{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T08:29:41Z","timestamp":1765268981702,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T00:00:00Z","timestamp":1725235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The fractional advection\u2013dispersion equation is used in groundwater hydrology for modeling the movements of contaminants\/solute particles along with flowing groundwater at the seepage velocity in porous media. This model is used for the prediction of the transport of nonreactive dissolved contaminants in groundwater. This paper establishes the existence and the uniqueness of solutions represented as fractional bi-variate power series of some initial-value problems and boundary-value problems for the fractional advection\u2013dispersion equation. Moreover, a method to approximate the solutions using fractional polynomials in two variables and to evaluate the errors in a suitable rectangle is designed. Illustrative examples showing the applicability of the theoretical results are presented.<\/jats:p>","DOI":"10.3390\/sym16091137","type":"journal-article","created":{"date-parts":[[2024,9,2]],"date-time":"2024-09-02T06:51:17Z","timestamp":1725259877000},"page":"1137","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Existence and Uniqueness of Solution Represented as Fractional Power Series for the Fractional Advection\u2013Dispersion Equation"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2142-836X","authenticated-orcid":false,"given":"Alexandru-Nicolae","family":"Dimache","sequence":"first","affiliation":[{"name":"Department of Hydraulics and Environment Protection, Technical University of Civil Engineering Bucharest, Lacul Tei 122\u2013124, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6840-9713","authenticated-orcid":false,"given":"Ghiocel","family":"Groza","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Lacul Tei 122\u2013124, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2718-8133","authenticated-orcid":false,"given":"Marilena","family":"Jianu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, Lacul Tei 122\u2013124, 020396 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4596-5661","authenticated-orcid":false,"given":"Iulian","family":"Iancu","sequence":"additional","affiliation":[{"name":"Department of Hydraulics and Environment Protection, Technical University of Civil Engineering Bucharest, Lacul Tei 122\u2013124, 020396 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,2]]},"reference":[{"key":"ref_1","unstructured":"Freeze, R.A., and Cherry, J.A. 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