{
  "abstract": "This paper studies the finite-time stability of linear stochastic fractional-order systems with time delay, in which the dynamics are generated by the proportional Caputo fractional derivative of order ς∈(12,1) with proportional parameter ϖ∈(0,1]. Starting from the integral representation of solutions and using the Itô isometry, we derive a mean-square estimate in which the exponential factor of the proportional kernel sharpens the constants. Combining this estimate with the classical Gronwall inequality and with a generalized one, we establish two finite-time stability criteria: an iterative criterion built on the delay intervals and a closed-form criterion of the Mittag–Leffler type. For ϖ=1, both criteria reduce exactly to known results for Caputo-type stochastic delay systems, so the present results extend the theory to the whole proportional family. The two criteria are compared analytically, a weighted refinement that reduces their conservatism is given, and the influence of the memory of the fractional operator and of the proportional parameter is discussed. Numerical simulations, based on a Monte Carlo scheme for the integral representation of the solution with quantified sampling error, illustrate the two criteria, compare their certified stability horizons, and confirm the theoretical findings.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia",
    "Department of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia",
    "Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia",
    "Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia",
    "Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia",
    "Mathematics Education Section, Faculty of Education and Arts, Sohar University, P.O. Box 44, Sohar 311, Oman"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 6,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Raouf Fakhfakh"
    },
    {
      "affs": [
        1
      ],
      "name": "Rabab Alzahrani",
      "orcid": "0009-0005-8167-2064"
    },
    {
      "affs": [
        2
      ],
      "name": "Fatimah Alshahrani",
      "orcid": "0000-0003-1306-7688"
    },
    {
      "affs": [
        3
      ],
      "name": "Hend Aljahani"
    },
    {
      "affs": [
        4
      ],
      "name": "Foued Mtiri"
    },
    {
      "affs": [
        5
      ],
      "name": "Abdellatif Ben Makhlouf",
      "orcid": "0000-0001-7142-7026"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ejccydq1q3dq8.bib",
  "cited_by": 0,
  "doi": "10.3390/fractalfract10100697",
  "fulltext_url": null,
  "id": "ejccydq1q3dq8",
  "issns": [
    "2504-3110"
  ],
  "landing_url": "https://www.mdpi.com/2504-3110/10/10/697",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Nonlinear Differential Equations Analysis"
    ]
  ],
  "openalex": "W7220368612",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/ejccydq1q3dq8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:32Z",
      "native_id": "10.3390/fractalfract10100697",
      "source": "crossref",
      "url": "https://www.mdpi.com/2504-3110/10/10/697"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2504-3110",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Finite-Time Stability of Linear Stochastic Proportional Fractional-Order Systems with Time Delay",
  "title_text": "Finite-Time Stability of Linear Stochastic Proportional Fractional-Order Systems with Time Delay",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:32Z",
  "url": "https://research.draconitic.com/paper/ejccydq1q3dq8",
  "venue": "Fractal and Fractional",
  "versions": []
}