Fractal and Fractional, Volume 10, Issue 2

In this article, the $\alpha$-fractal interpolation function $f^{\alpha }$ corresponding to any function f belonging to the weighted Sobolev space ${\mathcal{W}}_{\rho }^{r,2}\left(I\right)$ is defined. The convergence of sequences of $\alpha$-fractal interpolation functions correspondin...

Generative Adversarial Networks (GANs) perform well on natural images but often fail in domains governed by strict geometric or symbolic constraints. This work focuses on convolutional GANs and studies how their inductive biases interact with two con...

Low-maturity shale is a prime target for in situ conversion (ICP), yet heating window selection remains largely empirical because pore evolution and hydrocarbon generation are rarely quantified in tandem. Nenjiang Formation shale from the Songliao Ba...

Beamforming plays a central role in enhancing the performance of communication systems; however, suppressing sidelobes in planar antenna arrays (PAAs) while maintaining a compact aperture remains a challenging nonlinear optimization problem. This art...

This paper studies the issues of human balancing and stability in the sagittal plane using fractional and integer order time delay feedback control. The neural-mechanical model of human balance is represented as an inverted pendulum controlled by tor...

Accurately determining the T₂ cutoff value is critical for evaluating fluid mobility in deep tight reservoirs, yet strong pore structure heterogeneity challenges traditional methods. This study proposes a non-destructive prediction method based on mu...

The primary objective of this research article is to investigate the concept of extended $\mathcal{F}$-metric spaces and to establish a series of fixed point theorems for generalized contractions within this framework. We further introduce and analyze the notion...

Clarifying the complex interaction between climate risks and the new energy industry chain is of key significance to advancing the energy transition and strengthening industrial chain robustness. This research pairwise-matches the climate physical ri...

Motivated by the close relationship between the matrix square root and the matrix sign function, this paper develops a new high-order iterative framework for computing the principal square root of a matrix and its inverse. The proposed approach is de...

This paper develops a unified analytical approach for pricing a broad class of volatility-linked financial derivatives under the sub-mixed fractional geometric Brownian motion model. The proposed framework captures key empirical features of financial...

Fractional calculus approach is used to analyze the model of surfactant transport by anomalous diffusion and its adsorption on an interface in a liquid-liquid system. The anomalous diffusion is modeled by time-fractional partial differential equation...

Person re-identification (Re-ID) using infrared surveillance cameras has attracted increasing attention due to its robustness under low-light conditions. However, infrared images generally suffer from a low spatial resolution, which degrades Re-ID pe...

This article examines the behavior of seismic noise fields over the Japanese islands recorded by the F-net seismic network for 1997–2025. This paper uses nonlinear noise statistics: the entropy of the wavelet coefficient distribution, the Donoh...

The Kuramoto–Sivashinsky (KS) equation and its fractional form (FKS) are widely used across scientific fields, including fluid dynamics, plasma physics, and chemical processes, to model nonlinear phenomena such as shock waves. It is worth empha...

This article is devoted to constructing the fractional powers of operators and their matrix approximations. A key feature of this study is the use of a spectral approach that remains applicable even when the base operator does not generate a semigrou...

We propose a fast L2-1_σ finite element method for solving the time fractional Keller–Segel equations with a Caputo fractional derivative of $\alpha \in (0,1)$. Firstly, the fast L2-1_σ scheme on the graded mesh is used to discretize...

Rock pore–fracture systems exhibit inherent fractal characteristics, which exert a significant influence on fluid transport. In this study, coal rock is selected as the representative medium. Based on fractional calculus in physical fractal spa...

We present a conceptual and computational investigation of vacuum memory within a discrete toy-model framework. In this phenomenological approach, we introduce an effective memory field that records virtual events and nonlocal couplings on a lattice,...

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...

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 bri...

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 drivin...

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-fr...

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...

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...

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 fract...

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 acc...

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 Pri...

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...

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 ini...

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 turbul...

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...

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 serve...

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 invo...

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-D...

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 o...

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 f...

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. Cru...

The roughness of rock fractures has complex features that affect how fluids move through them. This research looks at how gas and water flows change in rough fractures when they are moved using a model based on fractal geometry. Rough surfaces are cr...

Conventional high-pressure mercury injection (HPMI) alone inadequately characterizes the fractal properties of nanopores in tight sandstone reservoirs, and existing fractal models for HPMI, low-temperature nitrogen adsorption (LTNA), and nuclear magn...

High-water material (HWM) is widely used for roadside filling in gob-side entry retaining (GER), where its creep behavior under sustained loading critically influences the long-term stability of the roadway. To enhance the long-term mechanical perfor...

Given a finite set of interpolation data $\{(x_i,y_i)\in I\times \mathbb{R},i=0,1,\dots ,N\}$, $I=[x_0,x_N]$, we construct a class of nonlinear fractal interpolation functions whose graphs are realized as attractors of appropriately defined iterated function system...

In this paper, we investigate the finite-time consensus problem of a fractional-order multi-agent system with repetitive motion. The system under consideration consists of robotic agents with a leader and a fixed communication topology. A distributed...

This paper develops a fractional six-compartment model to describe malware spread in wireless sensor networks. To represent actual network activity, the model is constructed using generalized proportional-Caputo operators that incorporate memory and...

This paper concentrates on fractional-order fuzzy cellular neural networks (FOFCNNs) with bounded uncertainties and transmission delays. To better capture real-world dynamic behaviors, the fuzzy AND and OR operators are employed to construct drive-re...

In this article, we propose an expanded mixed finite element method for variable-coefficient fractional dispersion equations (FDEs). By introducing two intermediate variables, $p=Du$ and $\sigma =-I_{\theta }^{\beta }p$, the FDEs are reformulated into a...

This article presents a detailed study of the two-dimensional quaternion fractional Fourier transform (2D QFRFT) and investigates its role in the probabilistic analysis of quaternion-valued signals. The 2D formulation is constructed by applying fract...

Brain electrical activity, as recorded through electroencephalography (EEG), displays scale-free temporal fluctuations indicative of fractal behavior and complex dynamics. This study explores the use of the Higuchi Fractal Dimension (HFD) as a proxy...

In this study, we address a coupled system of nonlinear fractional delay differential equations subject to cross-coupled multi-point boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretica...

Mock theta functions, introduced by Ramanujan in his last letter to Hardy, play a significant role in q-series theory and have natural connections to fractional q-calculus. In this paper, we study bilateral hypergeometric series of the form ${}_2\psi _2$=...

We investigate the existence of positive normalized (mass-constrained) solutions for the fractional nonlinear Schrödinger equation ${(-\Delta )}^{s}v+V\left(x\right)v=\lambda v+{\mu \left|v\right|}^{p-2}v+{\left|v\right|}^{2_s^{*}-2}v\mathrm{in}{\mathbb{R}}^N,{\parallel v\parallel }_2^2=b^2,$ where $N>2s$, $s\&is$...