{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:08:47Z","timestamp":1760242127335,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,12,29]],"date-time":"2018-12-29T00:00:00Z","timestamp":1546041600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006528","name":"Faculti Sains, Universiti Putra Malaysia","doi-asserted-by":"publisher","award":["GP-IPS\/2017\/9526600"],"award-info":[{"award-number":["GP-IPS\/2017\/9526600"]}],"id":[{"id":"10.13039\/501100006528","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this study, fifth-order and sixth-order diagonally implicit Runge\u2013Kutta type (DIRKT) techniques for solving fourth-order ordinary differential equations (ODEs) are derived which are denoted as DIRKT5 and DIRKT6, respectively. The first method has three and the another one has four identical nonzero diagonal elements. A set of test problems are applied to validate the methods and numerical results showed that the proposed methods are more efficient in terms of accuracy and number of function evaluations compared to the existing implicit Runge\u2013Kutta (RK) methods.<\/jats:p>","DOI":"10.3390\/a12010010","type":"journal-article","created":{"date-parts":[[2018,12,31]],"date-time":"2018-12-31T07:22:30Z","timestamp":1546240950000},"page":"10","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Diagonally Implicit Runge\u2013Kutta Type Method for Directly Solving Special Fourth-Order Ordinary Differential Equations with Ill-Posed Problem of a Beam on Elastic Foundation"],"prefix":"10.3390","volume":"12","author":[{"given":"Nizam","family":"Ghawadri","sequence":"first","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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 UPM, Selangor, Malaysia"},{"name":"Department of Mathematics, Universiti Putra Malaysia, Serdang 43400 UPM, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Firas","family":"Adel Fawzi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Computer Science and Mathematics, University of Tikrit, Tikrit P.O. Box 42, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fudziah","family":"Ismail","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400 UPM, Selangor, Malaysia"},{"name":"Department of Mathematics, Universiti Putra Malaysia, Serdang 43400 UPM, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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 UPM, Selangor, Malaysia"},{"name":"Department of Mathematics, Universiti Putra Malaysia, Serdang 43400 UPM, Selangor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,29]]},"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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