffraction, physical property testing, gradient centrifugation, and gradient drying NMR monitoring—were employed to investigate the micro-pore structure and fluid storage mechanisms of the lacustrine shale reservoir. The following insights were gained from this research. The organic matter pores (OMP) and inorganic pores (IP) developed within the Da’anzhai lacustrine shale reservoir together create the storage space for shale oil, while micro-fractures further enhance the reservoir’s storage capacity and flow performance. Lacustrine shale oil exists in three storage states: mobile oil, bound oil, and adsorbed oil. Mobile oil is primarily located within the micro-fractures and large pores (greater than 350 nm) of the shale reservoir and is the main target for industrial extraction. Bound oil is mainly found in the meso-pores, micropores, and narrow pore structures between rock grains (30 nm to 350 nm), and, theoretically, could potentially be developed through engineering methods such as hydraulic fracturing. Adsorbed oil, due to its close binding with organic matter and clay mineral surfaces, is difficult to release effectively using conventional techniques. The OM abundance, the mineral composition of lacustrine shale, and the pore structure all influence the storage states of shale oil. While a high TOC value increases the amount of mobile oil, the strong adsorption properties of kerogen and organic matter lead to the accumulation of adsorbed oil, which inhibits oil flow. Clay minerals further restrict oil flow by enhancing adsorption, while brittle minerals facilitate the movement of mobile oil by expanding pore space. Based on fractal geometry theory and multi-scale testing results, the large pores in the Da’anzhai lacustrine shale have a high fractal dimension and exhibit complex shapes. However, as pore complexity increases, the amount of adsorbed oil rises significantly, which in turn reduces the proportion of movable oil. Full article

During coal mining, the development of joint fractures in overlying rock strata is one of the key factors that degrade the mechanical properties of rock masses, form water-conducting fracture zones, and induce safety hazards. To investigate the fracture evolution characteristics of overlying strata during coal extraction under thick and hard roof conditions, this study established a mining physical model based on similarity simulation technology, tracked the fracture evolution process, and performed quantitative analysis using fractal theory. The results show that fracture development is significantly correlated with the mining advance distance: the fractal dimension of fractures is small in the initial mining stage and gradually increases as the working face advances. When the mining width exceeds the ultimate span of the roof, local fractures expand rapidly with a sharp rise in the fractal dimension to 1.436; further increasing the mining width triggers large-scale sudden fracture expansion, resulting in severe degradation of rock mass integrity, with the maximum fractal dimension reaching 1.445. The research findings provide theoretical references for safety management and disaster prevention in coal mining under thick and hard roof conditions. Full article

In this study, the bending behavior of beams is investigated using the fractional Euler–Bernoulli beam model. This model is developed based on fractional calculus, particularly employing the Riesz–Caputo derivatives, and is capable of accurately accounting for nonlocal and size-dependent effects in structural beams. Unlike classical models that rely on integer-order derivatives, the present model uses fractional-order derivatives, which offer greater precision in analyzing beam behavior at small scales such as micro and nano levels. In this work, various beams with different boundary conditions and loading types are analyzed. To solve the governing fractional equations, a numerical algorithm based on the finite difference method is developed, which also allows for the incorporation of a variable characteristic length function along the beam. The numerical simulation results demonstrate that the order of the fractional derivative and the characteristic length have a direct impact on the amount of beam deflection. These findings indicate that the Euler–Bernoulli model based on Riesz–Caputo derivatives has high potential for realistic simulation of beam bending behavior at small scales, making it an effective tool for accurate analysis of microscale structures. Full article

In this paper, we mainly investigate the radial symmetry and monotonicity of positive solutions for a nonlinear system involving pseudo-relativistic operators and fractional derivatives of order $(0,1)$. First, we prove a more general Narrow Region Principle and a Decay at Infinity Principle, which are essential for nonlocal pseudo-relativistic operators. Then, by using the direct method of moving planes, we prove the radial symmetry and radial monotonicity of positive solutions for the nonlinear system in the bounded domain $B_1\left(0\right)$ and the whole space, respectively. Finally, we show that the positive solutions of the system are strictly monotonically increasing in a Lipschitz coercive epigraph. Full article

Multilayered geological structures are common in geotechnical engineering, where cooling shrinkage induces polygonal cracks in interlayers, compromising rock mass strength, permeability, and long-term stability. Existing thermo-mechanical studies on layered rock cracking insufficiently address the combined effects of weak interlayer geometry or interface-regulated mechanisms. To address this gap, based on meso-damage mechanics and thermodynamics, this study adopts a thermo-mechanical coupling simulation method considering rock heterogeneity, innovatively focusing on the understudied stress transfer effect at strong–weak interlayer interfaces. Systematic investigations were conducted on the initiation, propagation, and saturation of polygonal cracks in plate-like layered rocks under surface cooling, analyzing the influences of weak interlayer thickness, number, and position, and comparing surface vs. inner interlayer behaviors. Results showed that stress transfer interruption at weak–strong layer interfaces can inhibit crack propagation. Inter weak interlayers produce significantly more cracks and fragments than surface weak interlayers, with a stratified crack length distribution, and the maximum fragment area increases exponentially with weak interlayer thickness. Crack development is strongly influenced by weak interlayer thickness, with thinner layers dominated by non-penetrating cracks and thicker layers tending to develop penetrating lattice-like cracks. The inter layer stress and crack distribution exhibit fractal characteristics, with crack density decreasing layer by layer and no new cracks forming after saturation. This study clarifies the regulatory mechanism of weak interlayer features and surface cooling on crack evolution, quantifies interface effects and fractal characteristics, and provides a theoretical basis for instability prediction of layered rock structures in low-temperature geotechnical engineering. Full article

