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Fractal and Fractional, Volume 9, Issue 8

August 2025 - 75 articles

Cover Story: This paper studies coherent upper conditional previsions—nonlinear functionals that extend expectations while maintaining consistency. We explore their integral representation via the countably additive Möbius transform, which applies when defined relative to a monotone set function of bounded variation. We further show that a representation on the Borel σ-algebra is possible even when using the Choquet integral with respect to Hausdorff measures, which lack bounded variation. This is due to Hausdorff outer measures being metric, making all Borel sets measurable. View this paper
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Articles (75)

  • Article
  • Open Access
1,272 Views
20 Pages

This paper presents a finite element implementation of a fractional rheological consolidation model in ABQUS, in which the fractional Merchant model governs the mechanical behavior of the soil skeleton, and the water flow is controlled by the fractio...

  • Article
  • Open Access
1 Citations
1,043 Views
28 Pages

This paper introduces an innovative artificial neural networks-based analytical solver for fractional partial differential equations (fPDEs), combining neural networks (NNs) with symbolic computation. Leveraging the powerful function approximation ab...

  • Article
  • Open Access
1 Citations
911 Views
29 Pages

Existence, Uniqueness, and Stability of Weighted Fuzzy Fractional Volterra–Fredholm Integro-Differential Equation

  • Sahar Abbas,
  • Abdul Ahad Abro,
  • Syed Muhammad Daniyal,
  • Hanaa A. Abdallah,
  • Sadique Ahmad,
  • Abdelhamied Ashraf Ateya and
  • Noman Bin Zahid

This paper investigates a novel class of weighted fuzzy fractional Volterra–Fredholm integro-differential equations (FWFVFIDEs) subject to integral boundary conditions. The analysis is conducted within the framework of Caputo-weighted fractiona...

  • Article
  • Open Access
766 Views
29 Pages

Automated First-Arrival Picking and Source Localization of Microseismic Events Using OVMD-WTD and Fractal Box Dimension Analysis

  • Guanqun Zhou,
  • Shiling Luo,
  • Yafei Wang,
  • Yongxin Gao,
  • Xiaowei Hou,
  • Weixin Zhang and
  • Chuan Ren

Microseismic monitoring has become a critical technology for hydraulic fracturing in unconventional oil and gas reservoirs, owing to its high temporal and spatial resolution. It plays a pivotal role in tracking fracture propagation and evaluating sti...

  • Article
  • Open Access
1 Citations
814 Views
23 Pages

The nanoporous structure of coal is crucial for the occurrence and development of coalbed methane (CBM). This study, leveraging the combined characterization of atomic force microscopy (AFM) and Gwyddion software (v2.62), investigated six anthracite...

  • Article
  • Open Access
4 Citations
802 Views
19 Pages

The dynamic process of water depletion plays a critical role in both surface coalbed methane (CBM) development and underground gas extraction, reshaping water–rock interactions and inducing complex permeability responses. Addressing the limited...

  • Article
  • Open Access
1 Citations
994 Views
30 Pages

Fracture Complexity and Mineral Damage in Shale Hydraulic Fracturing Based on Microscale Fractal Analysis

  • Xin Liu,
  • Jiaqi Zhang,
  • Tianjiao Li,
  • Zhengzhao Liang,
  • Siwei Meng,
  • Licai Zheng and
  • Na Wu

The geological structural complexity and microscale heterogeneity of shale reservoirs, characterized by the brittleness index and natural fracture density, exert a decisive effect on hydraulic fracturing’s effectiveness. However, the mechanisms...

  • Article
  • Open Access
808 Views
17 Pages

Understanding the intrinsic relationship between microscopic structures and macroscopic mechanical properties of rock under freeze–thaw (F-T) conditions is essential for ensuring the safety and stability of geotechnical engineering in cold regi...

  • Article
  • Open Access
1,245 Views
22 Pages

In the field of deep learning, the traditional Vision Transformer (ViT) model has some limitations when dealing with local details and long-range dependencies; especially in the absence of sufficient training data, it is prone to overfitting. Structu...

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Fractal Fract. - ISSN 2504-3110