Generalized Two-Hybrid One-Step Implicit Third Derivative Block Method for the Direct Solution of Second Order Ordinary Differential Equations

Abstract

Abstract:In this article, a general two-hybrid one-step implicit third derivative block method is developed for the direct solution of the second order initial value problems through interpolation and collocation approach. To derive this method, the approximate basis function is interpolated at the values {xn, xn+r} while its second and third derivatives are collocated at all points {xn, xn+r, xn+s, xn+1} in the given interval. The new developed method produces better accuracy if compared to the existing methods when solving the same problems.

On Modeling the Double and Multiplicative Binomial Models as Log-Linear Models

“…Comparing the new method with the existing method, the result presented in the Tables 1 and 2 shows that the new method performs better than the existing method, and even the order of new method is higher than the order of the existing method [11,18]. In this article, a one-step block method with three off-step points is derived via the interpolation and collocation approach.…”

On a class of four-step modified backward differentiation formula (MBDF) for solving second-order ordinary differential equations

“…Adopting Taylor's series expansion in Adeyeye and Omar (2016), a 3-step second derivative numerical method to approximate (1) has been presented. In Anake et al, (2012), a continuous one-step implicit hybrid block method for the direct solution of initial value problems (1), using power series were presented. A multistep collocation technique is used in Ehigie (2011) to develop 3-point explicit and implicit block methods, which has presented as suitable for generating solutions of the general secondorder ordinary differential equations of the form (1).…”

“…As in the cases of the technique adopted in (Adeyeye and Omar, 2016;Anake et al, (2012); Chollom et al, 2007), a new four-step Modified Backward Differentiation Formula has been presented in this research for developing the method, used in block for the solution of (1).…”