This paper presents a three-step block predictor-corrector method for solving general second order ordinary differential equations, addressing limitations in existing methods. The proposed method enhances accuracy by allowing evaluations at selected grid points without overlap and eliminates the need for developing starting values. It has been tested against numerical examples and demonstrated better approximation outcomes than traditional approaches.

1. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2297-2301
Three Steps Block Predictor-Block Corrector Method For The
Solution Of General Second Order Ordinary Differential
Equations
*Adesanya, A. Olaide, Odekunle, **M. Remileku, ***Alkali, M. Adamu
*Department of Mathematics, Modibbo Adama University of Technology, Yola, Adamawa State, Nigeria
**Department of Mathematics, Modibbo Adama University of Technology,Yola, Adamawa State, Nigeria
***Department of Mathematics, Modibbo Adama University of Technology,Yola, Adamawa State, Nigeria
Abstract:
We consider three steps numerical series to mention few. The major disadvantage of this
integrator which is derived by collocation and method is that the predictors are reducing order
interpolation of power series approximation predictors hence it has a lot of effect on the accuracy
solution to generate a linear multistep method of the method. Among authors that proposed
which serves as the corrector. The corrector is predictor corrector method are [6, 7, 12]. Other
then implemented in block method. We then setbacks of this method are extensively discussed by
propose another numerical integrator which is [7].
implemented in block method to serve as a In order to cater for the setbacks of the
constant order predictor to the corrector. Basic predictor-corrector method, scholars adopted block
properties of the corrector viz, order, zero method for direct solution of higher order ordinary
stability and stability interval are investigated. differential equation. This method has the properties
The new method was tested on some numerical of being self starting and gives evaluation at selected
examples and was found to give better grid points without over lapping. It does not require
approximation than the existing methods. developing predictors or starting values moreover it
evaluates fewer functions per step. The major setback
Keyword: Collocation, Interpolation, of this method is that the interpolation points cannot
approximate solution, block method, order, zero exceed the order of the differential equation, hence
stability. this method does not exhaust all interpolation points
AMS Subject classification: 65L05, 65L06, 65D30. therefore methods of lower order are developed.
Among these authors are [1, 7, 9, 10, 15, 13, 14, 15,
1 Introduction 16]. Scholars later adopted block predictor-block
This paper considers method of solving corrector method to cater for some of the setbacks of
general second order initial value problems of block method. They proposed a method which is
ordinary differential equation of the form capable of exhausting all the possible interpolation
point as the corrector and then proposed a constant
y ''  f ( x, y, y ' ), y k ( x0 )  y0 , k  0,1
k
order predictor using block method. The method
forms a bridge between the predictor-corrector
(1) method and the block method. Among these authors
Where f is continuous within the interval of are [2, 3, 4]. The major setback of this method is that
integration. the evaluation at selected points are overlapping
Method of reduction of higher order hence the behaviour of the dynamical system at
ordinary differential equation to systems of first order selected grid points cannot be mentioned.
ordinary differential equation has been discussed by In this paper, we proposed a method in
many scholars. These authors suggested that the which the corrector is implemented in block method;
method increased the dimension of the resulting hence this method gives evaluation at selected grid
system of the first order; hence it wastes computer points without overlapping.
and human effort. Among such authors are [5, 9, 11].
Scholars later adopted predictor–corrector method to 2 Methodology
solve higher order ordinary differential equation 2.1 Development of the block corrector|
directly. These authors proposed implicit linear We consider a power series approximate solution of
multistep methods which serve as the corrector, the the form
r  s 1
a
predictors proposed to implement this corrector are in
reducing order of accuracy. These methods are y ( x)  j xj
j 0
developed by adopting different approximate solution
such as Power series, Chebychev polynomial,
Newton polynomial, Lagrange polynomial, Fourier (2)
Where 𝑟 and 𝑠 are the number of collocation and
interpolation points respectively.
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2. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2297-2301
2t  18t 5  55t 4  60t 3  27t  6
The second derivative of (2) gives 1 6
r  s 1 0  
y ''   j ( j  1)a j x j 2 6
j 2
1 
3
2t  18t 5  55t 4  60t 3  24t 
1 6
(3)
 2   2t 6  18t 5  55t 4  60t 3  21t 
Substituting (3) into (1) gives 1
r  s 1
6
f ( x, y , y ' )   j ( j  1)a x j 2
10t 6  93t 5  305t 4  410t 3  180t 2  8t 
j
1
j 2 0 
360
100t 6  89t 5  2675t 4  2820t 3  936t 
(4) 1
Collocating (3) at x n  r , r  0(1)3 and interpolating 1 
360
(2) at x n  s , s  0(1)2 gives a system of non linear
equation of the form 2 
1
360
10t 6  99t 5  335t 4  390t 3  144t 
AU  B
3 
1
360
3t 5  15t 4  20t 3  8t 
(5)
1 xn xn2
xn3
xn 4
xn5
xn 
6
  x  xn
0 xn1
2 3
xn1 xn1 4
xn1 54
xn1 6
xn1  t , y n  j  y ( xn  jh ) ,
h
0 x n  2 2 3
xn 2 xn 2 4
xn 2 5
xn 2 xn 2 
6
 f n j  f ( xn , y( xn  jh), y ' ( xn  jh))
4 
U  0 0 2 6 xn 12 xn 2
20 xn 3
30 xn  Evaluating (6) at t  4 gives
0 0 2 6 xn1 12 xn1 20 xn1 30 xn1 
2 3 4
  h2
y n3  3 y n 2  3 y n1  y n   f n3  9 f n 2  9 f n1  f n 
0 0 2 6 xn 2 12 xn 2 20 xn 2 30 xn 2 
2 3 4
12
0 0 2 6 xn3 12 xn3 20 xn3 30 xn3 
2 3 4 (7)
 
