{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:27:24Z","timestamp":1760239644640,"version":"build-2065373602"},"reference-count":64,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,30]],"date-time":"2020-11-30T00:00:00Z","timestamp":1606694400000},"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>This paper presents an efficacious analytical and numerical method for solution of fractional differential equations. This technique, here in named q-HATM (q-homotopy analysis transform method) is applied to a one-dimensional fractional Fornberg\u2013Whitham model and a two-dimensional fractional population model emanating from biological sciences. The overwhelming agreement of our analytical solution by the q-HATM technique with the exact solution indeed establishes the efficacy of q-HATM to solve the fractional Fornberg\u2013Whitham model and the two-dimensional fractional population model. Furthermore, comparisons by means of extensive analysis using numerics, graphs and error analysis are presented to affirm the preference of q-HATM technique over other methods. A variant of the q-HATM using symmetry can also be considered to solve these problems.<\/jats:p>","DOI":"10.3390\/sym12121976","type":"journal-article","created":{"date-parts":[[2020,11,30]],"date-time":"2020-11-30T20:10:22Z","timestamp":1606767022000},"page":"1976","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Efficacious Analytical Technique Applied to Fractional Fornberg\u2013Whitham Model and Two-Dimensional Fractional Population Model"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9658-5864","authenticated-orcid":false,"given":"Cyril D.","family":"Enyi","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Hafr Al Batin, Hafar Al Batin 39524, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1080\/00207390410001686571","article-title":"A brief historical introduction to fractional calculus","volume":"35","author":"Debnath","year":"2004","journal-title":"Int. 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