# Starting Two Steps-four Off Steps Method Accurately for the Solution of Second Order Initial Value Problems

DOI : 10.17577/IJERTV1IS5489

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#### Starting Two Steps-four Off Steps Method Accurately for the Solution of Second Order Initial Value Problems

Starting two steps-four off steps method accurately for the solution of second order initial value problems

lAdesanya, A. Olaide, lOdekunle, M. Remilekun, 2Udoh, M. Mfon

2Department of Mathematics and Computer Science, Cross River State

University of Technology, Cross River State, Nigeria

Abstract

We derive an order five hybrid method through collocation of the differential- system and interpolation of the approximate solution which is implemented in predictor- corrector mode. Continuous block method was used to generate the independent solution which served as predictor. The efficiency of our method was tested on some second order initial value problem and was found to give better approximation than the existing methods.

Keywords hybrid method, collocation, differential system, interpolation, approximate solution, continuous block method, independent solution, predictor- corrector mode

A.M.S Subject Classification 65L05, 65L06, 65D30

1. Introduction

This paper considers the approximate solution to second order initial value problem of the form

n

Y11 = f(x, Y, Y1), Yk(xO) = Yk, k = 0, 1 (1)

1

where f is continuous and differentiable within the interval of integration.

Convectionally, higher order ordinary differential equation are solved by method of reduction to syatem of first order ordinary differential equation. This method is extensively discussed by Awoyemi and Kayode[5], Adesanya, Anake and Udoh [4], Kayode and Awoyemi [14], Jator [11], Awoyemi and Idowu [7] to mention few. These authors suggested that the direct method for solving higher order ordinary differential equation are more efficient since the method of reduction increased the dimension of the resulting system fo first order ordinary differential equation; hence it waste alot of computer and human effort.

Many scholars have proposed method implemented in predictor corrector mode for the direct solution of (1), among them are Awoyemi [6], Kayode [13], Olabode [16], Adesanya, Anake and Oghoyon [3], Kayode and Adeyeye [12]. These authors proposed an implicit multistep method in which seperate predictors are needed to implement the corrector. The major setback of this method is that the predictors are reducing order of accuracy, therefore it has an great effect on the accuracy of the method.

Scholars later proposed block method to cater for some setbacks of the predictor- corrector method. Block method has the properties for Runge kutta method of being self starting and does not require developing seperate predictors and eval- uate fewer function per step. Among scholars that proposed block method are Jator [10], Jator and Li [9], Awoyemi et al. [8], Adesanya et al. [1], Majid,Azmi and Suleiman [15], Adesanya et al. [2], Omar et al. [17],Siamak [19], Omar and Suleiman [18]. It was observwd that in block method, the number of interpolation points must be equal to the order of the differential equation, hence this method does not exhaust all possible interpolation points therefore a method of lower order

2

is developed.

In this paper, we developed a method which is implemented in predictor- corrector method but the predictors are constant order predictors hence cater for the setbacks of the convectional predictor- corrector method. The predictors are developed adopting block method hence these methods combine the properties of both predictor-corrector method and block method, therefore address some of the setbacks of the method.

2. 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)

The second derivatives of (2) gives

Y11 =

r+s-l

j=2

j(j – 1)ajxj-l (3)

Substituting (3) into (1) gives

f(x, Y, Y1) =

r+s-l

j=2

j(j – 1)ajxj-l (4)

3

3

3

where r and s are the number of interpolation and collocation points respec- tively. Interpolating (2) at xn+r, r = 0 (2 ) 4 and collocating (4) at xn+s, s = 0 (2 ) 2,

