A New Block Method for Special Third Order Ordinary Differential Equations

Abstract

A linear multistep method for the direct solution of initial value problems of ordinary differential equations was presented in this article. Collocation approximation method was adopted in the derivation of the scheme and then the scheme was applied as simultaneous integrator to special third order initial value problem of ordinary differential equations. The new block method possessed the desirable feature of Runge-Kutta method of being self-starting and eliminated the use of predictors. The 3-step block method is P-stable, consistent and more accurate than the existing one. Experimental results confirmed the superiority of the new scheme over the existing method.

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

“…The method developed is symmetric, consistent and convergent which can handle oscillatory type of problems. The numerical tests results obtained were compared with block method of Olabode [13] which was found to perform favorably than the existing method.…”

“…Table 1 shows the maximum absolute error of our predictorcorrector method and that of Olabode [13] block method for Problem 1. It reveals that the new method performed creditably well than that of Olabode [13] of higher order. Theoretical solution:…”

“…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.…”

“…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.…”

“…(1). 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].…”