{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T16:55:31Z","timestamp":1778950531213,"version":"3.51.4"},"reference-count":45,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T00:00:00Z","timestamp":1769644800000},"content-version":"vor","delay-in-days":28,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100004695","name":"Inyuvesi Yakwazulu-Natali","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004695","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:p>This paper presents a one\u2010step multiderivative hybrid block method of Order 12 that incorporates an overlapping strategy, in which intrastep points from the previous block are reused in the current step to enhance accuracy and stability when solving linear and nonlinear initial value problems. The derivation incorporates a multistep collocation and interpolation technique, using power series as the basis function for the approximate solution. Within a one\u2010step block, three intrastep points are considered. As a foundational step, a non\u2010overlapping one\u2010step multiderivative scheme is first developed and expressed in matrix form. The overlapping aspect of the method is then introduced by incorporating the second\u2010to\u2010last intrastep point of the previous step into each integrating block. The accuracy, consistency, and stability properties of the method are analyzed. The features of the method are determined through an error analysis of the numerical solutions of linear and nonlinear second\u2010order initial value problems. The nonlinear initial value problems are converted into linear ones using a modified Picard iteration technique. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. The results are evaluated against other methods from the literature.<\/jats:p>","DOI":"10.1155\/jama\/9948007","type":"journal-article","created":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T06:43:47Z","timestamp":1769755427000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Overlapping One\u2010Step Multiderivative Hybrid Block Method for Solving Second\u2010Order Initial Value Problems"],"prefix":"10.1155","volume":"2026","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7638-9153","authenticated-orcid":false,"given":"Uthman O.","family":"Rufai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2115-4642","authenticated-orcid":false,"given":"Precious","family":"Sibanda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1383-068X","authenticated-orcid":false,"given":"Sicelo P.","family":"Goqo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3969-9598","authenticated-orcid":false,"given":"Sandile","family":"Motsa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2026,1,29]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.1080\/00207160108805054"},{"key":"e_1_2_11_2_2","first-page":"325","article-title":"One Step Hybrid Block Method for Solving Nonlinear Second Order Dirichlet Value Problems of Ordinary Differential Equations Directly","volume":"15","author":"Aabdelrahim R.","year":"2024","journal-title":"International Journal of Nonlinear Analysis and Applications"},{"key":"e_1_2_11_3_2","doi-asserted-by":"publisher","DOI":"10.18280\/mmep.100207"},{"key":"e_1_2_11_4_2","article-title":"Implicit Hybrid Block Methods for Solving Second, Third and Fourth Orders Ordinary Differential Equations Directly","volume":"48","author":"Abolarin O. 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