{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T20:52:39Z","timestamp":1784235159426,"version":"3.55.0"},"reference-count":50,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T00:00:00Z","timestamp":1680048000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2023R183"],"award-info":[{"award-number":["PNURSP2023R183"]}]},{"name":"Princess Nourah bint Abdulrahman University","award":["No. 3159"],"award-info":[{"award-number":["No. 3159"]}]},{"name":"Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["PNURSP2023R183"],"award-info":[{"award-number":["PNURSP2023R183"]}]},{"name":"Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["No. 3159"],"award-info":[{"award-number":["No. 3159"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this study, we solve the fractional advection\u2013dispersion equation (FADE) by applying the Laplace transform decomposition method (LTDM) and the variational iteration transform method (VITM). The Atangana\u2013Baleanu (AB) sense is used to describe the fractional derivative. This equation is utilized to determine solute transport in groundwater and soils. The FADE is converted into a system of non-linear algebraic equations whose solution leads to the approximate solution for this equation using the techniques presented. The proposed approximate method\u2019s convergence is examined. The suggested method\u2019s applicability is demonstrated by testing it on several illustrative examples. The series solutions to the specified issues are obtained, and they contain components that converge more quickly to the precise solutions. The actual and estimated results are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed strategy. The innovation of the current work is in the application of an effective method that requires less calculation and achieves a greater level of accuracy. Furthermore, the proposed approaches may be implemented to prove their utility in tackling fractional-order problems in science and engineering.<\/jats:p>","DOI":"10.3390\/sym15040819","type":"journal-article","created":{"date-parts":[[2023,3,29]],"date-time":"2023-03-29T02:03:53Z","timestamp":1680055433000},"page":"819","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Comparative Analysis of Advection\u2013Dispersion Equations with Atangana\u2013Baleanu Fractional Derivative"],"prefix":"10.3390","volume":"15","author":[{"given":"Azzh Saad","family":"Alshehry","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0199-6850","authenticated-orcid":false,"given":"Humaira","family":"Yasmin","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Preparatory Year Deanship, King Faisal University, Al-Ahsa 31982, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fazal","family":"Ghani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rasool","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,29]]},"reference":[{"key":"ref_1","unstructured":"Machado, J.T., Kiryakova, V., Kochubei, A., and Luchko, Y. 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