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Fractal and Fractional

Fractal and Fractional is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering, published monthly online by MDPI.

Quartile Ranking JCR - Q1 (Mathematics, Interdisciplinary Applications)

All Articles (3,792)

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 equations in the bulk phases. The adsorption of surfactant is described by the corresponding time-fractional Neumann boundary conditions at the interface. The adsorption process is considered under mixed barrier-diffusion control, described by first-order ordinary differential equation, which relates the subsurface concentration with that on the interface. A second relation between these concentrations is derived in terms of a fractional equation by application of Laplace transform technique. By combining both relations the subsurface concentration is eliminated and a single multi-term fractional ordinary differential equation for the surfactant concentration on the interface is derived. Different adsorption kinetic models are considered. In the case of Henry adsorption isotherm the model is linear and possesses analytical solution in terms of multinomial Mittag-Leffler functions. In the cases of Volmer and van der Waals adsorption isotherms nonlinear differential equations of fractional order are obtained. They are reformulated in equivalent integral form, which is used for computer simulation of the process of adsorption. Numerical results are presented and compared with analytical asymptotic predictions.

13 February 2026

Schematic sketch of barrier-diffusion controlled adsorption at liquid/liquid interface.

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 performance. To address this issue, this study proposes a part attention and contrastive loss-based super-resolution reconstruction network (PCSR-Net) and a unified infrared-only Re-ID framework. The proposed PCSR-Net consists of a correlation-based super-resolution reconstruction network (CoSR-Net), a feature extractor for Re-ID, and a part attention mechanism that estimates the importance of different body regions. In addition, contrastive loss and part-aware reconstruction loss are incorporated to guide the super-resolution process toward identity-discriminative representations. Experimental results on DBPerson-Recog-DB1 and SYSU-MM01 demonstrate that the proposed method outperforms state-of-the-art approaches in terms of the equal error rate (EER), mean average precision (mAP), and rank-1 accuracy, validating its effectiveness for infrared-based person Re-ID.

12 February 2026

Overall workflow of the proposed method.

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 Donoho–Johnston (DJ) wavelet index, and the multifractal singularity spectrum support width. These parameters were chosen because their changes reflect the complication or simplification of the noise structure. Changes in the structure of seismic noise properties are analyzed in comparison with a sequence of strong earthquakes. Using a model of the intensity of interacting point processes, the effect of the leading of local noise property extrema relative to the seismic event times is estimated. Using the Hilbert–Huang decomposition, the synchronization of the amplitudes of the envelopes of noise property time series for different IMF levels is estimated. A sequence of weighted probability density maps of extreme values of noise properties is analyzed in comparison with the mega-earthquake of 11 March 2011 and the preparation of another possible strong seismic event.

12 February 2026

Purple dots represent the locations of F-net seismic stations on the Japanese islands. Gray circles are drawn with a radius of 250 km centered on each station. The union of these circles represents the network’s “area of influence,” which is being explored further.

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 emphasizing that this contribution is part (II) of a larger, systematic research program aimed at modeling, for the first time, completely nonintegrable, nonplanar, and fractional nonplanar evolutionary wave equations. This work focuses on the nonplanar KS framework and its applications to dust–acoustic shock waves in a complex plasma composed of inertial dust grains and inertialess nonextensive ions. This study analyzes both the nonplanar integer KS and nonplanar FKS equations, accounting for geometric effects. This is because the nonplanar model is most suitable for analyzing various nonlinear phenomena (e.g., shock waves) that arise and propagate in plasma physics, fluids, and other physical and engineering systems. Since the nonplanar KS equation is a fully non-integrable problem, its analysis poses a significant challenge for studying the properties of nonplanar shock waves in plasma physics. Therefore, the primary objective of this study is to analyze the nonplanar KS equation using the Ansatz method, thereby deriving semi-analytical solutions that simulate the propagation mechanism of nonplanar shock waves in various physical systems. Following this, we investigate the effect of the fractional factor on the profiles of nonplanar dust–acoustic shock waves to elucidate their propagation mechanism and assess the impact of the memory factor on their behavior. To achieve the second goal, we face a significant challenge because the model under study does not support exact solutions and is more complex than simpler physical models. Thus, the Tantawy technique is employed to overcome this challenge and to analyze this model for generating highly accurate analytical approximations suitable for modeling nonplanar fractional shock waves in various plasma models and in other physical and engineering systems.

12 February 2026

Profiles of planar (
  
    υ
    =
    0
    )
  
 shock-wave solutions (22) and (3) to the planar Burgers Equation (21) and the planar KS Equation (23), respectively, are compared: (a) the planar (
  
    υ
    =
    0
    )
  
 Burgers-shock-wave solution (22) in the plane 
  
    x
    ,
    t
  
 and (b) the planar (
  
    υ
    =
    0
    )
  
 KS-shock wave solution (3) in the plane 
  
    x
    ,
    t
  
. Here, 
  
    
      q
      ,
      η
      ,
      γ
      ,
      λ
    
    =
    
      0.7
      ,
      0.1
      ,
      1
      ,
      0.1
    
  
, which leads to 
  
    
      α
      ,
      β
    
    =
    
      −
      0.759257
      ,
      0.05
    
  
.

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Continuous/Discrete-Time Fractional Systems
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Continuous/Discrete-Time Fractional Systems

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Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering
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Complexity, Fractality and Fractional Dynamics Applied to Science and Engineering

Editors: Alexandra M. S. F. Galhano, Sergio Adriani David, António M. Lopes

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Fractal Fract. - ISSN 2504-3110