Evaluating the first derivative of (6) at t  0,1 gives
a0   yn 
a  y  2 h2
y n 2  8 y n1  y n   f n  117 f n1  18 f n2  f n3 
9
 1  n 1  hyn 
'
3 2 45
a 2   y n2  (8)
   
, A   a3  , B   f n   17 2
yn2  14 yn1  yn  47 f n  679 f n1  121 f n2  7 f n3 
11 h
hyn' 1 
a 4   f n 1  6 6 360
    (9)
 a5   f n2  Writing (7)-(9) in block form and solving for the
a  f 
 6  n 3  independent solution at selected grid points gives
Solving (5) for a j ' s , using Gaussian A ( 0 )Ym  ey n  cy n  df ( y n )  bF (Ym )
'
elimination method and substituting back into the
(10)
approximate solution, gives a continuous linear
Ym  yn1 , yn2 , yn3 
T
multistep method. Where
yn  yn1 , yn2 , yn 
2 3
y ( x)    j ( x) y n j  h 2   j ( x) f n j
T
j 0 j 0
(6)

yn  yn1 , yn , yn1
' ' '
T
f ( yn )   f n1 , f n2 , f n 
T
Where the coefficient of y n  j and f n  j gives
F (Ym )   f n1 , f n2 , f n3 
T
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3. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2297-2301
1 0 0 0 0 1
We then proposed a prediction equation of the form
A ( 0 )  0 1 0  ,
m
  e  0 0 1 ,
 
Ym  ey n   h    F  ( y n ),
 0
0 0 1 
  0 0 1
  (12)
 17 21  Where 2 is the order of the differential
0 38 38  equation. In this paper, m  2.
 14 24  d 
c  0 , F  ( yn )  
f ( x, y, y ' ) yn Substituting (10)
 19 19  dx
0 33 81  into (11) gives
 38 38 
   2

Ym  eyn  h  dF ( yn )  bh  F  eyn   h   f  f ( y) 
 851    0 
0 0 13680  (13)
 127  Writing (10) as
d  0 0 ,
 855  Ym  e * y n  chyn  d * h 2 f ( y n )  bF (Ym )
* ' *
0 0 267 

