Four-Step Block Method for Solving Third Order Ordinary Differential Equation

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Four-Step Block Method for Solving Third

Order Ordinary Differential Equation

K. O. Lawal #1, Y. A. Yahaya #2, S. D. Yakubu #3

#1, 2, 3

Federal University of Technology, Minna, Niger State, Nigeria

Abstract -In This paper we proposed a Four-Step Block- method of Hybrid Linear Multistep Method derived by collocation and interpolation technique with power series as basis function for direct solution of special third order initial value problem of ordinary differential equation. The basic properties of the block, which include consistency, Zero Stable, order, error constant and region of absolute stability were analysed. From the analysis the block derived was found to be Consistent, Zero stable, convergent and A-stable. Also, the block- method was tested on some numerical examples and the result computed shows the derived block method is more accurate than some existing methods considered in this paper.

Keywords – Zero Stable, Error constant, Region of Absolute stability and Convergence.

I. INTRODUCTION

This paper considered the development of a 4- Step Block Hybrid Linear Multistep method for the direct solution of special third order Initial value problem of ordinary differential equation(ODE) of the form,



,



, ( 0) 0 , ( 0) 0, ( 0) 0

y f x y y x y y x y y x y y

.

(1)

Third order differential equations usually arise in many physical problems such as electromagnetic waves, thin film flow and gravity- driven flows. Therefore, third order ODEs have attracted considerable attention. Many theoretical and numerical studies dealing with this type of equations have been discussed in literature. The popular approach for solving this type of equations is by converting it to a system of first order ODE, and solving it using any suitable method in literature.

Recent research in this area includes [3], a P-stable linear multistep method for general third order initial value problem of ODE, which is an improvement over Nystrom’s method, but the method was implemented in predictor-corrector mode. Like other linear multistep methods they have their draw backs, which includes not being self-starting, secondly, they advance the numerical integration of ODE in one-step at a time. [1] developed a hybrid formula of order four for starting the Numerov method applied to the initial value problem for solving special second order ODE. [10] derived a new multiple fnite difference methods through multistep collocation for special second order ODE. [8] developed a P- Stable block linear multistep method for solving third order ODE using collocation and interpolation technique on power series approximation method. The schemes were applied as a simultaneous integrator, and the derived block method possessed the desirable features of Runge-Kutta methods of being self-starting, that is eliminating the use of predictors, which is an improvement over [3].

This paper therefore proposes development of Block method that will be highly efficient in terms of accuracy, small error constant, possess better rate of convergence , and very easy to implement.

Definition 1: Initial Value Problem

This is any differential equation in which the initial condition of the problem is given, it is given in the form

1

( ) ( , , ... )

k k

y x f x y y y 

,

y x

 

₀ y_{0 ,} y x

 

₀ y₀ ...yk1(x₀)y₀k1

(2)

,

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Definition 2: Block Methods

A block method is formulated in Linear Multistep method form. According to Lambert in [2], block method preserve the advantage of one step methods, of being self- starting and permitting easy change of step length.

According to [2] a general k- block, r-point block method is a matrix of finite difference equation of the form

1 0

k k

m j m j j m j

j j

Y

A y

_

h

B F

_

 









_(2b)

where

Y

_m





y y

_n

,

_n1

,...,

y

_{n r 1}



T,

F

_m





f

_n

,

f

_n1

,...,

f

_{n r 1}



T

A s

_j

'

_and

B s

_j

'

_{are properly}

chosen r × r matrix coefficients and m= 0,1,2…represents the number

n



mr

is the first step number of the mth block and r is the proposed block size.

Definition 3: Consistency [7]

Linear Multistep Methods are said to be consistent if it has order

p



1

.

Definition 4:Convergence [7]

The necessary and sufficient conditions that a Linear Multistep Method be convergent are consistency and zero-stable.

Definition 5: Zero stability [7]

A block method is said to be zero-stable if the roots



_i

,

i



1, 2,3...

s

of the characteristic polynomial

 

 

defined by

 

0

0

s

i s i i

A

 













satisfies



_i 1 and for those roots with



_i 1, the

multiplicity must not exceed the order of the differential equation in consideration.

