FOUR STEPS IMPLICIT METHOD FOR THE SOLUTION OF GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

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FOUR STEPS IMPLICIT METHOD FOR THE

SOLUTION OF GENERAL SECOND ORDER ORDINARY

DIFFERENTIAL EQUATIONS

*1_{OSILAGUN, J. A.,}2_{ADESANYA, A.O.}2_{ANAKE, T. A. AND}2_{OGHOYON, G. J.} 1_{Department of Mathematical Sciences, Olabisi Onabanjo University,}

Ago Iwoye, Ogun State, Nigeria.

2_{Department of Mathematics, Covenant University, Sango Ota, Ogun State, Nigeria.}

*Corresponding author: [email protected] Tel: +2348086049694

plicit block method based on Newton’s

backward divided difference formula.

Adesanya et al. (2009) proposed a two step method for the general solution of second order which is self stating and adopted New-ton’s polynomial to generate the starting value. Awoyemi et al. (2009) recently pro-posed a self starting Numerov’s method. This method solves both initial and bound-ary value problems of ordinbound-ary differential equation. Yahaya (2007) constructed a Nu-merov method from a quadratic continuous polynomial solution. This process led to method which is applied to both initial and boundary value problems.

In this research work, we propose a block method for step length of four. This method adopts Newton’s polynomial approximation to generate the starting value and solves gen-eral second order ordinary differential

equa-ABSTRACT

Four steps implicit scheme for the solution of second order ordinary differential equation was derived through interpolation and collocation method. Newton polynomial approximation method was used to generate the unknown parameters in the corrector. The method was tested with numerical examples and it was found to be efficient in solving second order ordinary differential equations.

Keyword: collocation, interpolation, continuous, implicit, Newton’s polynomial, corrector.

INTRODUCTION

The second order initial value ordinary

dif-ferential equation of the form

,

(1) is considered in this paper. Differential equation is often used to model scientific and technological problems which most times, these equations do not have analytic solution, hence an approximate numerical method is required to solve these problems. Block methods for numerical solution of ordinary differential equations have been proposed by several researchers such as Milne (1953), Shampine and Watt (1969), Worland (1976) and Omar (1999). Rosser (1967) introduced the 3 point implicit block method based on integration formulae which is basically Newton’s cote type. Zanariah et al. (2006) proposed 3 points

im-) , ,

( '

''

y y x f

y  1

' 0, ( )

)

(a  y a  y

Journal of Natural Sciences, Engineer-ing and Technology ISSN - 2277 - 0593

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tion directly.

METHODOLOGY

We consider an approximate solution to (1) in power series

(2.2)

are constants to be determined. We Consider a linear multistep method of the form





 k

j j

j x

a x

y

0

) ( )

( 

1 2 ) 1 ( 0 ,

,  

 j a_j j k

j 



(2)

where , m=the distinct

collo-cation point, t is the interpolation point, for our method, the step length k=4













 



1

0

1

0 2

) ( )

( )

( t

r

m

r

r n r r

n

r x y h x f x

y  

] , [xn xn r

x 

, (3)

, r=0, 1, 2…, m-1 (4)

, r=0, 1, m-1 (5)

To get and , According to Yahaya (2007), he derived a matrix of the form

where: is an identity matrix of dimension .



 

 



1

0 , 1 ( )

) (

m i

i

i r i

r x  p x 

1 ,...., 2 , 1 ,

0 

 m

r



 

 



1

0 ,

1 ( ) ( )

) (

m t

i

i r

i x p x

x 



r n r

n y

x

y(  )  r[0,1,2...,t1] ( ) ,

''

r n f x

y  _

) (x j

 _j(x)

I

DC

I (tm)(tm)

          

          



 

  

  

   



 

.

. .

.

. .

.

. .

2 0

0

. 1

. .

.

