{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T00:58:24Z","timestamp":1776473904699,"version":"3.51.2"},"reference-count":88,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:00:00Z","timestamp":1584576000000},"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 manuscript investigates the fractional Phi-four equation by using    q   -homotopy analysis transform method (   q   -HATM) numerically. The Phi-four equation is obtained from one of the special cases of the Klein-Gordon model. Moreover, it is used to model the kink and anti-kink solitary wave interactions arising in nuclear particle physics and biological structures for the last several decades. The proposed technique is composed of Laplace transform and    q   -homotopy analysis techniques, and fractional derivative defined in the sense of Caputo. For the governing fractional-order model, the Banach\u2019s fixed point hypothesis is studied to establish the existence and uniqueness of the achieved solution. To illustrate and validate the effectiveness of the projected algorithm, we analyze the considered model in terms of arbitrary order with two distinct cases and also introduce corresponding numerical simulation. Moreover, the physical behaviors of the obtained solutions with respect to fractional-order are presented via various simulations.<\/jats:p>","DOI":"10.3390\/sym12030478","type":"journal-article","created":{"date-parts":[[2020,3,20]],"date-time":"2020-03-20T07:29:07Z","timestamp":1584689347000},"page":"478","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":80,"title":["New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach"],"prefix":"10.3390","volume":"12","author":[{"given":"Wei","family":"Gao","sequence":"first","affiliation":[{"name":"School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4468-3048","authenticated-orcid":false,"given":"Pundikala","family":"Veeresha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Karnatak University, Dharwad 580003, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6453-0308","authenticated-orcid":false,"given":"Doddabhadrappla Gowda","family":"Prakasha","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Davangere University, Shivagangothri, Davangere 577007, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4085-3625","authenticated-orcid":false,"given":"Haci Mehmet","family":"Baskonus","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Science Education, Faculty of Education, Harran University, Sanliurfa 63200, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5134-4431","authenticated-orcid":false,"given":"Gulnur","family":"Yel","sequence":"additional","affiliation":[{"name":"Faculty of Educational Sciences, Final International University, Mersin 10, Kyrenia 99370, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,19]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Memoire surquelques questions de geometrieet de mecanique, etsurun nouveau genre de calcul pour resoudrecesquestions","volume":"13","author":"Liouville","year":"1832","journal-title":"J. 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