Block Method With One Hybrid Point for the Solution of First Order Initial Value Problems of Ordinary Differential Equations

Abstract

This paper discussed the development of one -step, one hybrid block method for the solution of first order initial value problems. Two functions were combined to form the basis function which is collocated and interpolated at some selected grid and off-grid points to develop a linear multistep method which is implemented in block form. The paper further investigated the properties of the block method and found it to be convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to give better approximation than the methods we compared our results with.

High Order Continuous Extended Linear Multistep Methods for Approximating System of ODEs

“…Table 7 further shows the results at higher stepsize h = 0.1 for Akinfenwa and Jator [49] and the proposed methods for order 5 and 6. Given the IVPs in [51,52]…”

“…The HOCELMs2 is applied to Example 5 and the error (| y − y(t) |) in the various interval 0 < t ≤ 0.1 are reported in Table 8. It is clear from the numerical result and comparison in Table 8 that the HOCELMs2 is superior in terms of accuracy than the methods (p=5) in [51] and methods (p=6) of [52].…”

Continuous One Step Linear Multi-Step Hybrid Block Method for the Solution of First Order Linear and Nonlinear Initial Value Problem of Ordinary Differential Equations

“…We consider a nonlinear first order initial value problem of ordinary differential problem which was solved by [23]. y 0 x ðÞ¼À 10 y À 1 ðÞ 2 ; y 0 ðÞ ¼ 2, h ¼ 0:01 With exact solution given as yx ðÞ¼1 þ 1 1þ10x , the result is shown in Table 1, while the theoretical and numerical results are presented graphically in Figure 2.…”

Three-step optimized block backward differentiation formulae (TOBBDF) for solving stiff ordinary differential equations

“…Several numerical problems will be solved and comparison will be made with other methods to show the efficiency of the proposed method (Table 1). This paper considers an approximate method for the solutionof stiff differential equation of first-order initial value problem of the form, wheref (x, y) iscontinuousandsatisfiesthe existenceand uniquenesstheorem (Henrici, 1962).Recently many authors have applied hybrid block method with adifferent number of steps and hybrid points to find numerical solutionsfor the first-order differential equations (Sagir, 2014;Raymond et al, 2018;Ramos, 2017;Areo and Adeniyi, 2014;Yakusak and Adeniyi, 2015;Yahaya and Tijjani, 2015;Fotta and Alabi, 2015;Sunday et al, 2015). In this paper, we optimizedthe local truncation errors to find three off-points in one stepto obtain the most accurate solution.…”

“…The algorithm presented in this paper is based on block method and approximates the solution at several points (Raymond et al, 2018;Olanegan et al, 2015;Areo and Adeniyi, 2014;Yakusak and Adeniyi, 2015). Block methods were first introduced by Yahaya and Tijjani (2015) as a means of obtaining starting values for predictor-corrector algorithms and has since then been developed by several researchers (Milne, 1953;Fotta and Alabi, 2015;Sunday et al, 2015;Odejide and Adeniran, 2012), for general use. This paper presents a block method which preserves the Runge-Kutta traditional advantage of being self-starting and efficient.…”