. 1

1 1 2

1 1

1 1 2

1 1

1 2

m t

t n t

n t n

i t n n

n

i t n n

n

x x

x

x x

x

x x

x

(3)

Development value for the unknown

Theorem (3.0):

Assuming that and for k=0, 1, n are distinct values, then

, where is a polynomial that can be used to approximate

For Newton’s polynomial

, is the remainder and has the form

(Awoyemi et al.,2009)

Development of four steps method

In developing the method with step length k=4, we consider

D=                             1 , 2 0 , 2 1 , 1 , 0 , 1 , 2 2 0 , 2 2 1 , 2 1 , 2 0 , 2 1 , 1 2 0 , 1 2 1 , 1 1 , 1 0 , 1 . . . . . . . . . . . . . . . . . . . . m m t m t t m t m t m t m t m t h h h h h h c                ] , [ 1 b a c f n

 xk [a,b]

) ( ) ( )

(x y x R x

f   n y(x)

) (x f ) )...( )( ( ... ) )( ( ) ( )

(x a₀ a₁ xx₀ a₂ xx₀ xx₁  a_n xx₀ xx₁ xx_n_₁ y

) ( ) (x y x

f  R_n(x)

) )( )....( )( ( )! ( )

( ₀ ₁ ₁

1 1 n n n n

n x x x x x x x x

x f x

R     _ 

                                                    4 4 3 4 2 4 4 4 3 3 3 2 3 3 4 2 3 2 2 2 2 4 1 3 1 2 1 1 4 3 2 6 3 5 3 4 3 3 3 2 3 3 6 2 5 2 4 2 3 2 2 2 2 30 20 12 6 2 0 0 30 20 12 6 2 0 0 30 20 12 6 2 0 0 30 20 12 6 2 0 0 30 20 12 6 2 0 0 1 1 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x (7) (8) (9) (10)

(4)

This gives a continuous scheme

(11)

where:

Evaluating (11) at i.e. when , gives

(12)

(13)

(14) Evaluating the first derivative of (4,1) at t = -3, gives

h

t 2 18 55 60 21

1440 )

( 6 5 4 3

2

4     



h x x t  n3

4

n

xx_ t1,2,3

) 4

14 204

19 ( 240

480

240 4 3 2 1

2 2 3

4 n n n n n n n

n y y h f f f f f

y _  _  _  _  _  _  _ 

) 24

194 24

( 240

480

240y_n_₃  y_n_₂  y_n_₁ h2 f_n_₄  f_n_₃  f_n_₂  f_n_₁  f_n

) 17 252

402 52

3 ( 240

720

480y_n_₃  y_n_₂  y_n_₁ h2  f_n_₄  f_n_₃  f_n_₂  f_n_₁  f_n

) 475 1908

'

3 2

1 2

4 192 48 64 56 4

45 n n n n n

n

n f f f f hy

h y

y _   _  _  _  

(17)

(18)

(19)

(6)

Developing the unknown for k=4

We evaluate the first derivative of (8), and neglecting and higher values.

Solving (24)-(28) for , i = 1(1)4 gives,

Substituting for in (15) and substituting in (16)-(19) and solving for ,

gives,

6

a

n n

n y y y y y y

hy 12 75 120 220 300 137

60 ' ₅ ₄ ₃ ₂ ₁

 



 _ _ _ _ _

n n

n y y y y y y

hy 3 20 60 120 65 12

60 '_₁  _₅  _₄  _₃  _₂  _₁ 

n n

n y y y y y y

hy 2 15 60 20 30 3

60 ' ₅ ₄ ₃ ₂ ₁

2           



n n

n y y y y y y

hy 3 30 20 60 15 2

60 '_₃  _₅  _₄  _₃ _₂  _₁ 

n n

n y y y y y y

hy 15 65 120 60 20 3

60 '_₄  _₅  _₄  _₃  _₂  _₁

) 2259 247

2382 420

5386 ( 6348

' '

4 '

3 '

2 '

1

1 n n n n n n

n y y y y y

h y

y _   _  _  _  _ 

) 174 12

135 288

705 ( 529

' '

4 '

3 '

2 '

1

2 n n n n n n

n y y y y y

h y

y _   _  _  _  _ 

) 729 105

1578 2520

2598 ( 2116

' '

4 '

3 '

2 '

1

3 n n n n n n

n y y y y y

h y

y _   _  _  _  _ 

) 504 476

2580 1272

2188 ( 1587

' '

