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Journal of Advances in M athematics and C omputer Science
28(1) : 1- 7, 2018; Ar ticle no. JA MCS. 41557
IS SN: 245 6- 9968
(P as t n am e:
Br it is h Jou rn al o f Ma th em at ics & C om pu te r Sc ie nc e,
Pa st
IS SN : 22 31 -0 851 )
_____________________________________
*Corresponding author: E-mail: ajisco4live@yahoo.com;
Two-Step Hybrid Block Method for Solving First Order
Ordinary Differential Equations Using Power Series Approach
G. Ajileye
1*
, S. A. Amoo
1
and O. D. Ogwumu
1
1
Department of Mathematics and Statistics, Federal University, Wukari, Taraba State, Nigeria.
Authors’ contributions
This work was carried out in collaboration between all authors. Author GA designed the study, performed
the statistical analysis, wrote the protocol and wrote the first draft of the manuscript. Author SAA managed
the analyses of the study. Author ODO managed the literature searches. All authors read and approved the
final manuscript.
Article Information
DOI: 10.9734/JAMCS/2018/41557
Editor(s):
(1) Dariusz Jacek Jakóbczak, Assistant Professor, Chair of Computer Science and Management in this Department, Technical
University of Koszalin, Poland.
Reviewers:
(1) Abdu Masanawa Sagir, Hassan Usman Katsina Polytechnic, Nigeria.
(2)
Aliyu Bhar Kisabo, Center for Space Transport and Propulsion, National Space Reseach and Development Agency, Nigeria.
Complete Peer review History:
http://www.sciencedomain.org/review-history/25575
Received: 25
th
February 2018
Accepted: 3
rd
May 2018
Published: 16
th
July 2018
_______________________________________________________________________________
Abstract
In this paper, we consider the derivation of hybrid block method for the solution of general first order
Initial Value Problem (IVP) in Ordinary Differential Equation. We adopted the method of Collocation
and Interpolation using power series approximation to generate the continuous formula. The properties
and features of the methods are analyzed and some numerical examples are also presented to illustrate the
accuracy and effectiveness of the method.
Keywords: Collocation; interpolation; linear multistep method; hybrid and power series polynomial.
1 Introduction
In recent times, the integration of Ordinary Differential Equations (ODEs) is carried out using block methods.
In this paper, we propose an order five hybrid block integrator method for the solution of first order ODEs of
the form:
Original Research Article
Ajileye et al.; JAMCS, 28(1): 1-7, 2018; Article no.JAMCS.41557
2
=(,), ()=
,[.] (1.0)
Where is continuous within the interval of integration[,]. We assume that satisfies Lipchitz condition
which guarantees the existence and uniqueness of solution of (1.0).The discrete solution of (1.0) using linear
multistep method has being studied by authors like [1] and continuous solution of (1.0) [2] and [3,4]. One
important advantage of the continuous over discrete approach is the ability to provide discrete schemes for
simultaneous integration. These discrete schemes can be reformulated to general linear methods (GLM) [5].
The block methods are self-starting and can be applied to both stiff and non-stiff initial value problem in
differential equations. More recently, authors like [6,7,8,9] and [10] to mention few proposed methods
ranging from predictor- corrector to hybrid block method for initial value problem in ordinary differential
equation.
In this work, hybrid block method using Power series expansion will be considered. This will help in coming
up with a more computationally reliable integrator that can solve first order differential equations problems
of the form (1.0).
2 Derivation of Hybrid Method
In this section, we intend to construct the proposed two-step linear multistep method which will be used to
generate the main method and other methods required to set up the block method. We consider the power
series polynomial of the form:
()=∑
(2.0)
which is used as our basis to produce an approximate solution to (1.0) as
()=∑
(3.0)
and
()=∑
=(,) (4.0)
where
are the parameters to be determined, and are the points of collocation and interpolation
respectively. This process leads to (+−1) of non-linear system of equations with unknown coefficients,
which are to be determined by the use of Maple 17 Mathematical software.
3 Hybrid Block Method
Using equation (3.0) and (4.0), m=1 and t=5 our choice of degree of polynomial is (+−1). Equation
(3.0) is interpolated at the point =
and equation (4.0) is collocated at =(0,
,1,
,2) which lead to
system of equation of the form
∑
=
=0 (5.0)
∑
=
=(0,
,1,
,2) (6.0)
With the mathematical software, we obtained the continuous formulation of equations (5.0) and (6.0) of the
form
()=
+ℎ[
+
+
+
+
] (7.0)
Ajileye et al.; JAMCS, 28(1): 1-7, 2018; Article no.JAMCS.41557
3
After obtaining the values of
and
, =0 and =(0,
,1
,2) in (7.0)
We evaluated at the point =
,=(1,
,
,2) which gives the following set of discrete schemes to
form our hybrid block method.
=
+29
180ℎ
+31
45ℎ
/
+2
15ℎ
+1
45ℎ
/
−1
180ℎ
/
=
+251
1440ℎ
+323
720ℎ
/
−11
60ℎ
+53
720ℎ
/
−19
1440ℎ
/
=
+
ℎ
+
ℎ
/
+
ℎ
+
ℎ
/
−
ℎ
(8.0)
=
+7
45ℎ
+32
45ℎ
/
+4
15ℎ
+32
45ℎ
/
+7
45ℎ
Equations (8.0) are of uniform order five, with error constant as follows
[1
5760,3
10240,3
10240,− 1
15120]
4 Consistency
Definition: The Linear Multistep method is said to be consistent if it is of order P≥ 1 and its first and second
characteristic polynomial defined as ()=∑
and ()=∑
where Z satisfies
()∑
=0,()
(1)=0,()
(1)=2!(1), See [1].