3

gives a non linear system of equation

AX = U (5)

where

1 xn x2 3 4 5 6

n

x

x

x

x

I

n

n

n n

1 xn+ 2

3

1 x 4

x2 2

3

n+ 3

x3 2

3

n+ 3

x4 2

n+ 3

x5 2

n+ 3

x6 2 I

3

n+ 3

n+ n+

I

1 x 4

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

n+

I

I

I

n+ 3

I n+

x2 4

x3 4

x4 4

x5 4

x6 4

X = 2

3

I 0 0 2 6xn 12x

20×3

n+ 3

30×4

n+ 3

n

I

3

n+ 3

3

n+ 3

0 0 2 6xn+2 12×2

I 0 0 2 6xn+ 2 12×2 2

n

20×3 2

n+ 3

n

I

30×4 2

I

n+ 3

0 0 2 6xn+ 4 12×2 4

20×3 4

30×4 4

n+2

20x

3

n+2

30x

4

n+2

A = aO al a2 a3 a4 a5 a6

U =

Yn Yn+ 2

Yn+ 4

fn fn+ 2

fn+ 4

fn+2

3

3

3

3

solving (5) for a1j s and substituting into (2) gives a continuous hybrid linear multistep method in the form

Where

2

Y(x) = o:O + o: 2 Yn+ 2 + o: 4 Yn+ 4 + h

2

/32j fn+2j + /3 2 fn+ 2 + /3 4 fn+ 4 \

(6)

3 3 3 3

j=O

3 3 3 3

o:O

= -1 (243t6 – 1458t5 + 2970t4 – 2160t3 + 432t – 64)

64

4

o: 2

3

o: 4

3

= 1 32

(

(

= 1 64

243t6

243t6

– 1458t

5

5

– 1458t

+ 2970t4

+ 2970t4

– 2160t

– 2160t

+ 384t)

3

3

( )- –

+ 336t)

/3O

= 1 1215t6 7533t5 + 16470t4 14760t3 + 4320t2 + 128t 8640

/3 2

3

/3 4

3

= 1

(

960

( )- –

=

960

1 (

1350t6

135t

– 891t

6

– 8019t

5

+ 2010t

5

+ 16050t4

– 1560t

4

– 11280t

3

+ 1664t)

3

+ 256t

)

/32

= 1 243t5 810t4 + 720t3 128"t 8640

h

t = x-xn , Yn+j = Y(xn + jh), fn+j = f (xn, Y (xn + jh) , Y1 (xn + jh))

evaluating (6) at t = 2 gives a discrete scheme

(

3

3

p Yn+2 = 3Yn+ 4 – 3Yn+ 2 + Yn + 27

fn+2 + 9fn+ 2 – 9fn+ 2 – fn

(7)

3

3

equation (7) is our corrector

2.2 Development of the predictor

3

Interpolating (2) at xn+r, r = 0, 1 and collocating (4) at xn+s, s = 0 (l ) 2 gives

equation (5) where

A = aO al a2 a3 a4 a5 a6 a7 a8

3

3

3

3

U =

Yn Yn+l fn fn+ l

fn+ 2

fn+l fn+ 4

fn+ 5

fn+2

U =

Yn Yn+l fn fn+ l

fn+ 2

fn+l fn+ 4

fn+ 5

fn+2

5

1 xn x2 x3 x4

x5 x6 x7 x8

n n n

n+l

n+l

n+l

I

 0 2 0 2 0 2 0 2 0 2 0 2 0 2

I 1 xn+l x2 x3 x4

0

n n n n

n+l

n+l

n+l

n+l

x5 x6 x7 x8

n

n

n

n

n

I

I

I

20×3

30×4

42×5

56×6

0

6xn 12×2

20×3

30×4

42×5

56×6

2

I 0 6xn+ l 12x l

3

20×3 l 30×4 l 42×5 l 56×6 l

3

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

I 0

I

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

n+ 3

I

I

I

n+ 3

X = 0

6xn+ 2

12×2 2

n++l

20×3 2

20x

3

n++l

30×4 2

30x

4

n++l

42×5 2

42x

5

n+l

56×6 2

56x

6

n++l

6xn+l 12×2

20x

3

n++l

30x

4

n++l

42x

5

n+l

56x

6

n++l

0

I

6xn+ 4 12×2 4 20×3 4

3

30×4 4

n+ 3

42×5 4

n+ 3

56×6 4 I

3

n+ 3

n+ n+

I

0

n+2

3

n+ 3

n+ 3

0

3

3

I

I

n+ 3

n+ 3

n+ 3

n+ 3

n+

I

I

n+ 3

6xn+ 5 12×2 5 20×3 5

30×4 5

42×5 5

56×6 5

6xn+2 12×2

20x

3

n+2

30x

4

n+2

42x

5

n+2

56x

6

n+2

solving for a1j s using Guassian elimination method and substituting into (2) givesa continuous hybrid linear multistep method in the form

