{
  "abstract": "In this article, we utilize -Dickson polynomials to improve a method for spectral collocation that is used to numerically solve mixed classical and fractional-order integro-differential equations of the Fredholm type with variable coefficients under boundary conditions (IFDEs-CF) in the Caputo sense. The approximate solutions are assumed in the form of the truncated Dickson polynomial series. This innovative method transforms the equation with conditions into a linear algebraic system for the unknown Dickson coefficients using a matrix technique. Solving this system determines the coefficients of the assumed solution. We ultimately analyze the results with respect to the α parameter in the Dickson polynomials. The behavior of the solutions is monitored in terms of the -Dickson coefficients depending on different inputs of . . This approach is computationally attractive, and we provide illustrative examples with usage explanations. Additionally, we provide specific examples to demonstrate the method's accuracy, and we use least-squares error minimization to reduce error terms within the given domain. Finally, we develop the most common application of numerical approaches in a Python program.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, College of Science, University of Sulaimani, Sulaymaniyah, Iraq"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Shazad Shawki Ahmed",
      "orcid": "0000-0002-9409-4743"
    },
    {
      "affs": [
        0
      ],
      "name": "Kurdistan Hama Ali Hama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8yexymcy45rvp.bib",
  "cited_by": 0,
  "doi": "10.25130/tjps.v31i5.1925",
  "fulltext_url": "https://tjpsj.org/index.php/tjps/article/view/1925",
  "id": "8yexymcy45rvp",
  "issns": [
    "1813-1662",
    "2415-1726"
  ],
  "landing_url": "https://doi.org/10.25130/tjps.v31i5.1925",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "QC1-999"
    ],
    [
      "lcc",
      "QD1-999"
    ],
    [
      "openalex",
      "2611|Fractional Differential Equations Solutions"
    ]
  ],
  "openalex": "W7224122951",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": "Tikrit University",
  "ris": "https://research.draconitic.com/paper/8yexymcy45rvp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T13:46:27Z",
      "native_id": "e05e31c88b634ec4b13ca2477b190966",
      "source": "doaj",
      "url": "https://doi.org/10.25130/tjps.v31i5.1925"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2611|Fractional Differential Equations Solutions",
      "subfield": "general"
    }
  ],
  "title": "Solving Fredholm Integro-Differential Equations of Mixed-Order by α-Dickson Matrix-Collocation Method",
  "title_text": "Solving Fredholm Integro-Differential Equations of Mixed-Order by α-Dickson Matrix-Collocation Method",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T13:46:27Z",
  "url": "https://research.draconitic.com/paper/8yexymcy45rvp",
  "venue": "Tikrit Journal of Pure Science",
  "versions": []
}