A four-point fully implicit method for the numerical integration of third-order ordinary differential equations

Abstract

In this paper, we derived a continuous linear multistep method (LMM) with step number k = 4 through collocation and interpolation techniques using power series as basis function for approximate solution. An order-seven scheme is developed which was used to solve the third-order initial value problems (IVPS) in ordinary differential equation without first reducing to a system first-order. Taylor's series algorithm of the same order was developed to implement our method. The result obtained was compared favourably with existing methods.

Symmetric 2-Step 4-Point Hybrid Method for the Solution of General Third Order Differential Equations

“…Applying the truncation error formula in Awoyemi et al [6], associated with eqn. (6) by the difference operator eqn.…”

“…Attempts have also been made by these scholars [12][13][14]6,7,15]. Olabode [13] proposed a 5-step block scheme for the solution of special type of eqn.…”

“…The order of accuracy in Olabode et al [13] improves more than that of Olabode et al [13]. Awoyemi et al [6], developed a four-point implicit method for the numerical integration of third order ODEs using power series polynomial function [16]. Kuboye and Omar [7] proposed numerical solution of third order ordinary differential equations using a seven-step block method to improve on Awoyemi et al [6] and Olabode [13] which are of lower order of accuracy.…”

“…Awoyemi et al [6], developed a four-point implicit method for the numerical integration of third order ODEs using power series polynomial function [16]. Kuboye and Omar [7] proposed numerical solution of third order ordinary differential equations using a seven-step block method to improve on Awoyemi et al [6] and Olabode [13] which are of lower order of accuracy. Furthermore, a symmetric hybrid linear multistep method of order six having two off-step points for the solution of eqn.…”

“…The direct approach for solving this type of ordinary differential equations have been studied and appeared in different literatures [1][2][3][4][5][6][7]. This direct approach has demonstrated advantages over the popular approach (reduction to system of first order approach) in terms of speed and accuracy [8,9].…”