l

Y(x) =

o:jYn+j+p

2

/3j fn+j + /3 l fn+ l + /3 2 fn+ 2 + /3 4 fn+ 4 + /3 5 fn+ 5 \

(8)

j=O

j=O

3 3 3 3 3 3 3 3

h

where t = x-xn , Yn+j = Y(xn + jh), fn+j = f (xn, Y (xn + jh) , Y1 (xn + jh))

o:O = 1 – t o:l = t

1 /I 243t8 – 2268t7 + 8820t6 – 18522t5 + 22736t4 – 16464t3 \I

/3O =

13440

-1 ( 8

+6720t2

7 6

– 1265t

5 4 3 )

/3 l 3

/3 2

3

=

2240

=

4480

1215t

1 (

243t

– 10260t

8

– 2160t

7

+ 7872t

+ 14616t

+ 34524t

– 58086t

6

+ 14616t

+ 49140t

5

– 6720t

– 16800t

4

+ 267t

+ 267t

3 )

/3l

= -1 (1215t8 – 9720t7 + 30492t6 – 46872t5 + 35560t4 – 11200t3 + 525t)

3360

6

/3 4

3

= 1

(

7

4480

1215t8 – 9180t

+ 26964t6 – 38682t

+ 27320t4 – 8400t

+ 363t)

/3 5

3

= -1

7

5

3

( )- – –

2240

243t8 – 1728t

– 4788t – 6552t

+ 2536t4 – 1344t

+ 57t)

(

6

5

3

/32

= 1 243t8 1620t7 + 4284t6 5670t5 + 3836t4 1120t3 + 47t 13440

3

Solving for the independent solution yn+s, s = 1 (l ) 2, gives a continuous hybrid

block formula

l

(jh)m

(m)

2 \

2

Y(x) =

Yn +h

\Jjfn+j + \J l fn+ l + \J 2 fn+ 2 + \J 4 fn+ 4 + \J 5 fn+ 5

m!

j=O

j=O

3 3 3 3

3 3 3

3

(9)

where

\JO

= 1 243t8 2268t7 + 8820t6 18522t5 + 22736t4 16464t3 + 6720t2 13440

\J l

3

\J 2

3

( )- – –

= -1

(

2240

=

4480

1215t

1 (

243t8

– 2160t

7

– 10260t

8

+ 7872t6

+ 34524t

7

+ 14616t5

– 58086t

6

+ 14616t4

5

– 6720t3)

+ 49140t

– 16800t

4 3)

(

7

5

\Jl

= -1 (1215t8 – 9720t7 + 30492t6 – 46872t5 + 35560t4 – 11200t3)

\J 4

3

\J 5

3

3360

= 1

4480

( )- – –

=

2240

-1 (

1215t8

243t

– 1728t

8

– 9180t

7

+ 26964t6

– 4788t

– 6552t

6

– 38682t

+ 2536t

5

+ 27320t4

4

– 8400t3)

– 1344t

3)

\J2

= 1 243t8 1620t7 + 4284t6 5670t5 + 3836t4 1120t3 13440

7

3

evaluating (9) at t = 0 (l ) 2, gives a discrete block formula in the form

m

A(O)Ym = eYn + pdf (Yn) + pbF(Y ) (10)