This paper proposes an iterative learning framework for a class of fractional-order nonlinear multi-agent systems operating under directed iteration-varying switching topologies. To suppress trial-to-trial fluctuations in initial states, a P-type initial condition learning mechanism is integrated into the update law, enabling each agent to actively compensate for its own startup offset in each iteration. The study designs a distributed iterative learning protocol using only local neighbor information, and this protocol can simultaneously achieve fault estimation and diagnosis. By constructing a fractional-order system model and adopting the contraction-mapping analysis method, sufficient conditions are derived in this paper, which guarantee that both the fault error and initial condition error converge asymptotically to zero as the number of iterations approaches infinity. The proposed scheme, based on iterative learning fault estimation, can effectively handle unknown nonlinearities without relying on an accurate system model. Numerical simulation results further verify the effectiveness of the designed fault observer in achieving fault estimation. Full article

The Kuramoto–Sivashinsky (KS) equation and its fractional generalizations (FKSs) arise as canonical models for a wide class of nonlinear dissipative–dispersive systems, including thin-film flows, combustion fronts, drift–wave turbulence in plasmas, and chemically reacting media, where shock-like and strongly localized structures play a central role in the dynamics. Despite their apparent simplicity, KS-type models become analytically intractable once higher-order dissipation, geometric effects, and memory (fractional) operators are incorporated, and standard perturbative or transform-based schemes often lead to cumbersome recursive structures, slow convergence, or severe restrictions on the initial data. In this work, a novel direct approximation procedure, referred to as the Tantawy Technique (TT), is developed and implemented to solve and analyze planar fractional KS-type equations and their Burgers-type reductions in a systematic manner. The central difficulty is to construct, for a given physically motivated initial profile, a rapidly convergent series in fractional time that remains stable for a broad range of the fractional order and transport coefficients, while still retaining a clear link to the underlying shock-wave physics. To overcome this, the TT combines (i) a Tanh-based exact shock solution of the planar integer-order KS equation, obtained first as a reference via the standard Tanh method, with (ii) a carefully designed fractional-time ansatz in powers of $t^{\rho }$, where the spatial coefficients are determined recursively from the governing equation in the Caputo sense. This construction yields closed-form expressions for the first few terms in the approximation hierarchy and allows one to monitor convergence through residual and absolute error measures. Full article

This study investigates the critical role of resonance capture dynamics in determining the energy dissipation performance of nonlinear energy sinks (NES). A fluid inerter combining mass amplification and damping characteristics is proposed as a core component, based on which two configurations of fractional-order NES (configured in series and parallel) are systematically constructed. The applicability of the complex averaging method in fractional-order systems has been addressed through fractional calculus (such as Leibniz properties), enabling it to be analyzed like integer-order systems. Employing the multi-scale perturbation method, the energy transfer mechanism between the primary oscillator and the NES is derived, leading to the analytical determination of optimal cubic stiffness and maximum energy transfer efficiency. Comparative simulation shows that the parameters of the inerter directly affect the magnitude of critical damping. The optimal cubic stiffness design method is more reliable than traditional methods and can ensure effective target energy transfer triggering. Further analysis of dissipation time shows that the performance of fractional-order NES is superior to integer-order NES; notably, the dissipation time of series fractional-order NES is significantly shorter than that of parallel and traditional NES. In summary, this study provides theoretical guidance for the design of lightweight and high-performance NES and will also promote the application of fractional calculus theory in the field of engineering vibration reduction. Full article

This paper studies a class of p-valent analytic functions in the open unit disk using a modified Tremblay operator based on the Riemann–Liouville fractional integral and derivative. We derive a series representation of the operator, which serves as a useful tool to explore its analytic and geometric properties. New third-order differential subordination results are obtained by defining suitable classes of admissible functions. We provide sufficient conditions to ensure subordination relations with a given dominant function, leading to inclusion results in general complex domains. In addition, several applications are given for specific choices of the target domain, showing how the framework is flexible and able to produce unified results in the theory of differential subordination for multivalent analytic functions. Full article

