Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

Half-Step Implicit Linear Multistep Hybrid Block Third Derivative Methods of order Four for the Solution of Third order Ordinary Differential Equations Skwame, Yusuf α , Raymond, Dominic σ , Tumba, Pius ρ & Adiku, Lydia Ѡ Abstract- This paper proposes a half-step third derivative hybrid block method with two cases of order four for solution of third Order Ordinary Differential Equations. Method of interpolation and collocation of power series approximate solution was used to generate the continuous hybrid linear multistep method, which was then evaluated at non- interpolating points to give a continuous block method. The discrete block method was recovered when the continuous block was evaluated at all step points. The basic properties of the methods were investigated and were found to be zero-stable, consistent and convergent. The develop half-step method is applied to solve some third order initial value problems of ordinary differential equations and from the numerical results obtained, it is observed that our methods gives better approximation than the existing method compared with. Keywords: half-step, hybrid block method, third derivative. I. I ntroduction This paper consider third order initial value problems of the form 0' 0 ' ,0 0 , ' , , ''' y xy y xy xyxyxf y = = =                                 (1) Where f is continuous within the interval of integration, solving higher order derivatives method by reducing them to a system of first-order approach involves more functions to evaluate which then leads to a computational burden as in Kayode and Obuarhua (2013) and James et al. (2013). Various approaches have been proposed to find the analytic solution of (1) ranging from predictor-corrector method to hybrid methods. Despite the success recorded by the predictor-corrector methods, its major setback is that the predictor are in reducing order of accuracy especially when the value of the step-length is high and moreover the result are at overlapping interval. The hybrid method was established to circumvent the Dahlquist barrier theorem which gives a better approximation to solutions of initial value problems of stiff ordinary differential equations than the k-step method. Scholars who recently adopted the hybrid method other than the direct method in approximating (1) include among others Adesanya et al. (2013), Adebayo and 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 5 ( F ) © 2021 Global Journals Version I Author α : Department of Mathematical Sciences, Faculty of Science, Adamawa State University, Mubi-Nigeria Author σ Ѡ : Department of Mathematics and Statistics, Faculty of Pure and Applied Sciences, Federal University Wukari Nigeria. e-mails: domehbazza@yahoo.co.uk, raymond@fuwukari.edu.ng Author ρ : Department of Mathematics, Faculty of Science, Federal University Gashua, Nigeria. N otes