{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T07:34:20Z","timestamp":1770881660228,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T00:00:00Z","timestamp":1660089600000},"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>Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg\u2013de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method\u2019s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.<\/jats:p>","DOI":"10.3390\/sym14081653","type":"journal-article","created":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T09:47:06Z","timestamp":1660124826000},"page":"1653","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3633-4603","authenticated-orcid":false,"given":"Mariam","family":"Sultana","sequence":"first","affiliation":[{"name":"Department of Mathematica, Federal Urdu University of Arts, Sciences & Technology, Karachi 75300, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3115-0816","authenticated-orcid":false,"given":"Uroosa","family":"Arshad","sequence":"additional","affiliation":[{"name":"Department of Mathematica, Federal Urdu University of Arts, Sciences & Technology, Karachi 75300, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2959-4212","authenticated-orcid":false,"given":"Ali Hasan","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq"},{"name":"Institute of Mathematics, University of Debrecen, Pf. 400, H-4002 Debrecen, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy"},{"name":"Department of Mathematics, Faculty of Science, Hadhramout University, Al Mukalla 50512, Hadhramout, Yemen"},{"name":"Department of Mathematics, Faculty of Education, Seiyun University, Al Mukalla 50512, Hadhramout, Yemen"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Areej A.","family":"Al-Moneef","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,10]]},"reference":[{"key":"ref_1","unstructured":"Dumitru, B., and Ant\u00f3nio, M.L. (2019). Application of variable-order fractional calculus in solid mechanics. Volume 7, Applications in Engineering, Life and Social Sciences, Part A, De Gruyter."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"248175","DOI":"10.1155\/2012\/248175","article-title":"Application of integer and fractional models in electrochemical systems","volume":"2012","author":"Jesus","year":"2012","journal-title":"Math. Probl. 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