{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:29:03Z","timestamp":1760149743234,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T00:00:00Z","timestamp":1692921600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-RP23095"],"award-info":[{"award-number":["IMSIU-RP23095"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>Invoking the matrix transfer technique, we propose a novel numerical scheme to solve the time-fractional advection\u2013dispersion equation (ADE) with distributed-order Riesz-space fractional derivatives (FDs). The method adopts the midpoint rule to reformulate the distributed-order Riesz-space FDs by means of a second-order linear combination of Riesz-space FDs. Then, a central difference approximation is used side by side with the matrix transform technique for approximating the Riesz-space FDs. Based on this, the distributed-order time-fractional ADE is transformed into a time-fractional ordinary differential equation in the Caputo sense, which has an equivalent Volterra integral form. The Simpson method is used to discretize the weakly singular kernel of the resulting Volterra integral equation. Stability, convergence, and error analysis are presented. Finally, simulations are performed to substantiate the theoretical findings.<\/jats:p>","DOI":"10.3390\/fractalfract7090649","type":"journal-article","created":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T08:42:20Z","timestamp":1692952940000},"page":"649","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Matrix Transform Technique for Distributed-Order Time-Fractional Advection\u2013Dispersion Problems"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6464-7338","authenticated-orcid":false,"given":"Mohammadhossein","family":"Derakhshan","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, Apadana Institute of Higher Education, Shiraz 7187985443, Iran"},{"name":"Faculty of Technology and Engineering, Zand Institute of Higher Education, Shiraz 8415683111, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3266-7243","authenticated-orcid":false,"given":"Ahmed S.","family":"Hendy","sequence":"additional","affiliation":[{"name":"Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia"},{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"additional","affiliation":[{"name":"LAETA\/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8262-1369","authenticated-orcid":false,"given":"Alexandra","family":"Galhano","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias Naturais, Engenharias e Tecnologias, Universidade Lus\u00f3fona do Porto, Rua de Augusto Rosa 24, 4000-098 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3376-7238","authenticated-orcid":false,"given":"Mahmoud A.","family":"Zaky","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,25]]},"reference":[{"key":"ref_1","unstructured":"Podulbny, I. 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