where A(O) = 6 x 6 identity matrix

Ym =

Yn+ l Yn+ 2 Yn+l Yn+ 4 Yn+ 5 Yn+2

3

3

3

3

f (Yn) =

Yn-l Yn-2 Yn-3 Yn-4 Yn-5 Yn

3

3

3

3

F(Ym) =

 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

I

e = I

I

fn+ l

fn+ 2

fn+l fn+ 4

I

I

I

fn+ 5

fn+2

d =

28549

lO8864O

lO27

l7OlO

253

2688

lO88

85O5

35225

2l7728

4l

2lO

8

I

b = I

I

275 5l84

57l7 l2O96O

lO62l 272l6O

77O3

36288O

4O3 6O48O

l99 2l7728

l94 945

8

8l

788 85O5

97 l89O

46 2835

l9 85O5

l65 448

267

448O

5 32

363

448O

57 224O

47 l344O

l5O4 2835

8

945

2624

85O5

8

8l

32 945

8

l7Ol

8375 l2O96

3l25 72576

3l25 72576

625

24l92

275 5l84

l375 2l7728

6

7

I

3 35

68 lO5

I

3 7O

6

35 I

0

3

3

Substituting (10) into the first derivative of (9) at t = l (l ) 2 gives

3

( )

3

( )

3

( )

Y1

= Y1

+ h

l8l44O

756O

3

6O48O

3

2835

I ( l9O87 ) fn + (27l3 ) fn+ l – (l5487 ) fn+ 2 + ( 586 ) fn+l I

n+ l

n

6737

6O48O

fn+ 4 +

263

756O

fn+ 5 –

863

l8l44O

fn+2

n+ l

n

6737

6O48O

fn+ 4 +

263

756O

fn+ 5 –

863

l8l44O

fn+2

3

( )

3

( )

3

( )

Y1

= Y1

+ h

ll34O

l89

3

378O

3

2835

I ( ll39 ) fn + ( 94 ) fn+ l + ( ll ) fn+ 2 + ( 332 ) fn+l I

n+ 2

n

269

378O

fn+ 4 +

22

945

fn+ 5 –

37

ll34O

fn+2

n+ 2

n

269

378O

fn+ 4 +

22

945

fn+ 5 –

37

ll34O

fn+2

( )

3

( )

3

( )

Y1

= Y1

+ h

l344

56

3

224O

3

lO5

I ( l37 ) fn + (27 ) fn+ l + ( 387 ) fn+ 2 + ( 34 ) fn+l I

n+l

n

243 224O

fn+ 4 +

9 28O

fn+ 5 –

29 672O

fn+2

n+l

n

243 224O

fn+ 4 +

9 28O

fn+ 5 –

29 672O

fn+2

3

( )

3

( )

3

( )

Y1

= Y1

+ h

2835

945

3

945

3

2835

I ( 286 ) fn + (464 ) fn+ l + (l28 ) fn+ 2 + (l5O4 ) fn+l I

n+ 4

n

+

58

945

fn+ 4 +

l6

945

fn+ 5 –

8

2835

fn+2

n+ 4

n

+

58

945

fn+ 4 +

l6

945

fn+ 5 –

8

2835

fn+2

( )

( )

( )

Yn1 + 5

= Yn1

+ h

36288

+

l5l2

fn+ 4 +

3

235

l5l2

l2O96

fn+ 5 –

3

275

36285

567

fn+2

I ( 37l5 ) fn + ( 725 ) fn+ l + ( 2l25 ) fn+ 2 + (25O ) fn+l I

3

3875

l2O96

3

3875

l2O96

3

3

( )

( )

( )

Yn1 +2

= Yn1

+ h

42O

35

fn+ 4 +

3

l8 35

l4O

fn+ 5 +

3

4l 42O

lO5

fn+2

I ( 4l ) fn + (l8 ) fn+ l + ( 9 ) fn+ 2 + ( 68 ) fn+l I

+

9 l4O

+

9 l4O

3

3

9

3. Analysis of the basic properties of the block

1. Order of the method

We defined a linear operator on (6) to give

2 \l

2

Â£yY(x) h = Y(x)- o:O + o: 2 Yn+ 2 + o: 4 Yn+ 4 + h

/32j fn+2j + /3 2 fn+ 2 + /3 4 fn+ 4

3 3 3 3

j=O

3 3 3 3

(11)

Expanding Yn+j and fn+j in Taylor series and comparing the coefficient of h

gives

Â£yY(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 Â£ and the associated continuous linear multistep method

(6) 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)

328O5

The order of our discrete scheme is 5, with error constant Cp+2 = -8

2. Consistency

A linear multistep method (6) 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.