Definition 6: Region of Absolute Stability

Hairer and Wanner in [9] defined the region of absolute stability as the set of W equals







;

all the roots

 

_i

 

of the characteristics equation satisfying

 

_i

 



1

is called the stability region or region of absolute stability of the method.

Definition 7: A-Stability [7]

An A-stable method is a numerical method in which the stability region contains the entire left half plane.

Definition 7: Order and Error constant

A linear multistep method of the form

3

0 0

k k

j n j j n j

j j

y h f

 _  _

 







(3)

3

k is said to be of order P if

C

₀



C

₁



C

₂

 

...

C

_P



C

_P1



C

_P2



0

but C_p3 0,

3 p

C _ is called the error constant and

C

_p3

h

p3

y

p3

 

x

_n is the principal local truncated error at the point

x

_n .The local truncated error associated with the equation (27) was defined by a linear difference operator L as



( );



0

 

1

 

... q q q

 

...

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Where

q



3, 4,...

were given as

0 0 k

j j

C









1 1 k

j j

C

j









2 2

1 k

j j

C

j









3 3

1 1

1

3!

k k

j j

j j

C

j





 



 







_

 

_{ }





_





 





4 4

1 1

1

4!

k k

j j

j j

C

j



j



 



 







_

 

_{ }





_





 





5 2

5

1 1

1

1

5!

2!

k k

j j

j j

C

j



j



 



 







_

 

_{ }





_





 





(29)





3

1 1

1

1

!

3 !

k k

q q

q j j

j j

C

j

j

q



q





 



 







_

 

_{ }





_







 





II. METHODOLOGY

A power series of a single variable

x

_{in the form:}

 

0 j j j

P x

a x









(4)

is used as the basis or trial function, to produce the approximate solution given as

 

1

0 r s

j j j

y x

a x

 







a

_j



R y

,



C

M

( , )

a b

 

(5)

where s and r are the number of collocation and interpolation points respectively. The derivatives are given as follows

 

1 1

1 r s

j j j

y x

ja x

 











,

(6)

 

1 2

2

(

1)

r s

j j j

y

x

j j

a x

 













, (7)

 

1 3

3

(

1)(

2)

r s

j j j

y

x

j j

j

a x

 















. (8)

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 

1 3

3

,

(

1)(

2)

r s

j j j

f x y

j j

j

a x

 













, (9)

where

a

_j, are the parameters to be determined.

Collocating (9) at both step and off-step points

x



x

_{n p} ,

p



0, ...

v k

, where k is number of steps , v are the off-step points used in the collocation. Then interpolating (5) at both step and off -step points

x



x

_{n q} ,q0, ,u w where u and w are the off-step points used in the interpolation, Then gives a systems of nonlinear equation in the form

AX



U

(10)

which is solved to obtain the values of parameters

a

_j’s j0,1..., (r s 1),that are then substituted into (5), which after some manipulation, yields the new continuous method expressed in the form

 

 

 

3

0

0 ( )

k

n u n u w n w j n j v n v j

y x x y x y _ x y _ h  f _  f _



 

 

      _  _

 









(11)

We express the coefficients (



's and



'

s

) of the continuous scheme as continuous function of t by letting

n u

x

x

t

h







(12)

and noting that

1

dt

dx



h

and

2

2 2

1

d t

dx



h

Derivation of Four Steps Method with four Off- Step Points at Collocation and two Off- Step Points at Interpolation

For this method we considered three interpolation points and nine collocation points, we have

1

3

u



,

2

3

w



,

1 2 4

, ,

3 3 3

v



and

5

3

From (8) for r3and

s



9

we obtained a polynomial of degree

r

 

s

1

as follows;

 

11

0 j j j

y x

a x







(13)

Third derivative is given as

 

11 3

3

(

1)(

2)

_j j

j

y

x

j j

j

a x















(14)

From (14) we obtain

 

11 3

 

3

(

1)(

2)

_j j

,

j

y

x

j j

j

a x



f x y















(15)

