A New Block-Predictor Corrector Algorithm for the Solution of y’’’=f(x, y, y’, y’’)

doi:10.4236/ajcm.2012.24047

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American Journal of Computational Mathematics, 2012, 2, 341-344

http://dx.doi.org/10.4236/ajcm.2012.24047 Published Online December 2012 (http://www.SciRP.org/journal/ajcm)

A New Block-Predictor Corrector Algorithm for the

Solution of





,, ,



 

yy y



Adetola O. Adesanya1, Mfon O. Udo2, Adam M. Alkali1

1Department of Mathematics, Modibbo Adama University of Technology, Yola, Nigeria

2Department of Mathematics and Statistics, Cross River University of Technology, Calabar, Nigeria

Email: [email protected], [email protected]

Received May 10, 2012; revised September 16, 2012; accepted September 25, 2012

ABSTRACT

We consider direct solution to third order ordinary differential equations in this paper. Method of collection and inter-

polation of the power series approximant of single variable is considered to derive a linear multistep method (LMM)

with continuous coefficient. Block method was later adopted to generate the independent solution at selected grid points.

The properties of the block viz: order, zero stability and stability region are investigated. Our method was tested on

third order ordinary differential equation and found to give better result when compared with existing methods.

Keywords: Collection; Interpolation; Power Series; Approximant; Linear Multistep; Continuous Coefficient; Block

Method

1. Introduction

This paper considers the general third order initial value

problems of the form



 

00 00

,, ,,,

yfxyyyyx y

yx yyx y

 



 

 (1)

Conventionally, higher order ordinary differential equ-

ations are solved directly by the predictor-corrector me-

thod where separate predictors are developed to im-

plement the correct and Taylor series expansion adopted

to provide the starting values. Predictor-corrector me-

thods are extensively studied by [1-5]. These authors

proposed linear multistep methods with continuous co-

efficient, which have advantage of evaluation at all

points within the grid over the proposed method in [6]

The major setbacks of predictor-corrector method are

extensively discussed by [7].

Lately, many authors have adopted block method to

solve ordinary differential equations because it addresses

some of the setbacks of predictor-corrector method

discussed by [6]. Among these authors are [8-10].

According to [6], the general block formula is given





mn nm

yhyhy



 Y edfbF



(2)

where is e

s vector, is r-vector and is

vector,

d b



is the interpolation points and is the

collection points.

is a k-vector whose entry is thj





njnj nj

ffty







is the order of the differential

equation.

Given a predictor equation in the form



mn n

yh y



df. (3) Ye

Putting (3) in (2) gives







nn n

y hyhy

 

 bFe df

yhYe df



(4)

Equation (4) is called a self starting block-predictor-

corrector method because the prediction equation is

gotten directly from the block formula as claimed by

[11,12].

In this paper, we propose an order six block method

with step length of four using the method proposed by

[11] for the solution of third order ordinary differential

equation.

2. Methodology

2.1. Derivation of the Continuous Coefficient

We consider monomial power series as our basis fun-

ction in the form







x

(5)

The third derivative of (5) gives



 

jjj ax

sr







  (6)

A. O. ADESANYA ET AL.

342

The solution to (1) is soughted on the partition π

012 1nnN within the

integration interval [a,b] with constant step length

given as

axx xxxxb







,01.

ni n

hx xiN



 h

Substituting (6) into (1) gives









 

,, ,

sr j

fxyxyx y x

jjj ax

 





.

;

(7)

Interpolating (5) at collocating (7)

gives a system of equations



,113





1sr



,014



ax y







(8)



 

sr j.

jjj axf

 





 

 (9)

Solving (8) and (9) for

a’s and substituting back into

(5) gives a LMM with continuous coefficients of the

form

 

yh f











nj

(10)

where

a’s and



’s are given as



=;=

=32;

tttttt

ttt











76 542

12 71135 49 26;

10080

t tttttt







7654 2

=421141012652 ;

5040

tttttt



 



765 42

12147105532432 ;

1680

tttttt







765 432

5040

435701758401078452 ;

ttt tttt





  



765 4 3

1221 77 105105 58.

10080

tttttt





where 3.







2.2. Derivation of the Block Method

The general block formula proposed by [6] in the nor-

malized form is given by



 

mn nm

yhy hy

 



 AY edfbF (11)

Evaluating (10) at 1,0,4

xx i

;





,014;



the first and

second derivative at and substituting

into (10) gives the coefficient of (11) as

113331 14312483675314756251292514,

1120 63011201051440 90160 45 720 90 80 45







111100000000

123411110000 ,

12 812341111











012 12

 identity matrix,

107332 1863 21763811764232625164

10083155603158540153604540 45

1038243 3247271698

1114

1680215601052403 801530 1510 15

435245128 398364 5364

221

168 31511210536045 845360454045

47 1981871

10080 63011206348030

b 





9197

30 160720 12045

 

 



 

 

3. Analysis of the Properties of the Block

3.1. Order of the Method

We define a linear operator on the block (11) to give











:mm mm

yx hyhyhy

 



 Ye dfbF(12)

Expanding





xih and





xjh in Taylor

series , (12) gives













 

yxhCyxChyx

ChyxCh yx











(13)

The block (11) and associated linear operator are said

to have order if

p011 2

0, 0

CCC C





 .

