On Some Properties Of The Block Linear Multi-step Methods

(1)

Page

11

FULL LENGTH RESEARCH ARTICLE

The block methods together with their hybrid forms were studied (Onumanyi, et al. 2001) and further investigated (Chollom & Onumanyi, 2004; Dauda, et al. 2005) and found to be useful for the direct solution of initial and boundary value problems.

The advantages of these methods include (i) overcoming the issue of overlap of pieces of solutions usually associated with the multistep finite difference methods and (ii) they are self starting thus eliminating the use of other methods to obtain starting solutions. Before now, the determination of the order of these block methods, their convergence and plotting their absolute stability regions have been done for the single members of the block and whose result may not be assumed for the entire block. In this paper, we consider some of the properties of the entire block.

Convergence of the block linear multistep method: In the discreet case, a linear multistep method is said to be convergent if for all initial value problem (1.0)

y



x



-y n



x



→

0

as

h

ON SOME PROPERTIES OF THE BLOCK LINEAR MULTI-STEP METHODS

*Chollom, J. P.; Ndam, J. N. & Kumleng, G. M.

Department of Mathematics University of Jos, Nigeria

*(Corresponding author)

[email protected]

ABSTRACT

The convergence, stability and order of Block linear Multistep methods have been determined in the past based on individual members of the block. In this paper, methods are proposed to examine the properties of the entire block. Some Block Linear Multistep methods have been considered, their convergence, stability and order have been determined using these approaches.

Keywords: Block Linear multistep method, Region of absolute stability,

Convergence

INTRODUCTION

Continuous integration algorithms for the numerical solution of the initial value problem (IVP) of the form

y ′

( )

x

=

f x y

( , ),

a

<

x

<

b y a

, ( )

=

y

0 …1.0 has been the area of focus in recent times. The block linear multistep method was developed by Onumanyi, et al. (1994), Onumanyi, Sirisena & Jatau (1999) through continuous interpolants based on the work of Lie & Norsett (1986).

0,

x

∈



x

0,

x



.

→

Previously, the convergence of these methods is done considering only the individual members of the block. This paper determines the convergence of the entire block using the Fatunla (1994) approach where each block integrator is represented as a single step block, r-point multi-step method of the form:

1 1

2 2

1 1

.

,

.

,

.

,

.

n n r

n n

m m m m

n n

n r n _{n r} _n

f

y

f

Y

F

y

f

y

f

+ −

+ +

− −

+ ₊

⎛

⎞

⎛

⎞

⎛

⎞

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

=

⎜

⎟

=

⎜

⎟

=

⎜

⎟

=

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎛

⎞

⎜

⎝

⎠

⎜

⎝

⎟

⎠

⎜

⎝

⎠

⎟⎟⎟

⎟

…3.0

⎝

⎠

Definition (1.0) for

n



mr

, for some integer

m

the block method (2.0) is said to be zero stable if the roots

0

≥

R  det

∑

i0

k

A∗i_Rk−i _₀

,

1(1

j

R J

=

)

k

of the first characteristic polynomial satisfies

|

R

j

|

≤

1

and for the roots with

|

the multiplicity must not

exceed two

ORDER OF THE BLOCK LINEAR MULTI-STEP METHOD: The linear multi-step method is said to be of order

p

if

. This approach is normally used to determine the order of the individual members of the block. We extend this approach to determine the order of the entire block. To achieve this, the block linear multi-step method is expressed in the form

0 1 2 ... p 0, p 1 0

C =C =C = C = C + ≠

0 0

k k

ij n j h ij n j

i i

y

α

+ =

β

+

= =

∑

f

where

j



0, 1, . . .

k

_{is a positive integer. Equation (5) is} expanded to give the following system of equation.

R

j

|

≤

1

…4.0

…5.0

(2)

⎟⎟⎟_⎟ …6.0

0 01 11 21 1

02 12 22 2 1 03 13 23 3 2

01 1 2

. . . . . . . . .

. . . .

. . .

k n

k k kk n k

y y y

h

y β

α α α α

+

⎛ ⎞⎛_⎟ ⎞_⎟

⎜ _⎟⎜ _⎟

⎜ ⎟⎜ ⎟

⎜ ⎟_⎟⎜ ⎟_⎟

⎜ _⎟⎜ _⎟

⎜ _⎟⎜ _⎟₌

⎜ ⎟⎜ ⎟

⎜ ⎟_⎟⎜ ⎟_⎟

⎜ _⎟⎜ _⎟

⎜ ⎟_⎟⎜ ⎟_⎟

⎜ ⎜

⎝ ⎠⎝ ⎠

1 11 21 1 01 12 22 1 1 01 13 23 1 2

01 1 2

. . . . . . . . . . . . . . . . . . . . .

