{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T22:49:11Z","timestamp":1770331751006,"version":"3.49.0"},"reference-count":15,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,16]],"date-time":"2024-03-16T00:00:00Z","timestamp":1710547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we propose a new numerical scheme based on a variation of the standard formulation of the Runge\u2013Kutta method using Taylor series expansion for solving initial value problems (IVPs) in ordinary differential equations. Analytically, the accuracy, consistency, and absolute stability of the new method are discussed. It is established that the new method is consistent and stable and has third-order convergence. Numerically, we present two models involving applications from physics and engineering to illustrate the efficiency and accuracy of our new method and compare it with further pertinent techniques carried out in the same order.<\/jats:p>","DOI":"10.3390\/a17030123","type":"journal-article","created":{"date-parts":[[2024,3,18]],"date-time":"2024-03-18T04:25:15Z","timestamp":1710735915000},"page":"123","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["An Efficient Third-Order Scheme Based on Runge\u2013Kutta and Taylor Series Expansion for Solving Initial Value Problems"],"prefix":"10.3390","volume":"17","author":[{"given":"Noori Y.","family":"Abdul-Hassan","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zainab J.","family":"Kadum","sequence":"additional","affiliation":[{"name":"Basrah Education Directorate, Ministry of Education, Basrah 61001, Iraq"}],"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"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1090\/S0025-5718-1962-0150954-0","article-title":"Runge-Kutta Methods with Minimum Error Bounds","volume":"16","author":"Ralston","year":"1962","journal-title":"Math. Comput."},{"key":"ref_2","unstructured":"Ralston, A., and Rabinowitz, P. (1978). A First Course in Numerical Analysis, McGraw-Hill. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1080\/00207169408804245","article-title":"A Comparison of Modified Runge-Kutta Formulas Based on a Variety of Means","volume":"50","author":"Wazwaz","year":"1994","journal-title":"Int. J. Comput. Math."},{"key":"ref_4","first-page":"145","article-title":"Differential Equation Solver Simulator for Runge-Kutta Methods","volume":"21","author":"Hatun","year":"2016","journal-title":"Uluda\u011f Univ. J. Fac. Eng."},{"key":"ref_5","unstructured":"Burden, R.L., and Faires, J.D. (2011). Numerical Analysis, Brooks\/Cole Publishing Company. [9th ed.]."},{"key":"ref_6","first-page":"21134","article-title":"A Comparative Study on Numerical Solution of Ordinary Differential Equation by Different Method with Initial Value Problem","volume":"8","author":"Ahmad","year":"2017","journal-title":"Int. J. Recent. Sci. Res."},{"key":"ref_7","first-page":"307","article-title":"The Adomian Decomposition Method for Numerical Solution of First-Order Differential Equations","volume":"6","author":"Nhawu","year":"2016","journal-title":"J. Math. Comput. Sci."},{"key":"ref_8","unstructured":"Nagle, R.K., Saff, E.B., and Snider, A.D. (2018). Fundamentals of Differential Equations, Pearson. [9th ed.]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"121","DOI":"10.4236\/ajcm.2013.32020","article-title":"A Three-stage Multiderivative Explicit Runge-Kutta Method","volume":"3","author":"Wusu","year":"2013","journal-title":"Am. J. Comput. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.7153\/mia-2022-25-69","article-title":"Taylor-type Expansions in Terms of Exponential Polynomials","volume":"25","author":"Ali","year":"2022","journal-title":"Math. Inequalities Appl."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kadum, Z.J., and Abdul-Hassan, N.Y. (2023). New Numerical Methods for Solving the Initial Value Problem Based on a Symmetrical Quadrature Integration Formula Using Hybrid Functions. Symmetry, 15.","DOI":"10.3390\/sym15030631"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.1007\/s11075-018-0624-x","article-title":"Optimal Residuals and the Dahlquist Test Problem","volume":"81","author":"Corless","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_13","unstructured":"Lambert, J.D. (1973). Computational Methods in Ordinary Differential Equations, John Wiley & Sons Inc."},{"key":"ref_14","first-page":"2224","article-title":"A Hybrid Numerical Method with Greater Efficiency for Solving Initial Value Problems","volume":"10","author":"Ram","year":"2020","journal-title":"Math. Theory Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"127","DOI":"10.3844\/jmssp.2016.127.134","article-title":"Numerical Solution of First Order Initial Value Problems Using a Self-Starting Implicit Two-Step Obrechkoff-Type Block Method","volume":"12","author":"Omar","year":"2016","journal-title":"J. Math. 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