DOI : 10.17577/IJERTV1IS5063
- Open Access
- Total Downloads : 818
- Authors : Adesanya, A. Olaide, Awoyemi, D. Oni. , Famewo, Moyosoreoluwa
- Paper ID : IJERTV1IS5063
- Volume & Issue : Volume 01, Issue 05 (July 2012)
- Published (First Online): 02-08-2012
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Half Step Constant Predictor-corrector Method for the Solution of Second Order Ordinary Differential Equation
Half step constant predictor-corrector method for the solution of second order ordinary differential equation
lAdesanya, A. Olaide, 2Awoyemi, D. Oni. and 3Famewo, Moyosoreoluwa
lDepartment of Mathematics, Modibbo Adama University of Technology, Yola, Adamawa State, Nigeria
2Department of Mathematical Sciences, Federal University of Technology, Akure,
Ondo State, Nigeria
3Department of Mathematics, Covenant University, Sango Ota, Ogun State, Nigeria
Abstract
We consider a half step numerical integrator which is derived by collocating the differential system and interpolating the approximate solution to generate a continuous hybrid linear multistep method which serves as the corrector. The predictors are derived using block method hence a constant order predictors are developed. The properties of the corrector viz; order, consisitency, zero stability and convergence are verified. The new method was tested on some numerical examples and was found to give better approximation than the existing method.
Keyword: half step, collocation, differential system, interpolation, approxi- mate solution, predictor, corrector
A.M.S Subject Classification: 65L05, 65L06, 65D30
1
-
Introduction
This paper considers the approximate solution to the general second order initial value problems of the form
n
y11 = f(x, y, y1) yk (xn) = yk, k = 0, 1 (1)
Equation (1) are convectionally solved by reducing to system of first order ordinary differential equation, then any approximate method of solving first order can be adopted to solve the resulting system of first order equation. This method is extensively discussed by Adesanya, Anake and Udoh [5], Awoyemi and Kayode [6], Jator [11] to mention few. These authors suggested that the direct method for solving higher order ordinary differential equations are more efficient since the method of reduction increased the dimension of the resulting system of first order; hence it wastes alot of computer and human efforts.
Scholars have worked on predictor-corrector method for the solution of implicit linear multistep method, among them are Kayode and Adeyeye[12], Adesanya, Anake and Oghoyon [4], Awoyemi [7], Olabode [14]. They individually proposed method in which reducing order predictors are adopted to implement the corrector. The major setback of this method is that the predictors are reducing order of accuracy, therefore it has a great effect on the accuracy of the method. Other setbacks of this method are discussed by Awoyemi [7] and Awoyemi et al. [8].
Scholars later proposed block method to cater for some of the setbacks of predictor-corrector method. Block method has the properties of being self starting and gives evaluation at selected grid points without overlapping. They do not
2
require developing seperate predictors and starting values. moreover it evaluates fewer function per step. Among these authors are Jator [10], Jator and Li [9], Simiak [16], Abbas [1], Adesanya et al. [2], Awoyemi et al. [8], Omar and Suleiman
[15] Majid et al. [13].It was observed that in block method, the number of interpolation points cannot exceed the order of the differential equation, hence this method does not exhaust all possible interpolation points, therefore method of lower order are developed.
In this paper, we developed a method which is implemented in predictor cor- rector method in which the predictors are constant order of accuracy. This method combines the properties of both predictor-corrector and block method.
