{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T11:44:10Z","timestamp":1782301450238,"version":"3.54.5"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T00:00:00Z","timestamp":1742774400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Deanship at Hail University","award":["RG-23 077"],"award-info":[{"award-number":["RG-23 077"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper introduces a novel Runge\u2013Kutta (RK) pair of orders 8(6) designed specifically for solving linear inhomogeneous initial value problems (IVPs) with constant coefficients. The proposed method requires only 11 stages per iteration, a significant improvement over conventional RK pairs of orders 8(7), which typically demand 13 stages. The reduction in stages is achieved by leveraging a smaller set of order conditions tailored to linear inhomogeneous problems, where traditional simplification techniques are not applicable. To address the complexity of deriving such methods, the authors employ the Differential Evolution algorithm, a global optimization technique, to solve the resulting system of equations. The new RK pair, named NEW8(6)Lin, is tested on several benchmark problems, including scalar, linear inhomogeneous, and larger systems, demonstrating a superior performance in terms of accuracy and computational efficiency. The method\u2019s high phase-lag accuracy and efficiency make it particularly suitable for problems requiring high precision over extended intervals. The coefficients of the method are provided with high precision, enabling direct implementation in computational environments like Mathematica. The results highlight the method\u2019s potential as a robust tool for solving linear inhomogeneous IVPs, offering a balance between computational cost and accuracy. This work contributes to the ongoing development of specialized numerical methods for differential equations, particularly in scenarios where traditional approaches struggle with efficiency or stability.<\/jats:p>","DOI":"10.3390\/axioms14040245","type":"journal-article","created":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T06:21:38Z","timestamp":1742797298000},"page":"245","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On High-Order Runge\u2013Kutta Pairs for Linear Inhomogeneous Problems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1816-3767","authenticated-orcid":false,"given":"Houssem","family":"Jerbi","sequence":"first","affiliation":[{"name":"Department of Industrial Engineering, College of Engineering, University of Ha\u2019il, Hail 81481, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sanaa","family":"Maali","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Applied College, University of Ha\u2019il, Hail 81481, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sondess Ben","family":"Aoun","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, College of Computer Science and Engineering, University of Ha\u2019il, Hail 81481, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arwa N.","family":"Aledaily","sequence":"additional","affiliation":[{"name":"Department of Information and Computer Science, College of Computer Science and Engineering, University of Ha\u2019il, Hail 81481, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0026-2636","authenticated-orcid":false,"given":"Vijipriya","family":"Jeyamani","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, College of Computer Science and Engineering, University of Ha\u2019il, Hail 81481, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9220-6924","authenticated-orcid":false,"given":"Theodore E.","family":"Simos","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Hangzhou Dianzi University, Er Hao Da Jie 1158, Xiasha, Hangzhou 310018, China"},{"name":"Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, West Mishref, Kuwait City 32093, Kuwait"},{"name":"Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6801-8117","authenticated-orcid":false,"given":"Charalampos","family":"Tsitouras","sequence":"additional","affiliation":[{"name":"General Department, National & Kapodistrian University of Athens, GR34-400 Euripus Campus, 34400 Psachna, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,24]]},"reference":[{"key":"ref_1","first-page":"3","article-title":"An Algorithmic approach to Runge\u2013Kutta\u2013Nystrom pairs","volume":"14","author":"Kovalnogov","year":"2023","journal-title":"TWMS J. 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