{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T23:03:25Z","timestamp":1778627005978,"version":"3.51.4"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T00:00:00Z","timestamp":1651190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Foundation of Yibin University, China","award":["2019QD07"],"award-info":[{"award-number":["2019QD07"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The algebras of the symmetry operators for the Klein\u2013Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for providing the solution to a class of linear and nonlinear fractional Klein\u2013Gordon equations arising in classical relativistic and quantum mechanics. We study the Yang homotopy perturbation transform method (YHPTM), which is associated with the Yang transform (YT) and the homotopy perturbation method (HPM), where the fractional derivative is taken in a Caputo\u2013Fabrizio (CF) sense. This technique provides the solution very accurately and efficiently in the form of a series with easily computable coefficients. The behavior of the approximate series solution for different fractional-order \u2118 values has been shown graphically. Our numerical investigations indicate that YHPTM is a simple and powerful mathematical tool to deal with the complexity of such problems.<\/jats:p>","DOI":"10.3390\/sym14050907","type":"journal-article","created":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T08:21:25Z","timestamp":1651652485000},"page":"907","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Approximate Solution of Nonlinear Time-Fractional Klein-Gordon Equations Using Yang Transform"],"prefix":"10.3390","volume":"14","author":[{"given":"Jinxing","family":"Liu","sequence":"first","affiliation":[{"name":"Faculty of Science, Yibin University, Yibin 644000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9349-4729","authenticated-orcid":false,"given":"Muhammad","family":"Nadeem","sequence":"additional","affiliation":[{"name":"Faculty of Science, Yibin University, Yibin 644000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustafa","family":"Habib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Engineering and Technology, Lahore 54890, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,29]]},"reference":[{"key":"ref_1","unstructured":"Miller, K.S., and Ross, B. 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