4 '

3 '

2 '

1

4 n n n n n n

n y y y y y

h y

y _   _  _  _  _ 

2



n

y f_n__i i1(1)4

) 48 37 48

1 6

1 4

3 6

1 ( 1 4

1 ' '

4 '

3 '

2 '

1

1 n n n n n n

n y y y y y

h f

f _   _  _  _  _ 

(21) (22)

(23)

(24)

(25)

(26)

(27)

(28)

(7)

Substituting (25)-(28) into the second derivative of (8) and comparing with (29)-(32) give

Analysis of the scheme Convergence

The convergence analysis of this scheme is determined using the approach of Fatunla (1992), where each block integrator is represented as a single step block r- point multistep method of the form

where: are r by r matrix respectively with element

specifically, is an identity matrix,

are vectors of numerical estimate describe below. With r-vector

) 48 29 16

3 6

7 4

9 2

3 ( 1 41

1 ' '

4 '

3 '

2 '

1

3 n n n n n n

n y y y y y

h f

f _   _  _  _  _ 

) 3 7 3

7 3

16 6

3 16 (

1 ' '

4 '

3 '

2 '

1

4 n n n n n n

n y y y y y

h f

f            

n n

n y hf

y'_₁  ' 0.222193

n n

n y hf

y'_₂  ' 0.111768

n n

n y hf

y'_₃  ' 1.188485

n n

n y hf

y'_₄  ' 1.311393

(31)

(32)

(33) (34)

(35)

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Awoyemi (2009) for details

Analysis of the block

det

det

Hence the block is zero stable.

Numerical example

We test the efficiency of our scheme on linear and non linear second order differential equation.

Problem 1:

0 1 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

     

 

    

 

   



    

 

    

 

    

 

    

 



0 1 0

0 0

1 0

0

1 0 0

     

 

    

 

   

   

0 ) 1 (

3 _ _

 

1 , 0 , 0 , 0

 

0 ) ( ' 2

''

 x y y

' 1

(0) 1, (0) , 0.1/ 40

2

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S/N _{Expected solution} New method _error

1 1.00125000065104 1.001250000186 4.650D-10

2 1.00250000520835 1.0025000039971 _1.211D-09

3 1.00375001757828 1.003750013474 _4.030D-09

4 1.00500004166729 1.005000047982 _6.314D-09

5 1.00625008138212 1.0062500803586 _8.462D-09

6 1.00750014062974 1.00750002579038 _1.148D-09

7 1.00875022331755 1.008750239662 _1993D-09

8 1.01000033335333 1.0100000783829 _2.550D-09

9 1.01125047464542 1.0112504890376 _4.256D-09

10 1.01250065110271 1.0125006101014 _4.100D-08

Problem 11

x

xe y y''   3

40 / 1 . 0 , 32

5 ) 0 ( , 32

3 ) 0

(  y'   h

y

) 3 exp( 32

3 4 ) (

x x x

y

  

Exact solution

S/N Expected result y(x) New method yn(x) _{Error y(x)-yn(x)}

1 -0.094140915761848 -0.0941409131568 _2.61D-09

2 -0.094532404142338 -0.0945324074753 _3.33D-09

3 -0.094924451608388 -0.0949244224215 _2.92D-08

4 -0.095317044390700 -0.0953170247449 _2.02D-08

5 -0.095710168480980 -0.0957101793552 _1.08D-08

6 -0.096103809629113 -0.0961039982252 _1.88D-07

7 -0.09649533403163 0.0964952920353 _4.23D-08

8 -0.096892584872264 -0.0968923659413 _2.18D-07

9 -0.097289689232184 -0.0972874625827 _2.22D-06

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differential equation, Wiley, New York.

Omar, Z. 1999. Developing parallel block

method for solving higher orders ODES directly. Thesis, University putra Malaysia

Rosser, J.B. 1967. A Runge Kutta for all

season, SIAM, P. 417-452.

Sham pine, L.F., Watts, H. 1969. Block

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of Math-comp 23: 731-740.

Worland, P.B. 1976. Parallel method for the

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Yahaya, Y. A., 2007. A note on the

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