. . .

k n

k k kk n k

f f f

f

β β β

β β β β

+

⎛ _⎞⎟ _⎟

⎜ _⎟⎜ _⎟

⎜ ⎟⎜ ⎟

⎜ ⎟⎜

⎛ ⎞

⎜ _⎟⎟⎜

⎜ _⎟⎜

⎜ _⎟⎜ _⎟

⎜ ⎟⎜ ⎟

⎜ ⎟_⎟⎜ ⎟_⎟

⎜ _⎟⎜ _⎟

⎜ ⎟⎜ ⎟

⎜ _⎟⎟⎜

⎜ ⎜

⎝ ⎠⎝ ⎠⎟⎟

x y x

′

=

.

pp

c

C

c

⎛

⎞

⎟

⎛

⎞

⎟

⎛

⎞

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

=

⎜

⎟

=

⎜

⎟

=

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎟

⎜

⎝

⎠

⎝

⎠

⎝

⎠

JJJG JJJG JJJJJG

⎟⎟⎟

⎟

The expression (6.0) is equivalent to

0 0

k k

ij n j h ij n j

i i

y

f

α

+ =

β

+

= =

∑

0 1

1

01 11

02 12 2

03 13 3

. , . ,... .

. . .

0 1

k

k k k

k k kk

α α α

⎛ ⎞

⎛ ⎞_⎟ ⎛ ⎞_⎟ _⎜ _⎟

⎜ _⎟ ⎜ _⎟ _⎜ _⎟

⎜ ⎟ ⎜ ⎟ _⎜ ⎟

⎜ ⎟ ⎜ ⎟ _⎜ ⎟_⎟

⎜ ⎟_⎟ ⎜ ⎟_⎟ _⎜

⎜ _⎟ ⎜ _⎟ _⎜

⎜ _⎟ ⎜ _⎟

=⎜_⎜ ⎟_⎟ =⎜_⎜ ⎟_⎟ =⎜_⎜ ⎜

⎜ ⎟_⎟ ⎜ ⎟_⎟ _⎜

⎜ _⎟ ⎜ _⎟ _⎜

⎜ ⎟ ⎜ ⎟ _⎜

⎜ ⎟_⎟ ⎜ ⎟_⎟

⎜ ⎜ ⎜

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

JJJG JJJG JJJG ⎟⎟⎟_⎟⎟⎟

⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟

0 1

1

01 11

02 12 2

03 13 3

. , . ,... .

. . .

0 1

k

k k k

k k kk

β β β

⎛ ⎞

⎛ ⎞_⎟ ⎛ ⎞_⎟ _⎜ _⎟

⎜ _⎟ ⎜ _⎟ _⎜ _⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎜ ⎟ ⎜ ⎟ _⎜ ⎟

⎜ ⎟ ⎜ ⎟ _⎜ ⎟_⎟

⎜ ⎟_⎟ ⎜ ⎟_⎟ _⎜

⎜ _⎟ ⎜ _⎟ _⎜

⎜ _⎟ ⎜ _⎟

=⎜_⎜ ⎟_⎟ =⎜_⎜ ⎟_⎟ =⎜_⎜ ⎜

⎜ ⎟_⎟ ⎜ ⎟_⎟ _⎜

⎜ _⎟ ⎜ _⎟ _⎜

⎜ ⎟ ⎜ ⎟ _⎜

⎜ ⎟_⎟ ⎜ ⎟_⎟

⎜ ⎜ ⎜

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

JJJG JJJG JJJG ⎟⎟⎟

⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟

where and

Adopting the order procedure used in the single case for the block method, we recall that

0

( )

(

)

(

, (

))

k

h j j

i

y x

α

y x

jh

h f x

β

jh y x

jh

=

⎡

⎤

=

∑

⎣

JJJG

+

−

JJJG

+

−

⎦

A

…7.0

where

y x

is the exact solution satisfying

y

f

Carrying a Taylor series expansion of (7.0) about x yields the equation

( )

( , ( )).