-
Methodology
2.1 Development of the corrector
We consider a power series approximate solution of the form
y(x) =
r+s-l
j=O
ajxj (2)
where r and s are the number of interpolation and collocation respectively. The second derivative of (2) gives
y11(x) =
r+s-l
j=O
j(j – 1)ajxj-2 (3)
3
substituting (3) into (1) gives
f(x, y, y1) =
r+s-l
j=O
j(j – 1)ajxj-2 (4)
Equation (4) is called the differential system. Interpolating (2) at xn+r, r =
8
8
8
2
0 (l) 3
and collocating xn+s, s = 0 (l ) l , gives a non linear system of the form
AX = U (5)
A = aO al a2 a3 a4 a5 a6 a7 a8
U =
yn yn+ 1
yn+ 1
yn+ 3
fn fn+ 1
fn+ 1
fn+ 3
fn+ 1
8
4
8
8
4
8
2
n
x
x
x
x
x
x
n
n
n
n
n
n
1 xn x2 3 4 5 6 7 8
I
8
I
I
1 xn+ 1
1 x 1
x2 1
n+ 4
x3 1
n+ 4
x4 1
n+ 4
x5 1
n+ 4
x6 1
n+ 4
x7 1
n+ 4
n+ 8
n+ 8
n+ 8
n+ 8
n+ 8
n+ 8
n+ 8
x8 1
n+ 4
I n+
8
x2 1
8
x3 1
8
x4 1
x5 1
x6 1
x7 1
x8 1
8
I 1 xn+ 3
8
x2 3
8
x3 3
8
x4 3
x5 3
20×3
x6 3
30×4
x7 3
42×5
x8 3 I
8
56×6
n+ n+
I n
4
X = 0 0 2 6xn 12×2
I
n+ 8
n+ 8
20×3
n+ 8
30×4
n+ 8
42×5
n+
I
I
I
n
n
n
n
n
I
0 0 2 6xn+ 1 12×2 1
20×3 1
30×4 1
42×5 1
56×6 1
8 n+ 8
I
4
n+ 4
I
n+ 8
n+ 8
n+ 4
n+ 8
n+ 4
n+ 8
n+ 4
n+ 8
n+ 8
I
I
I
n+ 4
0 0 2 6xn+ 1 12×2 1
20×3 1
30×4 1
42×5 1
56×6 1
I
n+ 8
0 0 2 6xn+ 3
8
2
12×2 3
n+ 2
20×3 3
n+ 2
30×4 3
n+ 8
n+ 2
42×5 3
n+ 8
n+ 2
56×6 3
n+ 2
8
0 0 2 6xn+ 1
12×2 1
20×3 1
30×4 1
42×5 1
56×6 1
Solving (5) using Guassian elinination method and substituting into (2) gives
4
a continuous hybrid linear multistep method of the form
y(x) = a y
+ a 1 y 1 + a 1 y 1 + a 3 y 3 + h
8
8
4
4
(6)
8
8
2
2
2 /I 6Ofn + 6 1 fn+ 1 + 6 1 fn+ 1 \I
O
n
O
n
8
8
4
4
8
8
+6n+ 3 f3 + 6 1 fn+ 1
where yn+j = y (xn + jh) , f ((xn + jh), y (xn + jh) y1 (xn + jh))
1
aO =
21
(2097152t7 – 3670016t6 + 2408448t5 – 716800t4 + 86016t3 – 596t + 21)
a 1 =
1 /I 176160768t8 – 329252864t7 + 24221056t6 – 89112576t5 \I
/
8 217
+17002496t4 – 1462272t3 + 6912t
I – –
1 352321536t8 593494016t7 + 370671616t6 103563264t5
217
4
a 1 = –
+11784192t4 – 258048t3 – 2916t
\I
a 3 =
1 /I 528482304t8 – 857735168t7 + 499122176t6 – 118013952t5 \I
8 651
+6565888t4 + 946176t3 – 11008t
6 =
1 /I 16515072t8 – 65404928t7 + 84926464t6 – 51351552t5 \I
O 312480
+15829184t4
– 2421664t3
– 2421t
I – –
1 / 24772608t8 14303232t7 21489664t6 + 23466240t5