 1520  
(14)
Substituting (13) in to (14) gives
  1653 93  11  Ym  e * y n  chyn  d * h 2 f ( y n )  bF (Ym )
* '
 13680 13680 13680
 399 111 7 
b  (15)
 855 855 855  Equation (15) is our block predictor-block
 1539 1701 93  corrector method.
 1520
 1520 1520   4.0 Analysis of basic properties of the method
4.1 Order of the method
2.2 Development of Block predictor We define a linear operator on the block (10) to give
[7] developed 3 steps block method of the form (10) ℒ 𝑦 𝑥 :ℎ =
which is writing explicitly to give
h2 y ( x)  A ( 0 )Ym  ey n  chyn  df ( y n )  bF (Ym )
'
y n1  y n  hy 
'
n 97 f n  114 f n1  39 f n2  8 f n3  (16)
360 Expanding (16) in Taylor series gives
h2 ℒ 𝑦 𝑥 : ℎ = 𝑐0 𝑦 𝑥 + 𝑐1 ℎ𝑦 ′ 𝑥 + 𝑐2 ℎ2 𝑦 ′′ 𝑥 +
y n 2  y n  2hyn  28 f n  66 f n1  6 f n 2  2 f n3 
'
⋯ + 𝑐 𝑝 ℎ 𝑏 (𝑥)
45
Definition: the block (10) and associated linear
h2
y n3  y n  3hyn  39 f n  108 f n1  27 f n 2  6 f n3 
' operator are said to have order 𝑝 if
40 c0  c1  c2  ...  c p 1  0 and c p  2  0 . The
y n1  y n  9 f n  19 f n1  5 f n 2  f n3 
' ' h term c p  2 is called the error constant and implies that
24 the local truncation error is
( p  2) p2 p3
y n 2  y n   f n  4 f n1  f n 2 
' ' h t n k  c p  2 h y xn  O(h )
3 Hence the block (10) has order five with error
 
y n3  y n  3 f n  9 f n 1  9 f n 2  3 f n3 
' ' h 97 41 9
constant  , ,
8  383040 23940 42560 

For the derivation and analysis of the block predictor, 4.2 Zero Stability
readers are referred to [7]. A block method is said to be zero stable if as h  0 ,
2.3 Implementation of the Method the root r j ; j  1(1)k of the first characteristic
Given the general block formula proposed by [7] in polynomial  ( R)  0 that is
 ( R)  det A R  0
the form k 1
satisfying R  1
( 0)
Ym  ey n  h  df ( y n )  bh  F (Ym )
(11)
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4. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2297-2301
and for those roots with R  1 must have 11) 08)
1.10 3.74
multiplicity equal to unity {see [9] for details} 0. 1.42364593 1.42364893
13(- 47(-
8 0193602 0303739
1 0 0 0 0 1 10) 08)
   
Hence R 0 1 0  0 0 1  0 0. 1.48470027 1.48470027
1.88 5.95
    9 8594052 8782232
18(- 07(-
0 0 1 0 0 1
    10) 08)
3.22 9.29
R 3  R 2  0 , R 2 R  1  0 R  0,0,1 1. 1.84930614 1.54930614
11(- 40(-
0 4334055 4656174
10) 08)
5.0 Numerical Experiments
We test our method with second order initial value Table II for problem II
problems.
Problem 1: Consider a non linear initial value
problem
1
y' ' x( y' )  0, y(0)  1 , y ' (0) 
, h  0.01
2
1 2 x
Exact solution y ( x)  1  log e  
2 2 x
This problem was solved by [1] where a block of
0.1
order four was proposed for h  and [7] where
32
a 3 steps block method was proposed for the same
step size. The result is shown in table I
Problem II: We consider a highly oscillatory test
problem
y ' '2 y  0 , where   2 , with initial condition
y(0)  1, y' (0)  2, h  0.01
Exact solution; y( x)  cos2 x  sin 2 x
We compare our result with [7] where a block
method with step length of three was proposed Conclusion:
Table I for problem I We have proposed a 3 steps block predictor-
Erro block corrector method in this paper. This method
Computed Erro
x Exact result r in has addressed some of the setbacks of block predictor
result r
[1] corrector method proposed by [2, 3, 4]. The result has
1.33 4.98 shown clearly that our method gives better
0. 1.05004172 1.05004172
00(- 27(- approximation than the existing method.
1 9278491 9278624
13) 11)
9.94 4.10 References
0. 1.10033534 1.10033534
98(- 43(- [1] Anake, T. A., Awoyemi, D. O., Adesanya,
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03(- 42(- Alkali, M. A., (2012): Order six block
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12) 09) predictor corrector for the solution of
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1.83 7.24
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6 4203111 4338198 Cowell more accurately by hybrid Adams
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5. Adesanya, A. Olaide, Odekunle, M. Remileku, Alkali, M. Adamu / International Journal of
Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue4, July-August 2012, pp.2297-2301
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