2

0 1 2

( )

...

q q

( )

h

y x

C y x

C hy x

C h y x

q

′ ′′

=

JJJG

+

JJJG

+

JJJG

+

JJJG

A

…8.0

where

C

01 11 1

02 12 2

03 13 3

0 1

.

,

.

...

.

p

p p

c

The block linear multi-step method is said to be of order

p

if

C

and the local truncation

error is expressed as

T

C

0

=

1

=

C

2

=

...

C

p

=

0,

C

p+1

≠

0

JJJG JJJG JJJG JJJG JJJJJJJG

h

+

y

+

x

+

=

JJJJJJJG

)

B

1 1

1 p p

( ).

n p n

STABILITY REGIONS OF THE BLOCK LINEAR MULTI-STEP METHODS: To determined the absolute stability regions of the block methods, they are reformulated as General Linear Methods of Burage & Butcher (1980) where they used a partition

matrix containing

A A

and

B

expressed in the form.

(

s

+

r

) (

×

s

+

r

₁

,

₂

,

₁ ₂

₁

( )

1 1

2 2

1,2,...,

i i

Y hf Y

y y

A

B

i

N

A

B

−

⎡

⎤ ⎡

⎤

⎡

⎤

⎢

⎥ ⎢

⎥

⎢

⎥

=

⎢

⎥ ⎢

=

⎥

⎢

⎥

⎣

⎦

⎢

⎣

⎥

⎦

⎣

⎦

…9.0

(3)

1 1 1

1 1 1 1 1 1 1

[

...

s

...

s

] ,

T

[

...

s

... ]

i i i i i i i i

Y

=

Y

−

Y Y

− −

Y

−

y

=

Y

−

Y y

− −

y

T

I

where

1 1 2 2

0 0 0 0

0

0 1

T T

I u e u

A B

I

A B A B

u e u

A B

v ω θ θ

⎡ ⎤ _⎡ ₋ _⎤

⎢ ⎥ ⎢ ⎥

⎡ ⎤ ⎡ ⎤ _⎢ _⎥ _⎢ _⎥

⎢ ⎥ ⎢ ⎥ _⎢ _⎥

= _⎢ _⎥ = _⎢ ₋ _⎥ =_⎢ _⎥ = ⎢ ⎥

⎢ ⎥

⎢ ⎥ ⎣ ⎦

⎣ ⎦ _⎢ _⎥ _⎢ ₋ _⎥

⎢ ⎥ ⎣ ⎦

⎣ ⎦

Applying (9.0) to the test equation

y

leads to the recurrence equation

, where the stability matrix

,

x

0

C

λ

′

=

≥

∈

h

1 [i 1]

( )

[ ]i

1,2,...,

1,

y

−

=

M z y i

=

N

−

z

=

λ

1

2 2 1

( )

(

)

M z

=

B

+

zA I

−

zA

−

B

_…10

and the stability polynomial of the method

. …11

( , )

z

det(

I

M

( ))

ρ η

=

η

−

z

The absolute stability region of the method is defined as

:

( , )

1

A

=

x

∈

C

ρ η

z

=

⇒

η

≤

For the purpose of this paper, the matrices

A A

are replaced by

A B

and

U

respectively. The coefficients of these matrices indicate the relationship between the various numerical quantities that arise in the computation.

1

,

2

,

B B

1

,

2

V

=

, ,

ANALYSIS OF THE BLOCK LINEAR MULTI-STEP METHODS: In this section, the analysis of the convergence, order and stability properties of the block linear multi-step methods is carried out for some block linear multi-step methods.

CONVERGENCE ANALYSIS: Considering the block hybrid Adams Moulton method for k

4

with one off grid point at 7 2

α

as

given below:

=

7 2

1 3 90 1 2 3 4

2 3 2520 1 2 3 4

3 280 1 2 3 4

3 161280

[ 34 114 34 1728 ]

[ 11 105 1211 1981 704 126 ]

[87 399 147 357 192 42 ]

[ 83

h

n n n n n n n n

h

n n n n n n n n

h

n n n n n n n n

h n

n

y y f f f f f f

y y f

+ + + + + + +

+ + + + + +

+ +

= + − − − + +

= + − + − − + −

= + + + − − +

= + − 7

2

7 2

1 2 3 4

4 3 2520 1 2 3 4

(5

6727 3038 44072 41600 2583 ]

