{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T01:20:16Z","timestamp":1773364816154,"version":"3.50.1"},"reference-count":65,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T00:00:00Z","timestamp":1751500800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents a spectral method to solve nonlinear distributed-order diffusion equations with both time-distributed-order and two-sided space-fractional terms. These are highly challenging to solve analytically due to the interplay between nonlinearity and the fractional distributed-order nature of the time and space derivatives. For this purpose, Hexic-kind Chebyshev polynomials (HCPs) are used as the backbone of the method to transform the primary problem into a set of nonlinear algebraic equations, which can be efficiently solved using numerical solvers, such as the Newton\u2013Raphson method. The primary reason of choosing HCPs is due to their remarkable recurrence relations, facilitating their efficient computation and manipulation in mathematical analyses. A comprehensive convergence analysis was conducted to validate the robustness of the proposed method, with an error bound derived to provide theoretical guarantees for the solution\u2019s accuracy. The method\u2019s effectiveness is further demonstrated through two test examples, where the numerical results are compared with existing solutions, confirming the approach\u2019s accuracy and reliability.<\/jats:p>","DOI":"10.3390\/axioms14070515","type":"journal-article","created":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T08:22:07Z","timestamp":1751530927000},"page":"515","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Hexic-Chebyshev Collocation Method for Solving Distributed-Order Time-Space Fractional Diffusion Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6980-9786","authenticated-orcid":false,"given":"Afshin","family":"Babaei","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, University of Mazandaran, Babolsar 47415-95477, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4449-8109","authenticated-orcid":false,"given":"Sedigheh","family":"Banihashemi","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Mazandaran, Babolsar 47415-95477, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4957-9028","authenticated-orcid":false,"given":"Behrouz Parsa","family":"Moghaddam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, La.C., Islamic Azad University, Lahijan 44169-39515, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4407-4314","authenticated-orcid":false,"given":"Arman","family":"Dabiri","sequence":"additional","affiliation":[{"name":"Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,3]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. 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