Collocating (15) at

x

_{n p} where

0, , ,1, , , 2,3

1 2

4 5

3 3

3 3

p



and interpolating (13) at

x

_{n q} , where

1

0,

3

q



and

2

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2 3 4 5 6

2 3 4 5 6

1 1 1 1 1 1

3 3 3 3 3 3

2 3 4 5 6

2 2 2 2 2 2

3 3 3 3 3 3

2 3

2 3

1 1 1

3 3 3

2 3

2 2 2

3 3 3

2 3

1 1 1

1 1 1

0 0 0 6 24 60 120

0 0 0 6 24 60 120

0 0 0 6 24 60 120

0 0 0 6 24 60 120

0 0 0 6 24

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                      

7 8 9 10 11

7 8 9 10 11

1 1 1 1 1

3 3 3 3 3

2

2 3

4 4 4

3 3 3

2 3

5 5 5

3 3 3

2 3

2 2 2

2 3

3 3 3

2 3

4 4 4

60 120

0 0 0 6 24 60 120

0 0 0 6 24 60 120

0 0 0 6 24 60 120

0 0 0 6 24 60 120

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

                                           

7 8 9 10 11

2 2 2 2

3 3 3 3 3

4 5 6 7 8

4 5 6 7 8

1 1 1 1 1

3 3 3 3 3

4 5 6 7 8

2 2 2 2 2

3 3 3 3 3

4 5 6 7 8

1 1 1 1 1

4 4

3

210 336 504 720 990

210 336 504 720 990

210 336 504 720 990

210 336 504 720 990

210 336

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                    

5 6 7 8

4 4 4 4

3 3 3 3

4 5 6 7 8

5 5 5 5 5

3 3 3 3 3

4 5 6 7 8

2 2 2 2 2

4 5 6 7 8

3 3 3 3 3

4 5 6 7 8

4 4 4 4 4

504 720 990

210 336 504 720 990

210 336 504 720 990

210 336 504 720 990

210 336 504 720 990

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

                          1 0 3 1 2 3 2 3 1 3 4 2 5 3 6 1 7 4 3 8 5 9 3 10 2 11 3 4 n n n n n n n n n n n n y y a a _y a f a f a f a a f a f a f a a f a f f            _{ }       _{ }    _{ }         _{ }    _{ }    _{ }         _{ }    _{ }    _{ }         _{ }    _{ }    _{ }         _{ }    _{ }        _{ }    

(16)

The coefficient matrix can be solved either by inversion technique or Gaussian elimination technique to obtain the following coefficients of the continuous scheme.

Substituting the coefficients into (11) yield the following continuous scheme

 

2 2 2 3 2

1 2

3 3

9 3 9 3 92041 332053

( ) 1 9

2 2 n n 2 2 n 129330432 117573120

y x _ t  t y_   t y _ _ t  t y_{ } h  __ t t

    

4 5 6 7 8 9

11 3299 12563 12029 79 229

648t 64800t 172800t 201600t 2688t 26880t

     

10 11 2 3

17 3 11752063 263 1

12800t 35200t fn 658627200t 15360t 6t 

 

  _ _  

 

4 5 6 7 8 9

1979 633 39291 11403 5211 1431

10560t 35200t 140800t 35200t 28160t 24640t

     

10 11 2 4

1 3

243 243 32 27259 11

25600t 387200st fn 17325t 725760t 28t

 

  _ _  

 

5 6 7 8 9 10

659 1401 8433 891 27 81

2800t 3200t 11200t 1792t 160t 2800t

     

11 2 4 5

2 3

98281 941531 55 1319

26943840 29393280 162 3240

243

123200

t fn t t t t

 _{ }

 _ _    





6 7 8 9 10 11

1

3271 4913 989 181 31 3

8640t 5040t 1344t 672t 640t 880t fn 

      _



2 4 5 6 7

690071 21277 11 513 537 33921

239500800t 967680t 48t 1600t 2560t 44800t 

_     



8 9 10 11 2

4 3

2295 4509 243 243 331 2767

3584t 17920t 5120t 70400t fn 249480t 290304t 

 

    _ _ 

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4 5 6 7 8 9 10

11 1649 489 549 81 297 2349

112t 11200t 6400t 1600t 256t 2240t 89600t

      