10

For our method,

3

p(r) = r2 + 3r 4 – 3r 2 + 1

3

27

3

3

and o-(r) = l (r2 + 9r 4 – 9r 2 – 1 .

Clearly p(1) = p1(1) = 0 and p11(1) = 2!o-(1).

Hence our method is consistent

3. 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

Hence our method is not zero stable

4. Stability region

8y.

The method (13) 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 (6) and writing r = cos 0 + i sin 0 gives the stability region as shown in fig (1)

11

y

3

2

1

9

8

7

6

5

4

3

2

1

x

1

2

3

fig(1)

4. Numerical Experiments

1. 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 )

This problem was solved by Awoyemi [5] where a method of order 8 is proposed and it is implemented in predictor-corrector mode with h = 1/320. Jator [10] also solved this problem in block method where a block of order 6 and step-length of 5 is proposed with h = 0.05.We compared our result with these two results as shown in table 1. Though the result of Awoyemi [5] was not shown in this paper but

12

Jator [10] has shown that there method is better.

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 .

Exact solution (sin x)2

This problem was solved by Awoyemi [5] where a method of order 8 is proposed and it is implemented in predictor-corrector mode with h = 1/320. Jator [10] also solved this problem in block method where a block of order 6 and step-length of 5 is proposed with h = 0.049213. We compared our result with these two results as shown in table 2.Though the result of Awoyemi [5] was not shown in this paper

but Jator [10] has shown that there method is better.

I I

Error= Exact result-Computed result

Table 1 for problem 1

x Exact result Computed result Error Error in [10] 0.1 1.050041729278914 1.0500417292784907 6.6613(-16) 7.1629(-12)

0.2 1.1003353477310756 1.1003353477310676 7.9936(-15) 1.5091(-11)

0.3 1.1511404359364668 1.1511404359364292 3.7525(-14) 4.5286(-11)

0.4 1.2027325540540821 1.2027325540539671 1.1501(-13) 1.0808(-10)

0.5 1.2554128118829952 1.2554128118826986 2.9665(-13) 1.7818(-10)

0.6 1.3095196042031119 1.3095196042024151 6.9677(-13) 4.4434(-10)

0.7 1.3654437542713964 1.3654437542698326 1.5638(-12) 7.4446(-10)

0.8 1.4236489301936022 1.4236489301901405 3.4616(-12) 1.5009(-09)

0.9 1.4847002785940522 1.4847002785863199 7.7322(-12) 3.7579(-09)

1.0 1.549306144340554 1.5493061443162750 1.7780(-11) 4.7410(-09)

13

Table 2 for problem 2

 x Exact result Computed result Error Error in [10] 1.1069 0.7998266847638 0.799826684754 9.7840(-12) 2.8047(-10) 1.2069 0.8733436578646 0.873343657851 1.3090(-11) 2.7950(-10) 1.3069 0.9319765974804 0.931976597464 1.5780(-11) 2.1490(-10) 1.4069 0.9733879933357 0.972338799331 1.7581(-11) 5.4975(-11) 1.5069 0.9959269037589 0.995926903740 1.8360(-11) 1.1545(-10) 1.6069 0.9986947735170 0.998694773498 1.8103(-11) 4.4825(-10) 1.7069 0.9815812563774 0.981581256055 1.6880(-11) 7.7969(-10) 1.8069 0.9452681426358 0.945268614248 1.4862(-11) 1.1840(-09) 1.9069 0.8912045176254 0.891204517613 1.2232(-11) 1.6318(-09) 2.0069 0.8215443313867 0.821544331377 9.2473(-12) 2.0567(-09)
5. Conclusion

We have proposed a two-steps four off steps method of order five in this paper. Continuous block method which has the properties of evaluation at all points within the interval of integration is adopted to give the independent solution at non overlapping intervals as the predictor. 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 performed better than the existing method

1. block method and the predictor corrector method as shown in the numerical examples.

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14

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