Research Journal of Applied Sciences, Engineering and Technology 11(1): 1926, 2015
DOI: 10.19026/rjaset.11.1671
ISSN: 20407459; eISSN: 20407467
© 2015 Maxwell Scientific Publication Corp.
Submitted: December 10, 2014 Accepted: January 27, 2015 Published: September 05, 2015
Zurni Omar, Department of Mathematics, School of Quantitative Sciences, College of Art and
Sciences, Universiti Utara Malaysia, Malaysia, Tel.: +60194443993
This work is licensed under a Creative Commons Attribution 4.0 International License (URL: http://creativecommons.org/licenses/by/4.0/).
19
J.O. Kuboye and Zurni Omar
Department of Mathematics, School of Quantitative Sciences, College of Art and Sciences,
Universiti Utara Malaysia, Malaysia
! In this study, a new block method of order nine is proposed to solve second order initial value problems
of ordinary differential equations directly. The method is developed via interpolation and collocation approach
where the use of power series approximate solution as a basis function is considered. The properties of the
developed block method which includes zerostability, order, consistency and convergence are also established. The
numerical results reveal that the new method performs better than the existing method when applied for solving
second order ordinary differential equations.
"# Block method, collocation, interpolation, initial value problems, ordinary differential equations
$%&%$
In this study, numerical method for the direct
solution of initial value problems of ordinary
differential equations of the form
β α = ′ = ′ = ′ ′ ) 0 ( , ) 0 ( ) , , ( y y y y x f y
(1)
is examined. Efforts in the development of numerical
methods for solving higher order initial value problems
especially second order ordinary differential equations
have been made by eminent scholars such as Awoyemi
(2001), Adesanya et al. (2008), Kayode (2008) and
Yahaya and Badmus (2009).
Recently, much attention had been devoted in
developing predictorcorrector methods for solving (1)
directly. It is noted that this method is associated with
some limitations and these are: computational burden
which as a result of many functions to be evaluated,
combining of predictors that are of lower order with the
correctors. All these highlighted drawbacks affect the
accuracy of the method (Kayode, 2008; Adesanya
et al., 2008).
In order to advance the accuracy of numerical
methods, block method was introduced to
simultaneously generate numerical results (Adesanya
et al., 2013; Omar and Suleiman, 1999, 2003, 2005).
This method gives better approximation and found to be
cost effective because of the evaluation of few
functions involved. The derivation of block methods
with lower steplength for solving second order
ordinary differential equations have been considered by
researchers like Adesanya et al. (2008) and
Mohammed et al. (2010) in which the accuracy of the
methods are very low.
To bring improvement on the accuracy of block
method, this study considers higher steplength k = 8
for the development of block method that can solve
second order initial value problems of ordinary
differential equations directly.
%’()*
Power series approximate solution of the form
) (
1
0
∑
− +
=
=
s r
j
j
j
x a x y
(2)
is considered as an interpolation polynomial. Where r
and s are the number of interpolation and collocation
points, respectively. Equation (2) is differentiated twice
to give:
∑
− +
=
−
− = ′ ′
1
2
2
) 1 ( ) (
s r
j
j
j
x a j j x y
(3)
The approximate solution (2) is interpolated at x = x
n+i
, i
= 5(1)6 and we collocate Eq. (3) at x = x
n+i
, i = 0(1)8.
The interpolation and collocation equations give:
AX = B
(4)
where, X = [a
0
, a
1
, a
2
, ..., a
10
]
T
, B = [y
n+5
, y
n+6
, f
n
, f
n+1
,
..., f
n+8
]
T
the value of A is shown in Appendix A.