4
/
8
-8094464t
+ 1002624t3
– 4887t
6 1 = -19530
\I
I – –
1 177995776t8 287309824t7 + 164749312t6 37044224t5
17360
4
6 1 = –
+1231552t4 + 452928t3 – 4455t
\I
5
I – –
/
1 24772608t8 39698432t7 + 22951936t6 5531904t5
19530
8
6 3 = –
+377216t4 + 30464t3 – 423t
\I
6 1 =
1 /I 2359296t8 – 3538944t7 + 1974272t6 – 479232t5 \I
2
h
t = x-xn
44640
+39232t4 + 1248t3 – 27t
2
Evaluating (6) at t = l, gives a discrete scheme
2
yn+ 1 +
31
128
8
yn+ 3 –
31
318
4
8
yn+ 1 +
31
128
p /I
29760
23fn+ 1 + 688fn+ 3 + \I
yn+ 1 +yn =
2358fn+ 1 + 688fn+ 1 + 23fn
2
8
4 8
(7)
2.2 Development of predictors
In developing the predictor, we interpolate equation (2) at xn+r, r = l, 3
and
4 8
collocating (4) at xn+s, s = 0 (l ) l
to generate a system of non linear equation in
8 2
the form (5) where
A = aO al a2 a3 a4 a5 a6
4
8
8
4
8
2
U =
yn+ 1
yn+ 3
fn fn+ 1
fn+ 1
fn+ 3
fn+ 1
U =
yn+ 1
yn+ 3
fn fn+ 1
fn+ 1
fn+ 3
fn+ 1
6
4
I 1 xn+ 1
n+ 4
1 x 3
I
x2 1
n+ 4
n+ 4
n+ 4
n+ 4
n+ 8
x3 1
n+ 8
x4 1
n+ 8
x5 1
n+ 8
x6 1 I
n+ 8
I n+
x2 3
x3 3
x4 3
x5 3
x6 3
8
I 0 0 2 6xn 12×2
20×3
n+ 8
30×4 I
n+ 8
n
8
n+ 8
X = I 0 0 2 6xn+ 1 12×2 1
n
20×3 1
n
I
30×4 1
I
2
I 0 0 2 6xn+ 1 12x 1
20×3 1
n+ 8
30×4 1
n+ 8
4 n+ 4
I
8
n+ 8
I
n+ 4
n+ 8
n+ 4
I
I
n+ 8
0 0 2 6xn+ 3 12×2 3
20×3 3
30×4 3
2
n+ 2
n+ 2
n+ 2
0 0 2 6xn+ 1 12×2 1 20×3 1 30×4 1
Solving this equation using Guassian elinimation method and substituting into
(2) gives a continuous hybrid linear multistep method of the form
4
4
8
8
8
8
4
4
8
8
2
2
y(x) = a 1 y 1 + a 3 y 3 + p (6Ofn + 6 1 fn+ 1 + 6 1 fn+ 1 + 6n+ 3 f3 + 6 1 fn+ 1 3 (8)
–
a 1 = 3 8t a 3
4 8
= 8t – 2
( )- – –
6 = 1 262144t6 491520t5 + 358400t4 128000t3 + 23040t2 1900t + 51
O 46080
(-
1
6 = 1
8 11520
6 1
4
=
7680
262144t
1 (
262144t6
– 393216t
6
– 442368t
5
+ 194560t
5
+ 266240t4
– 30720t
4
– 61440t
3
+ 1908t – 189)
3
– 644t + 20t
)
(-
3
6 = 1
8 11520
6 1
2
=
46080
262144t
1 (
262144t6
– 294912t
6
– 344064t
5
+ 112640t
5
+ 143360t4
– 15360t
4
– 20480
3
+ 284t – 39)
3
+ 132t – 9
)
8
8
2
Solving for the independent solution yn+s, s = l (l ) l , gives a continuous hybrid
7
block formula of the form
l
(jh)m
(m) 2 ( 3
y(x) =
yn +h \J!Ofn + \J! 1 fn+ 1 + \J! 1 fn+ 1 + \J!n+ 3 f3 + \J! 1 fn+ 1
(9)
m!