11 2 4 5

5 3

243 77291 39041 11 343

123200t fn 269438400t 19595520t 540t 10800t

 

 _ _   

 

6 7 8 9 10 11

2

401 2459 641 71 21 9

28800t 33600t 8960t 2240t 3200t 17600t fn 

      _



2 4 5 6

7333 1009 11 907 11

1616630400t 33592320t 36288t 1814400t 69120t 

 _{    }

 

7 8 9 10 11

3

227 13 1 1 3

201600t 10752t 1680t 7168t 246400t fn 

    _ 



2 4 5 6

3653 47 1 169 17

23710579200t 47029248t 99792t 9979200t 3801600t

 _{    }

 

7 8 9 10 11

4

167 5 13 1 1

4435200t 118272t 591360t 179200t 5420800t fn

 

    _{ }

 

(17)

Evaluating (17) at _{1 4 5 2 3}

3 3

,

,

,

,

n _{n n} n n

x



x

_

x



x

_

x



x

_

x



x

_

x



x

_ and

x



x

_n4gives value of t

to be

2

,1,

4 5 8

, ,

3

3 3 3

and

11

3

.The following discrete schemes were obtained

3

1 1 2 1 2

3 3 3 3

2027 683233 2759

3 3

6889050 39916800 129600

n n _{n n} n _{n n}

y _ y  y _  y _ h _ f  f _  f _



1 4 5 2

3 3

65209 377 61 9803

+

22044960 fn 201600 fn 86400 fn 73483200 fn

   

3 4

1483 247

881798400 fn 489891200 fn 

 _



The order

P



9

and the Error constant

C

₁₂

 

7.054 10



10

16523 1541849 3

3 8 6 1

4 1 2 _{18370800 29937600 3}

3 3 3

y y_n y y h f_n f_n

n n n



      

   _

36971 2

453600 ₃ 1 4 5 2

3 3

20777 11153 31 1109

1837080 2721600 21600 4082400

f

n fn fn fn fn

 _  _  _  _  _

251 83

3 4

73483200fn 808315200fn



 _  _{ }

The order

P



9

and the Error constant

C

₁₂

 

1.4095 10



9 3

5 1 2 1 2

3 3 3 3

3

1763 206191 701

15 10

979776 1995840 3888

n _{n n} n _{n n}

n

y_ y  y_  y _ h _ f  f _  f _



1 4 5 2 3

3 3

2825 1027 139 331 11

45927 fn 40320 fn 54432 fn 612360fn 1632960 fn

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4 13

64665216 fn 

 _



The order

P



9

and the Error constant

C

₁₂

 

2.5892 10



9

3

2 1 2 1 2

3 3 3 3

6635 31279 28877

10 24 15

2204496 181440 90720

n n _{n n} n _{n n}

y_  y  y_  y _ h _ f  f _  f _



1 4 5 2

3 3

161213 15613 247 8203

1102248fn 1102248fn 18144fn 7348320fn

   

3 4

439 13

44089920 fn 484989120 fn 

  _



The order

P



9

and the Error constant

C

₁₂

 

3.7153 10



8

3

3 1 2 1 2

3 3 3 3

104933 106 1199

28 63 36

5248800 275 900

n n _{n n} n _{n n}

y_  y  y _  y _ h _ f  f _  f _



1 4 5 2 3

3 3

21827 65741 3029 249319 11869

131220 fn 43200 fn 5404fn 437400fn 1312200fn

    

4

14131

115473600

f

n





_



The order

P



9

and the Error constant

C

₁₂

 

9.2146 10



7

3

4 1 2 1 2

3 3 3 3

544669 2894681 5111

55 120 66

11022480 1995840 12960

n n _{n n} n _{n n}

y _  y  y _  y_ h _ f  f _  f _



1 4 5 2

3 3

4348657 45349 30197 12162167 1102248 fn 60480fn 30240fn 7348320 fn

   

3 4

21871021 3718123

44089920 fn 484989120fn 

  _



(18)

The order

P



9

and the Error constant

C

₁₂

 

1.1036 10



5

Improving the block by considering additional equations arising from first and second derivative. By letting

 

_{ }

 

