An accurate five-step trigonometrically-fitted numerical scheme for approximating solutions of second order ordinary differential equations with oscillatory solutions

Abstract

In this paper, class of second order ordinary differential equation with oscillatory solutions is considered. By employing the trigonometric basis function, a continuous five-step scheme known as five-step trigonometrically fitted scheme is derived to approximate solutions to the class of considered equation. Consistency and zero stability of the developed method were proved. Stability and convergence properties of this new scheme were also established. The scheme so obtained is used to solve standard initial value problems with oscillatory solutions. From the numerical results obtained, it was revealed that the proposed method performs better than some of the existing methods in the literature.

Solving Directly Second Order Initial Value Problems with Lucas Polynomial

“…The hybrid one step method for solving directly second order initial value problems generated in this paper is a accurate and efficient, b consistent and zero stable, c self-starting and requires only one grid functions evaluation at each integration step, d and can complete favorably with other existing methods [ [1], [2], [3], [4], [5], [6]] in the literature.…”

“…Many Authors such as Henrici [7], Simos [8] Awoyemi [9] Adeniran, Akindeinde and Ogundare [5], Adeniran, Odejide and Ogundare [4], Adeniran and Ogundare [2] have devoted lots of attention to the development of various methods for solving directly (1.1) without reducing it to system of first order.…”

“…Adeniran, Akindeinde and Ogundare [5] develop a five-step trigonometrically fitted scheme for approximating solutions of second order ordinary differential equation with oscillatory solutions. Stability and convergence properties of the scheme were established.…”