{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T07:24:52Z","timestamp":1772263492079,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,16]],"date-time":"2019-02-16T00:00:00Z","timestamp":1550275200000},"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 primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure      u  ( 4 )   = f  ( x , u ,  u \u2032  ,  u  \u2033   )      and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed.<\/jats:p>","DOI":"10.3390\/sym11020246","type":"journal-article","created":{"date-parts":[[2019,2,17]],"date-time":"2019-02-17T22:11:50Z","timestamp":1550441510000},"page":"246","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4) = f(x, u, u\u2032, u\u2033)"],"prefix":"10.3390","volume":"11","author":[{"given":"Nizam","family":"Ghawadri","sequence":"first","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8614-8281","authenticated-orcid":false,"given":"Norazak","family":"Senu","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"},{"name":"Department of Mathematics, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"}]},{"given":"Firas","family":"Adel Fawzi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Computer Science and Mathematics, University of Tikrit, Salah ad Din P.O. Box 42, Iraq"}]},{"given":"Fudziah","family":"Ismail","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"},{"name":"Department of Mathematics, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8924-0681","authenticated-orcid":false,"given":"Zarina Bibi","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"},{"name":"Department of Mathematics, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.amc.2006.05.068","article-title":"Numerical solution for high order differential equations using a hybrid neural network\u2014Optimization method","volume":"183","author":"Malek","year":"2006","journal-title":"Appl. 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