0

0

,du x du x

Z x Z

dx  dx 

 

_{ }

2

 

2

0 0

2 , 2

d u x d u x

Z x Z

dx dx

 

 

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Evaluating the first and second derivative of the continuous scheme at

x



x

₀using (11) gives t the

value

1

2



.Thus the following derivative discrete schemes were obtained

3

1 2 1 2

3 3 3 3

9 3 2692147 6174799 761

6

2 2 484989120 146361600 29568

o n _{n n} n _{n n}

hz  y  y_  y _ h _ f  f _  f _



1 4 5 2

3 3

198359 119981 7523 3246301

6928416 fn 5702400fn 798336 fn 1616630400 fn

    

3 4

302161 4481

9699782400fn 4267904256fn 

 _



The order

P



9

and the Error constant

C

₁₂

 

2.0421 10



8

2 3

1 2 1 2

3 3 3 3

2314559 14371477 519971

' 9 18 9

24494400 39916800 1814400

o n _{n n} n _{n n}

h z  y  y_  y_ h _ f  f _  f _



1 4 5 2

3 3

772517 843251 189251 121183

2449440 fn 3628800 fn 1814400 fn 5443200fn

   

3 4

33937 12601

97977600 fn 1077753600 fn 

  _



(20)

The order

P



9

and the Error constant

C

₁₂

 

2.45836 10



4

And imposing a continuity equation at

x



x

_n4

1488170869

2440841041

63

69

₃

66

-

1

4

₂

1

₂

2

_2424945600

_439084800

₃

3

3

hz

y

y

y

h

f

_n

f

_n

n

n

_n

_n











_



_



_

_

_

2 1

3 43 53

1454116087 484548391

46073729 8273785897

2851200 fn 242494560 fn 39916800 fn 19958400 fn

 _  _  _  _

2 3 4

8776922113 12895271941 5215046803

1616630400 fn 9699782400 fn 106697606400 fn

   

  

  

The order

P



9

and the Error constant

C

₁₂

 

7.71145 10



5

4 1 2 1

3 3 3

49841471 60381217

2 _{' 9 18 9} 3

24494400 3628800

n _{n n n}

h z _  y_n y _  y _ h _ f_n f _



2 1 4 5

3 3 3

103616221 273689179 464410381 154335421 3628800

1814400 fn 2449440 fn  fn 1814400 fn

 _  

2 3 4

19670039

235609967

23815499

777600

f

n

97977600

f

n

97977600

f

n









_

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(21)

The order

P



9

and the Error constant

C

₁₂

 

2.45836 10



4

Combining the discrete schemes in (18) and (20) forms the 4-step block method, which will be used simultaneously to generate solutions at both the steps and off-step points. And we explicitly obtain the initial conditions atx_n4,n0, 4,...N4 using computed values of

4 4

( _n ) _n

u x _ y _ ,z x( _n4)z_n4,z x( _n4)z_n_4,over subintervals



x x₀, ₄



,



x x₄, ₈



,…



x_N4,x_N



that do not overlap.

Stability Analysis

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1 3 2 3 1 4 3 5 3 2 3 4

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

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

n n n n n n n n y y y y y y y y                                   _                               7 6 5 4 3 3 2 1

1341139 2637631 11607931 4236559 953311 1100903 225815040 359251200 1454967360 718502400 359251200 193

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

n n n n n n n n y y y y h y y y y                                   _                             73657 191783 9956480 8314099200 640185638400 3203533 130583 1159406 21113 47423 273463 5111 4747 61746300 2806650 22733865 561330 2806650 75779550 90935460 2500725150

1679859 6777 799 18387 20709 10841600 70400 6336 197120



   

  13291 83 463

492800 1478400 591360 97574400 973376 194944 1152512 34964 110464 127616 29888 1588 3087315 1403325 4546773 200475 1403325 7577955 113669325 178623225 168517625 2512375 131479375

316141056 14370048 290993472

 

   

 7535875 1819375 10488875 983375 365875 28740096 14370048 387991296 2327947776 25607425536