j=O
8 8 4 4
8 8 2 2
Where
90
\J!O
= 1 (512t6 – 960t5 + 700t4 – 250t3 + 45t2)
\J! 1
8
1
5
= -45
(1024t6
5
– 1728t
+ 1040t4
– 240t3)
\J! 1
4
= 1 15
512t6
– 768t
+ 380t4
– 60t3)
\J! 3
8
1
5
(
= -45
(1024t6
– 1344t
5
+ 560t4
– 80t3)
\J! 1
2
= 1 45
256t6
– 288t
+ 110t4
– 15t3)
8
8
2
(
Evaluating (9) at t = l (l ) l , gives a discrete block formula in the form
4
8
2
A(O)Ym = eyn + pdf (yn) + pbF (Ym) (10)
8
Ym =
yn+ 1
yn+ 1
yn+ 3
yn+ 1
f(yn) =
yn-l yn-2 yn-3 yn
F (Ym) =
8
fn+ 1
4
fn+ 1
8
fn+ 3
2
I
I
I
I
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
fn+ 1 e =
II
8
b = I
I
367 92l6O
–
282
92l6O
ll6 92l6O
–
2l 92l6O
l44 576O
–
3O
576O
l6 576O
–
3
576O
468 lO24O
54 lO24O
6O lO24O
–
9 lO24O
24 36O
I
6 36O
I
8 36O
0
d =
367
92l6O
53
576O
l47
lO24O
7
36O
Evaluating the first derivative of (9) at t = l (l ) l
and substituting in (10)
gives
yn+ 1
1
1 h (
8 8 2
251fn + 646fn+ 1 – 264fn+ 1 + 10fn+ 3 – 19fn+ 1
3
8
= yn + 5760
yn+ 1
= yn + 720
29fn + 124fn+ 1 + 24fn+ 1 + 4fn+ 3 – fn+ 1
1 1 h (
8 4 8 2
3
4
yn+ 3
= yn + 640
27fn + 102fn+ 1 + 72fn+ 1 + 42fn+ 3 – 3fn+ 1
1 1 h (
8 4 8 2
3
8
yn+ 1
= yn +
7fn + 32fn+ 1 + 12fn+ 1 + 32fn+ 3 + 7fn+ 1
2
180
1 1 h (
8 4 8 2
3
8
4
8
2
-
Analysis of the basic properties of the block
-
Order of the method
We defined a linear operator on (7) to give
2
£{y(x) h} = y(x) – yn+ 1 +
128
8
31 yn+ 3 –
318
yn+ 1 +
31 4
128
8
31 yn+ 1 + yn –
p ( 3
23fn+ 1 + 688fn+ 3 + 2358fn+ 1 + 688fn+ 1 + 23fn
(11)
29760 2 8 4 8
Expanding yn+j and fn+j in Taylor series and comparing the coefficient of h
9
gives
£{y(x) h} = COy(x) + Clhy1(x) + … + Cphpyp(x) + Cp+lhp+lyp+l(x)
+Cp+2hp+2yp+2(x) + … (12)
Definition 1 Order
The difference operator £ and the associated continuous linear multistep method
(15) are said to be of order p if CO = Cl = … = Cp = Cp+l = 0 and Cp+2 is called the error constant and implies that the local truncation error is given by tn+k = Cp+2h(p+2)y(p+2)(x) + 0 (hp+3)
59996l6OO
The order of our discrete scheme is 8, with error constant Cp+2 = -79
-
Consistency
A linear multistep method (7) is said to be consistent if it has order p 1 and if p(1) = p1(1) = 0 and p11(1) = 2!o-(1) where p(r) is the first characteristic polyno- mial and o-(r) is the second characteristic polynomial.
For our method,
3l
2
3l
3l r 2 + 1
p(r) = r + l28r 3 – 3l8 r + l28 1
465
23r
+ 688r 2 + 2358r + 688r 2 + 23
.
and o-(r) = l ( 2 3 1 3
Clearly p(1) = p1(1) = 0 and p11(1) = 2!o-(1).
