Research Article A One-Point Third-Derivative Hybrid Multistep Technique for Solving Second-Order Oscillatory and Periodic Problems Mufutau Ajani Rufai , 1 Ali Shokri , 2 and Ezekiel Olaoluwa Omole 3,4 1 Dipartimento di Matematica, Universit` a Degli Studi di Bari Aldo Moro, Bari 70125, Italy 2 aculty of Mathematical Sciences, University of Maragheh, Maragheh, Iran 3 Department of Mathematics and Statistics, Joseph Ayo Babalola University, Ikeji Arakeji, Osun State, Nigeria 4 Department of Mathematics, aculty of Science, ederal University Oye•Ekiti, Oye•Ekiti, Ekiti State, Nigeria CorrespondenceshouldbeaddressedtoAliShokri;shokri@maragheh.ac.ir Received 31 October 2022; Revised 22 December 2022; Accepted 7 January 2023; Published 23 January 2023 AcademicEditor:KoladeM.Owolabi Copyright©2023MufutauAjaniRufaietal.TisisanopenaccessarticledistributedundertheCreativeCommonsAttribution License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly cited. Tis paper describes a third•derivative hybrid multistep technique (TDHMT) for solving second•order initial•value problems (IPs)withoscillatoryandperiodicproblemsinordinarydiferentialequations(ODEs),thecoefcientsofwhichareindependent of the frequency (omega) and step size (h). Tis research is signifcant because it has numerous applications to real•life phenomenasuchaschaoticdynamicalsystems,almostperiodicproblems,anddufngequations.Tecurrentmethodisderived from the collocation of a derivative function at the equidistant grid and of•grid points. Te TDHMTobtained is a continuous scheme for obtaining simultaneous approximations to the solution and its derivative at each point in the [x0, xN] interval integration.Tepresenceofhighderivativesincreasestheorderofthemethod,whichincreasestheaccuracymethod’sorderand the stability property, as discussed in detail. Finally, the proposed method is compared to existing methods in the literature on some oscillatory and periodic test problems to demonstrate the technique’s efectiveness and productivity. 1.Introduction Te numerical solution of general second•order IPs of ODEs of the form (1) is the focus of this study. u′′(x)� f(x, u, u′), ux 0 ( � u 0 , u′ x 0 ( � u 0 ′ , u, x, f ∈ R d , (1) whose solution u(x) is assumed to have an oscillatory or periodic behaviour. Problems with structure (1) arise much of the time in astrophysicsandspace,nuclearphysics,celestialmechanics, molecular dynamics, circuit theory, chemical kinetics, and other various areas of applications in science and engineering. Te study places emphasis on second•order IPs,wherethefrst•derivativedoesnotshowupexplicitly, duetotheirapplications,andmanyoftheseproblemsdonot have known analytical solutions. Because of real•life ap• plicationsofthisproblem,numerousscientistsarepropelled toexamineitsnumericalsolutions(see[1,2]andreferences therein). Teauthorsofreference[1]developeda3•pointvariable stepblockhybridmethod(3•pointSBHM)usingLagrange polynomialsasthebasisfunction.Te3•pointSBHMwas applied to difcult chemical problems such as the Belou• sov–Zhabotinsky reaction and Hires. It has been demon• strated that the method meets the basic requirement for convergence. Achar tenders symmetric multistep Obrechkof methods with eight•order algebraic accuracy and zero•phase lag for periodic initial•value problems of second•order diferential equations in reference [2]. Te Hindawi Journal of Mathematics Volume 2023, Article ID 2343215, 12 pages https://doi.org/10.1155/2023/2343215