13689 783 2494 648 783 13 43 4

16940 3850 3465 1925 3850 330 69300 190575

4074381 59049 5751 531441 59049 2168320 492800 4928 985600 49280

       1 3 2 3 1 4 3 5 3 2 3 4

46899 10359 1863 0 98560 985600 10841600

88128 3456 22016 972 3456 8576 192 1444

21175 1925 3465 385 1925 5775 385 190575

n n n n n n n n f f f f f f f f                                                            _{  }    _                     3 49458977 0 0 0 0 0 0 0

14549673600 1966241 0 0 0 0 0 0 0

113669325 31039 0 0 0 0 0 0 0

739200 8808496 0 0 0 0 0 0 0

113669325 72043375 0 0 0 0 0 0 0

581986944 3134 0 0 0 0 0 0 0

17325 105597 0 0 0 0 0 0 0

246400 3952 0 0 0 0 0 0 0

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The first characteristics polynomial is given as

 

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

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

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

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 det

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

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

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

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

P R R

    

    

    

    

    

    

 _{   } _

    

    

    

    

   

   



          

 _

(23)





7

₁

₀

R

R



 

, which implies

1

0,

2

0,

3

0,

4

0,

5

0,

6

0,

7

0

R



R



R



R



R



R



R



and 8

R



1

Therefore, the derived block method is zero stable since

R



1

is simple, and the magnitude of other roots

R



0

. Also the method is consistent since the method as an order of p1. Hence the proposed method is convergent.

Region of Absolute Stability

The region of Absolute stability of the derived block method was plotted using idea of [6]

0 5 10 15 20 25 30 35 40

-20 -15 -10 -5 0 5 10 15 20

Re(Z)

Im

(z

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Fig. 1 Region of Absolute Stability for Case k = 4 with four Off- Step Points at Collocation and two Off- Step Points at Interpolation

From definition 7 the block is A-stable.

Numerical Experiment Problem 1

x

y

 

e

(Non-Stiff)

(0) 3 , (0) 1 , (0) 5

y  y  y 

Exact solution is given as

y x

( ) 2 2

 

x

2



e

x Source:[11]

Problem 2

y

  

y

(Stiff problem.)

(0) 1 , (0) 1 , (0) 1

y  y   y  , 0 x 1 ,h0.1

Exact solution is given as

y x

( )



e

x Source: [8]

Problem 3

3cos( )

y  x (Non -Stiff problem)

(0) 1 , (0) 0 , (0) 2 ,

0.05

y



y





y





h



Exact solution is given as

y x

( )

  

x

2

3

x

3sin( ) 1

x



Source:[11

TABLE I

Result of problem 1 from Four Steps with four Off- Step Points at Collocation and two Off- Step Points at

Interpolation method

x Exact solution Computed results

P=9

Error in Derived Method

Error in [4] p=9

0.1 3.12517091807565 3.12517091807565 0.0000E+00 0.000E+00

0.2 3.30140275816017 3.30140275816017 0.0000E+00 2.8422E−13

0.3 3.52985880757600 3.52985880757600 0.0000E+00 1.6729E−12

0.4 3.81182469764127 3.81182469764127 0.0000E+00 2.9983E−11

0.5 4.14872127070013 4.14872127070013 0.0000E+00 3.1673E−11

0.6 4.54211880039051 4.54211880039051 0.0000E+00 9.1899E−11

0.7 4.99375270747048 4.99375270747048 0.0000E+00 8.9531E−11

0.8 5.50554092849247 5.50554092849247 0.0000E+00 1.9168E−10

0.9 6.07960311115695 6.07960311115695 0.0000E+00 2.1110E−10

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TABLE II

Result of problem 2 from four steps with four Off- Step Points at Collocation and two Off- Step Points at