The discrete Schemes derived are all of order greater than one and satisfy the condition (i)-(iii).
5 Zero Stability of the Block Method
The block method is defined by [11] as
=
+ℎ
ℎ
=[
,
,
,…,
]
T
=[
,
,
,…,
]
T
and
are chosen r x r matrix coefficient and =0,1,2… represents the block number, =, the
first step number in the m-th block and r is the proposed block size.
The block method is said to be zero stable if the roots of
,=1(1) of the first characteristics polynomial
is
()=det
=0,
=
satisfies |R
j
|≤ 1, if one of the roots is +1, then the root is called Principal Root of ().
Ajileye et al.; JAMCS, 28(1): 1-7, 2018; Article no.JAMCS.41557
4
2
2/3
1
2/1
1
2/1
2/3
2
2/3
2/1
1
45
7
45
32
15
4
45
32
160
3
80
21
20
9
80
51
1440
19
720
53
60
11
720
323
180
1
45
1
15
2
45
31
1000
1000
1000
1000
1000
0100
0010
0001
n
n
n
n
n
n
n
n
n
n
n
n
f
f
f
f
h
y
y
y
y
y
y
y
y
n
n
n
n
f
f
f
f
2/1
1
2/3
45
7
000
160
27
000
1440
251
000
180
29
000
where
()
=
1000
0100
0010
0001
,
()
=
1000
1000
1000
1000
,
()
=
45
7
45
32
15
4
45
32
160
3
80
21
20
9
80
51
1440
19
720
53
60
11
720
323
180
1
45
1
15
2
45
31
and
()
=
45
7
000
160
27
000
1440
251
000
180
29
000
The first characteristics polynomial of the scheme is
()=det[
−
]
()=
1000
1000
1000
1000
000
000
000
000
()=
1000
100
100
100
0
1000
100
100
100
(−1)=0
Ajileye et al.; JAMCS, 28(1): 1-7, 2018; Article no.JAMCS.41557
5
=
=
=0
=1
We can see clearly that no root has modulus greater than one (i.e.
≤1) ⍱. The hybrid block method is
zero stable.
6 Numerical Examples
Problem 1:
=,(0)=1,ℎ=0.1
Exact Solution: ()=exp()
Table 1. Comparison of approximate solution of problem 1
x Exact solution
Proposed scheme Error in proposed
scheme
Error in [2]
0.1 1.105170918075648 1.105170917860730 2.149179E-10 1.226221039551945e-05
0.2
1.221402758160170
4.7505E-10
1.355183832019158e-05
0.3
1.349858807576003
1.349858806788490
7.875129E-10
1.497709759790133e-05
0.4 1.491824697641270 1.491824696480820 1.16045E-09 1.655225270247307e-05
0.5 1.648721270700128 1.648721269097010 1.603118E-09 1.829306831546695e-05
0.6 1.822118800390509 1.822118798264440 2.126069E-09 2.021696710463594e-05
0.7 2.013752707470477 2.013752704729200 2.741277E-09 2.234320409577606e-05
0.8
2.225540928492468
5.989459E-09
2.469305938346267e-05
0.9 2.459603111156950 2.459603106852120 4.30483-09 2.729005110868599e-05
1.0 2.718281828459046 2.718281824122030 4.337016E-09 3.01601708376864e-05
Fig. 1. Plot of error in proposed scheme and error in [2]
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
3.50E-05
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Error in proposed Scheme
Error in [2]
Ajileye et al.; JAMCS, 28(1): 1-7, 2018; Article no.JAMCS.41557
6
Problem 2:
=0.5(1−),(0)=0.5,ℎ=0.1
Exact Solution: ()=1−0.5e−0.5x
Table 2. Comparison of approximate solution of problem 2
x Exact solution Proposed scheme Error in proposed scheme Error in [7]
0.1
0.524385287749643
0.524385287750861
1.218026E-13
5.574430e-012
0.2 0.547581290982020 0.547581290981880 1.399991E-13 3.946177e-012
0.3 0.569646011787471 0.569646011786286 1.184941E-12 8.183232e-012
0.4
0.590634623461009
0.590634623462548
1.538991E-12
3.436118e-011
0.5 0.610599608464297 0.610599608463187 1.110001E-12 1.929743e-010
0.6
0.629590889659141
0.629590889658614
5.270229E-12
1.879040e-010
0.7 0.647655955140643 0.647655955142752 2.10898E-12 1.776835e-010
0.8 0.664839976982180 0.664839976969201 1.297895E-11 1.724676e-010
0.9
0.681185924189113
0.681185924158290
3.08229E-11
1.847545e-010
1.0 0.696734670143683 0.696734670139561 4.121925E-11 3.005770e-010
Fig. 2. Plot of error in proposed scheme and error in [7]
7 Discussion of Result
We observed that from the two problems tested with this proposed block hybrid method the
results converges to exact solutions and also compared favourably with the existing similar methods (see
Tables 1, 2).
8 Conclusion
In this paper, we have presented Hybrid block method algorithm for the solution of first order ordinary
differential equations. The approximate solution adopted in this research produced a block method with
0.00E+00
5.00E-11
1.00E-10
1.50E-10
2.00E-10
2.50E-10
3.00E-10
3.50E-10
12345678910
Error in proposed Scheme
Error in [7]
Ajileye et al.; JAMCS, 28(1): 1-7, 2018; Article no.JAMCS.41557
7
stability region. This made it to perform well on problems. The block method proposed was found to be
zero-stable, consistent and convergent.
Competing Interests
Authors have declared that no competing interests exist.
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_______________________________________________________________________________________
© 2018 Ajileye et al.; This is an Open Access article distributed under the terms of the Creative Commons Attribution License
(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited.
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