{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T10:31:22Z","timestamp":1775817082073,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T00:00:00Z","timestamp":1668384000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of KwaZulu-Natal"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This study presents a new variant of the hybrid block methods (HBMs) for solving initial value problems (IVPs). The overlapping hybrid block technique is developed by changing each integrating block of the HBM to incorporate the penultimate intra-step point of the previous block. In this paper, we present preliminary results obtained by applying the overlapping HBM to IVPs of the first order, utilizing equally spaced grid points and optimal points that maximize the local truncation errors of the main formulas at the intersection of each integration block. It is proven that the novel method reduces the local truncation error by at least one order of the integration step size, O(h). In order to demonstrate the superiority of the suggested method, numerical experimentation results were compared to the corresponding HBM based on the standard non-overlapping grid. It is established that the proposed method is more accurate than HBM versions of the same order that have been published in the literature.<\/jats:p>","DOI":"10.3390\/a15110427","type":"journal-article","created":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T04:21:45Z","timestamp":1668399705000},"page":"427","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Overlapping Grid-Based Optimized Single-Step Hybrid Block Method for Solving First-Order Initial Value Problems"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3969-9598","authenticated-orcid":false,"given":"Sandile","family":"Motsa","sequence":"first","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3201, South Africa"},{"name":"Mathematics Department, University of Eswatini, Private Bag 4, Kwaluseni M201, Eswatini"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,14]]},"reference":[{"key":"ref_1","first-page":"107","article-title":"An optimized two-step hybrid block method for solving first-order initial-value problems in ODEs","volume":"19","author":"Ramos","year":"2017","journal-title":"Differ. 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Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5576891","DOI":"10.1155\/2022\/5576891","article-title":"A Family of A-Stable Optimized Hybrid Block Methods for Integrating Stiff Differential Systems","volume":"2022","author":"Singla","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_6","first-page":"124","article-title":"An efficient optimized adaptive step-size hybrid block method for integrating differential systems","volume":"362","author":"Singh","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1016\/j.matcom.2021.10.023","article-title":"An adaptive pair of one-step hybrid block Nystr\u00f6m methods for singular initial-value problems of Lane\u2013Emden\u2013Fowler type","volume":"193","author":"Ramos","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s40314-021-01729-7","article-title":"Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations","volume":"41","author":"Ramos","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1007\/s40819-021-01220-1","article-title":"Numerical Solution of Generalized Kuramoto\u2013Sivashinsky Equation Using Cubic Trigonometric B-Spline Based Differential Quadrature Method and One-Step Optimized Hybrid Block Method","volume":"8","author":"Kaur","year":"2022","journal-title":"Int. J. Appl. Comput. Math. Vol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1090\/S0025-5718-1964-0159424-9","article-title":"Implicit runge-kutta processes","volume":"18","author":"Butcher","year":"1964","journal-title":"Math. 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