Interpolation method

x Exact solution Computed results

P=9

Error in Derived method

Error in [4] P=9

1 0.90483741803595 0.90483741803596 1.0000E-14 0.0000E+00

0.2 0.81873075307798 0.81873075307798 0.0000E+00 2.7756E−14

0.3 0.74081822068172 0.74081822068172 0.0000E+00 1.5838E−12

0.4 0.67032004603564 0.67032004603564 0.0000E+00 2.7879E−11

0.5 0.60653065971263 0.60653065971263 0.0000E+00 2.9477E−11

0.6 0.54881163609403 0.54881163609403 0.0000E+00 8.5048E−11

0.7 0.49658530379141 0.49658530379141 0.0000E+00 8.0357E−11

0.8 0.44932896411722 0.44932896411722 0.0000E+00 1.6601E−10

0.9 0.40656965974060 0.40656965974060 0.0000E+00 1.1176E−10

1.0 0.36787944117144 0.36787944117145 1.0000E-14 1.4871E−10

TABLE III

Result of problem 3 from four steps with four Off- Step Poins at Collocation and two Off- Step Points at

Interpolation method

x Exact solution Computed result

P=9

Error in Derived method

0.05 1.00256249218797 1.00256249218797 0.0000E+00

0.1 1.01049975005952 1.01049975005952 0.0000E+00

0.15 1.02418560257920 1.02418560257920 0.0000E+00

0.20 1.04399200761482 1.04399200761482 0.0000E+00

0.25 1.07028812223643 1.07028812223643 0.0000E+00

.30 1.10343938001598 1.10343938001598 0.0000E+00

0.35 1.14380657763365 1.14380657763365 0.0000E+00

0.40 1.19174497307405 1.19174497307405 0.0000E+00

0.45 1.24760339766631 1.24760339766631 0.0000E+00

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III. CONCLUSION

The introduction off-step points at both collocation and interpolation in developing block methods results in a better approximation and higher order, thus enabling the schemes to bypass the Dalhquist Barrier theorem.

REFERENCES

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[2] Anake , T.A.(2011). Continuous Implicit Hybrid One-Step Methods for the solution of initial value problems of general second order ordinary differential equations. Ph.d thesis submitted to school of postgraduate studies, Covenant University ,Ota. Available: http://thesis.convenantuniversity.edu.ng/bitstream/handle/123456789/198/CUGpo60192-anake

[3] Awoyemi, D. O. (2003). A P-stable Linear Multistep method for solving general third order ordinary differential equations.

International Journal of Computermathematics ,vol. 80, pp. 987- 993, Available: 10.1080/0020716031000079572

[4] Awoyemi, D. O., Kayode, S. J. and Adoghe, L. O. (2014). A Five Step P-stable method for the numerical integration of third order ordinary differential equations. American Journal of Computational mathematics, 2014, pp.119-126. Available: 10.4236/ajcm2014.4301

[5] Badmus, A. M. and Yahaya, Y.A.(2014). A new Algorithm of obtaining order and error constants of third order Linear Multistep Method (LMM). Astan Journal of Fuzzy andApplied Mathematics, vol. 2, pp. 190-194.Available:

http://www.adjournline.com/index. journal=AJFAM& page=article&op=view&path[]=1855

[6] Chollom, J.P., Ndam, J.N. and Kumleng, G. M., (2007). On some properties of Block Linear Multistep Method. Science world Journal, vol. 2, pp. 11-17. Available: http://www.scienceworldjournal.org/article/view/2262

[7] Lambert, J.D. (1973). Computation methods in ordinary differential equations. New York John wiley and sons, 224. [8] Olabode, B.T. and Yusuph, Y. (2009). A new Block method for special third order ordinary differential equations. Journal of

mathematics and Statistics, vol. 5, pp. 167-170. Available:http://docsdrive.com/pdfs/science publication/jmssp/2009/167-170.pdf [9] Olabode, B. T.and Omole, E. O. (2015). Implicit Hybrid Block Numerov- type method for the direct solution of fourth-order ordinary

differential equations. American Journal of Computational and Applied Mathematics, vol. 5, pp. 129-139, Available: 10.5923/j.ajcam 20150505 01

[10] Yusuph Y. and Onumanyi P. (2015).New Multiple FDM through collocation for.

y

 

f x y

( , )

. Preeceeding of Conference organize by National Mathematical Center Abuja. Abacus, vol. 29, no. 2, pp. 92-100.

[11]Obarhua, F. O., & Kayode , S. J. (2016). Symmetric hybrid Linear Multistep method for General third order differential equations. Open Access Library Journal, 3, 1-8, Available: doi:10.4236/oalib.1102583