Hence our method is consistent
10
-
Zero stability
A linear multistep method is said to be zero stable, if the zeros of the first char- acteristic polynomial p(r) satisfies I r I:s 1 and for I r I= 1 is simple
Our method was found to be zero stable.
-
Region of absolute stability
8y.
The method (7) is said to be absolute stable if for a given h, all roots zs of the characteristic polynomial 1 (z, h) = p (z) + po- (z) = 0, satisfies I zs I< 1, s = 1, 2, …, n. where h = -,\2p and ,\ = 8f .
The boundary locus method is adopted to determine the region of absolute stability. Substituting the test equation y11 = – ,\2p into (7) and writing r = cos 0 + i sin 0 gives the stability region as shown in fig. (1), plotted using Scientific workplace software.
4 y
3
2
1
9
8
7
6
5
4
3
2
1
x
1
2
3
4
fig (1)
11
-
-
Numerical Experiments
-
Test Problems
2
We test our method with second order initial value problems Problem 1: Consider the non-linear initial value problem (I.V.P) y11 – x(y1)2 = 0, y(0) = 1, y1(0) = l, h = 0.05
2
2-x
Exact solution: y(x) = 1 + l ln (2+x )
Jator [10] solved this problem in block method where a block of order 6 and step-length of 5 is proposed with h = 0.05.Adesanya et al. [2] also solve this problem where the adopted constant predictor corrector method , wher a corrector of order 8 is proposed. Though we did not show the result of Jator [9] but Adesanya et al. [2] was better in term of accuracy. We compare our result with this result as shown in table 1
Problem 2: We consider the non-linear initial value problem (I.V.P)
2y
6
4
6
2
y11 = (yt)2 – 2y, y( 1T ) = l, y1( 1T ) = 3 , h = 0.05
Exact solution:(sin x)2
Jator [10] solved this problem in block method where a block of order 6 and step-length of 5 is proposed with h = 0.05.Adesanya et al. [2] also solve this problem where the adopted constant predictor corrector method , wher a corrector of order 8 is proposed. Though we did not show the result of Jator [9] but Adesanya et al. [2] was better in term of accuracy. we compare our result with this result as shown in table 2
Error=IExact result-computed resultI
12
table 1 for problem 1
x Exact result Computed result Error Error in [2] 0.1 1.050041729278 1.050041729278 5.5511(-15) 7.5028(-13)
0.2 1.100335347731 1.100335347731 2.0650(-15) 9.7410(-12)
0.3 1.151140435936 1.151140435936 5.0404(-14) 3.7638(-11)
0.4 1.202732554054 1.202732554054 9.6145(-14) 9.7765(-11)
0.5 1.255412811882 1.255412811882 1.7230(-13) 2.0825(-10)
0.6 1.309519604203 1.309519604203 2.8288(-13) 3.9604(-10)
0.7 1.365443754271 1.365443754271 4.6473(-13) 7.0460(-10)
0.8 1.423648930193 1.423648930192 7.5250(-13) 1.2095(-09)
0.9 1.484700278594 1.484700278593 1.2370(-12) 2.0511(-09)
1.0 1.549306144334 1.549306144332 2.0736(-12) 3.5066(-09)
table 2 for problem 2
x
Exact result
Computed result
Error
Error in [2]
1.1048
0.7981568789707
0.798156789000
3.0250(-12)
1.8811(-10)
1.2048
0.8719546393729
0.8719546393769
4.0059(-12)
2.4539(-10)
1.3048
0.9309237421478
0.9309237421528
5.0252(-12)
3.0306(-10)
1.4048
0.9727132751817
0.9727132751875
6.0277(-12)
3.5819(-10)
1.5048
0.9956572216671
0.9956572216741
6.9687(-12)
4.0838(-10)
1.6048
0.9988408788614
0.9988408788929
7.7953(-12)
4.5128(-10)
1.7048
0.9821373243990
0.9821373244077
8.4637(-12)
4.8473(-10)
1.8048
0.9462124762851
0.9462124762940
8.9351(-12)
5.0696(-10)
1.9048
0.8924985448466
0.8924985448558
9.1801(-12)
5.1697(-10)
2.0048
0.8231369350259
0.8231369350350
9.1735(-12)
5.1381(-10)
13
-
-
Conclusion
We have proposed a two steps-four hybrid points method in this paper. Contin- uous block method which has the properties of evaluation at all points with the interval of integration is adopted to give the independent solution at non over- lapping intervals as the predictor to an order eight corrector. This new method forms a bridge between the predictor-corrector method and block method. Hence it shares the properties of both method. the new method evaluate fewer function per step hence makes this performed better than the existing method i.e. block method and the predictor corrector method as shown in the numerical examples.