This investigation provides a comprehensive analytical framework for the topological morphology and global convergence dynamics governing a specific family of sixth-order iterative schemes designed for nonlinear equations with multiple roots. By invoking a Möbius conjugacy transformation upon the specialized polynomial class $f\left(z\right)={\left((z-p)(z-q)\right)}^m$, we project the iterative sequence onto the Riemann sphere $\widehat{\mathbb{C}}$, effectively recasting the algorithm as a discrete complex dynamic system. The core of this study lies in the bifurcation analysis of the associated parameter space. We meticulously chart the stability manifolds, tracing the evolution of critical orbits to distinguish between regions of predictable convergence and those characterized by chaotic instability. By examining the iterative methods generated by these rational endomorphisms, the research unveils the intricate fractal boundaries that delineate the basin of attraction, offering a profound insight into the structural robustness of higher-order methods. In the dynamical plane, the geometry of the basins of attraction is scrutinized to evaluate the robustness of the numerical flow and its sensitivity to the configuration of weight functions. By analyzing the fractal complexity of the boundaries within these basins, we provide a detailed characterization of the iterative morphology and its global reliability. The analytical findings are supported by high-resolution graphical representations and comparative numerical data, illustrating the superior performance and structural integrity of the proposed methods in solving nonlinear problems. Full article

This paper presents two enhanced variants of the Artificial Protozoa Optimizer (APO), namely the Adaptive Balanced Artificial Protozoa Optimizer (AB-APO) and the Fractional Calculus-Enhanced Artificial Protozoa Optimizer (FC-APO), for optimal multi-Distributed Energy Resources (DERs) planning in smart radial distribution networks. The proposed framework addresses the coordinated allocation of Electric Vehicle Charging Stations (EVCSs), photovoltaic (PV) units, and Battery Energy Storage Systems (BESS). The AB-APO introduces an adaptive balancing mechanism that dynamically regulates exploration and exploitation to improve convergence stability and robustness, while the FC-APO incorporates fractional-order dynamics to embed long-memory effects, enhancing numerical stability and search smoothness. The proposed optimizers are evaluated on the IEEE-33 and IEEE-69 bus systems under eight DERs penetration scenarios. Simulation results demonstrate significant reductions in real and reactive power losses, improved voltage profiles, and effective mitigation of EV-induced network stress. Real power loss reductions exceeding 54%, 38.53%, 53.78%, 38.20%, 61.68%, and 60.72% are achieved for the IEEE-33 system, while reductions of 64.32%, 63.51%, 64.33%, 63.51%, 67.31%, and 67.04% are obtained for the IEEE-69 system across Scenarios 3–8. Overall, the results highlight the effectiveness of adaptive balancing and fractional-order modeling in strengthening APO-based optimization and confirm the suitability of the AB-APO and FC-APO as efficient planning tools for future smart distribution networks. Full article

Some open problems regarding fractional powers of the negative generator of a discrete-time random walk and a Markov process are discussed. The suggested approach combines analytic and probabilistic ideas and may be useful for developing fractional operators with multidimensional and/or abstract discrete arguments. Full article

A Monte Carlo based fractal prediction model is proposed to describe the evolution of surface morphology during the running-in process. The model accounts for the random and fractal characteristics of worn surfaces. The Weierstrass–Mandelbrot function is employed to simulate rough surfaces and establish the correlation between fractal dimension and surface roughness. By integrating traditional sliding wear models with surface effect functions, a unified prediction framework is developed. Experiments are conducted to obtain worn surface parameters and calculate fractal dimensions at different running-in stages. Model parameters are optimized by minimizing the variance between experimental and predicted results. Monte Carlo simulations are then introduced to represent the stochastic nature of the friction system, thereby improving prediction accuracy and objectivity. The proposed model reveals locally random yet globally convergent patterns, which are consistent with experimental observations. It effectively captures the stochastic evolution of surface morphology and provides a reliable approach for predicting worn surface behavior during running-in. Full article

The coordination of switched fractional-order nonlinear heterogeneous multi-agent systems (FONHMASs) with cooperative and antagonistic interactions presents significant challenges due to the complex coupling of switched fractional-order dynamics. Crucially, existing control methods typically rely on integer-order assumptions and precise system modeling, which are inadequate for capturing the inherent non-local memory behaviors of fractional dynamics. Furthermore, they generally assume fixed agent dynamics, and cannot be applied to switched FONHMASs where the continuity of agents’ dynamics is violated at switching instants. Considering the constraints of precise modeling difficulties and limited task time for switched FONHMASs in practice, a distributed $D^{\alpha }$-type iterative learning control (ILC) protocol is proposed to achieve bipartite consensus in the presence of cooperative and antagonistic interactions. Also, without relying on repetitive initial conditions, based on a presented initial state learning mechanism and $D^{\alpha }$-type ILC protocol, the bipartite consensus error convergence property with each iteration is achieved. Additionally, in consideration of external disturbances, the robustness of the iterative bipartite consensus controller for the switched FONHMASs is analyzed. Simulation results confirm that the switched FONHMASs achieve the convergence and robustness of the bipartite consensus errors along the iteration direction. In addition, the proposed $D^{\alpha }$-type ILC protocol achieves a maximum root-mean-square-error (MRMSE) of 0.0168 in time domain, significantly outperforming the integer-order ILC (MRMSE = 0.3601) and fractional-order PID control (MRMSE = 0.7550), confirming its superiority. Full article