A. O. ADESANYA ET AL. 343

The term 2p is called the error constant and implies

that the local truncation error for the block is given by

C







22 3

pp p

p n

tChyxh

 







(14)

Hence the block (11) has order 6, with error

constant



0h

224











zz

91243 321078

320 45 448031510080 315

916313 8

315 16090 160945

C







3.2. Zero Stability of the Block

The block (11) is said to be zero stable if the roots

of the characteristic polynomial

satisfies

1, 2,

z



det





,A1z and the root

1z has multiplicity not exceeding the order of the

differential equation. moreover as



 

0,1 ,











 where



is the order of

the differential equation,



dimrA

For the block (11), 12,3r





 

91z





Hence our method is zero stable.

3.3. Convergence

A method is said to be convergent if it is zero stable and

has order .

1p

From the theorem above, our method is convergent.

4. Numerical Experiments

4.1. Test Problem

We test our schemes with third order initial value pro-

blems:

Problem 1. Consider a special third order initial value

problem

3sinyx















01, 0 0,0201

yy x







Exact solution:



3cos2

yxx 









This problem was solved by [13] using self-starting

predictor-corrector method for special third order diffe-

rential equations where a scheme of order six was pro-

posed.

Problem 2. Consider a linear third order initial value

problem

0yy



 

















01, 01,01,0,1yy yx

 

 

Exact solution:





21 cossin

xx x

This problem was solved by [14] where a method of

order six was proposed. They adopted predictor corrector

method in their implementation. Our result is shown in

Table 1.

4.2. Numerical Results

The following notations are used in the table.

XVAL: Value of the independent variable where nu-

merical value is taken;

ERC: Exact result at XVAL;

NRC: Numerical result of the new result at XVAL;

ERR: Magnitude of error of the new result at XVAL.

5. Discussion

We have proposed a new block method for solving third

order initial value problem in this paper. It should be

noted that the method performs better when the step-size

is chosen within the stability interval. The Tabl e s 1 and 2

had shown our new method is more efficient in terms of

accuracy when compared with the self starting predictor

Table 1. Showing result of pr oble m 1, h = 0.01.

XVAL ERC NRC NRC ERR IN [13]

0.1 0.990012495834077 0.990012495834077 0.0000+00 9.992007(–16)

0.2 0.960199733523725 0.960199733523724 9.99200(–16) 7.660538(–15)

0.3 0.911009467376818 0.911009467376816 1.55431(–15) 2.287059(–14)

0.4 0.843182982008655 0.843182982008652 3.10862(–15) 5.906386(–14)

0.5 0.757747685671118 0.757747685671113 4.66293(–15) 1.153521(–13)

0.6 0.656006844729035 0.656006844729028 6.88338(–15) 1.982858(–13)

0.7 0.539526561853465 0.539526561853456 9.10382(–15) 3.127498(–13)

0.8 0.410120128041496 0.410120128041484 1.14908(–14) 4.635736(–13)

0.9 0.269829904811992 0.269829904811978 1.42108(–14) 6.542544(–13)

1.0 0.120906917604418 0.120906917604401 1.74582(–14) 8.885253(–13)

A. O. ADESANYA ET AL.

344

Table 2. Showing result of pr oble m 2, h = 0.1.

XVAL ERC NRC ERR ERR IN [14]

0.1 0.004987516654767 0.004987518195317 1.54055(–09) 1.189947(–11)

0.2 0.019801063624459 0.019801073469968 9.84550(–09) 3.042207(–09)

0.3 0.043999572204435 0.043999595857285 2.36528(–08) 7.779556(–08)

0.4 0.076867491997406 0.076867535270603 4.32732(–08) 7.749556(–07)

0.5 0.117443317649723 0.117443356667842 3.90181(–08) 3.398961(–06)

0.6 0.164557921035623 0.164557928005710 6.97008(–08) 9.501398(–06)

0.7 0.216881160706204 0.216881108673223 5.20329(–08) 1.756558(–06)

0.8 0.272974910431491 0.272974775207245 1.35224(–07) 2.745889(–05)

0.9 0.331350392754953 0.331349917920840 4.74834(–07) 3.888082(–05)

1.0 0.390527531852589 0.390526462491195 1.06936(–06) 5.137153(–05)

corrector method proposed by [11] and [15]. It should be

noted that this method performs better when the step size

(h) is within the stability interval.

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