References
-
Abbas, S, Derivation of a new block method similar to the block trapezoidal rule for the numerical solution of first order IVPs, Scinec Echoes, 2006, 10-24
-
Adesanya, A. O., Odekunle, M. R. and Adeyeye, O., Continuous block method hybrid-predictor method for the solution of y11 = f(x, y, y1), International Journal of Mathematics and Scientific Computing, 2012, (In press)
-
Adesanya, A. O., Odekunle, M. R. and Alkali, M. A., Order six continuous co efficient method for the solution of second order ordinary differential equation, Canadian Journal of Science and Engineering Mathematics. 2012, (In Press)
-
Adesanya, A. O., Anake, T. A., Oghoyon, G. J., Continuous implicit method for the solution of general second order ordinary differential equation. Journal of Nigerian Association of Mathematical Physics, 15,(2009), 71-78
-
Adesanya, A. O, Anake, T. A and Udoh, M O., Improved continuous method for direct solution of general second order ordinary differential equation.
14
J. of Nigerian Association of Mathematical Physics, 13, 2008, 59-62.
-
Awoyemi, D. O and Kayode, S. J., A maximal order for the direct solution of initial value problems of general second order ordinary differential equation, Proceedinds of the conference organised by the National Mathematical center ,
Abuja, 2005
-
Awoyemi, D. O., A p-stable linear multistep method for solving third order ordinary differential equation, Inter. J. Computer Math. 80(8), 2003, 85 – 991
-
Awoyemi, D. O., Adebile, E. A, Adesanya, A. O, and Anake, T. A., Modified block method for the direct solution of second order ordinary differential equation.
Intern. J. of Applied Mathematics and Computation, 3(3), 2011, 181-188
-
Jator, S. N and Li, J., A self starting linear multistep method for the direct solution of the general second order initial value problems. Intern. J. of Comp.
Math., 86(5), 2009, 817-836
-
Jator, S. N., A sixth order linear multistep method for direct solution of
y11 = f(x, y, y1), Intern. J. of Pure and Applied Mathematics, 40(1), 2007, 457-472
-
Jator, S. N, Improvement in Adams-Moulton for the first order initial problems, Journal of Tenness Academy of science, 76(2), 2001, 57-60
-
Kayode, S. J. and Adeyeye, O. A 3-step hybrid method for the direct solu- tion of second order initial value problem. Australian Journal of Basic and Applied Sciences, 5(12), 2011, 2121-2126
-
Majid, Z. A, Azmi, N. A and Suleiman, M., Solving second order differential equation using two point four step direct implicit block method. European Journal of Scientific Research, 31(1), 2009, 29-36
-
Olabode, B. T., An accurate scheme by block method for the third order
15
ordinary differential equation, Journal of Science and Technology, 10(1), 2009, http:/www.okamaiuniversity.us/pjst.htm
-
Omar, Z and Suleiman, M., Parallel R-point implicit block method for solving higher order ordinary differential equation directly. Journal of ICT, 3(1), 2003, 53-66
-
Siamak, M., A direct variable step block multistep method for solving gen- eral third order ODEs. Journal of algor. 2010, DOI.10,1007/s11075-01009413-X
16