[ 7 21 371 1728 406 ]

n n n n n n

h

n n n n n n n n

f f f f f

y y f f f f f f

+ + + + +

+ + + + + + +

⎫⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎬ ⎪⎪ ⎪

+ − + + − _⎪⎪

⎪⎪ ⎪

= + − + + + + _⎪⎭

.1)

…12.0

The block method (12) is represented in the form (2.0) to give

n+1 n-4

n+2 n-3

n+3 n-2

7 n-1

n+ 2

n n+4

1 0 0 0 0

1 0 _{1 0 0 y} _y

y 0 1 0 0 0

0 1 _{1 0 0 y}

0 0 1 0 0

0 0 1 1 0

h= y y

y 0 0 0 1 0 y

0 0 1 0 1

y

0 0 0 0 0 _y 0 0 0 0 1

⎡ ⎤

− ⎡ ⎤

⎡ ⎤_⎢ _⎥ _⎢ _⎥⎡ ⎤

⎢ ⎥_⎢ _⎥ _⎢ _⎥⎢

⎢ − ⎥⎢ ⎥ _⎢ _⎥⎢

⎢ ⎥⎢ ⎥ _⎢ _⎥⎢

⎢ ⎥⎢ ⎥₌_⎢ _⎥⎢ ⎥

⎢ ⎥_⎢ _⎥ _⎢ _⎥

⎢ ⎥

⎢ ⎥_⎢ _⎥ _⎢ _⎥

⎢ ⎥

⎢ − ⎥⎢ ⎥ _⎢ _⎥⎢ ⎥

⎢ ⎥_⎢ _⎥ _⎢ _⎥

⎢ ⎥

⎢ ⎥_⎢ _⎥ _⎢_⎣ _⎥_⎦

⎢ ⎥ ⎢ ⎥

⎣ ⎦⎢_⎣ ⎥_⎦ ⎣ ⎦

⎥

34 114 34 1728 1

90 90 90 90 90

1981 704 126

20 520

399 147 357 192 42 280 280 280 280 280 6725 3038 44072 44600 2583 161280 161280 161280 161280 161280 7 21 371 1728 406 2520 2520 2520 2520 2520

h

− − −

Where

1 90

11 2520 87

(0) (1) (1)

280 83 161280

7 2520

0 0 0 0

1 0 0 0 0 0 0 0 0 0

0 0 0 0

0 1 0 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 0

0 0 0 1 0 ₀ ₀ ₀ ₀

0 0 0 0 0

0 0 0 0 1 ₀ ₀ ₀ ₀

A A B

−

⎡ ⎤

⎡ ⎤ ⎡ ⎤ _⎢ _⎥

⎢ ⎥ ⎢ ⎥ _⎢ _⎥

= ⎢ ⎥ = ⎢ ⎥ = _⎢ _⎥

⎢ ⎥ ⎢ ⎥ _⎢ _⎥

⎢ ⎥ ⎢_⎢ ⎥_⎥ _⎢ _⎥

⎢ ⎥ ⎣ ⎦

⎣ ⎦ ⎢⎣ ⎥⎦

105 1211

25 2520 2520 2 2520

⎡ ⎢ ⎢

⎢ ₋ ₋

⎢ + _⎢

⎢ ₋ ₋

⎢ ⎢ ⎢

− ⎢

⎢ − −

⎢ ⎢

⎣

1

2

3

7 2

4

n

f

f f

f

+ + +

+ +

⎤ ⎥

f

⎡ ⎤

⎥

⎢ ⎥

⎥

⎢ ⎥

⎥_⎢ _⎥

⎥

⎢ ⎥

⎥

⎢ ⎥

⎥_⎢ _⎥+

⎥_⎢ _⎥

⎥

⎢ ⎥

⎥

⎢ ⎥

⎥_⎢ _⎥

⎥

⎣ ⎦

⎥

⎢ ⎥_⎦

1

90 4

11 3 2520

87 2 280

83 ₁

161280 7 2520

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

n

f f f h

f f

− −

− − − ₋

−

⎡ _{⎤ ⎡} _⎤

⎢ _{⎥ ⎢} _⎥

⎢ _{⎥ ⎢}_⎣ _⎥_⎦

⎢ ⎥

⎣ ⎦

(4)

The characteristic polynomial corresponding to (12.0) is given as:

0 1

0 0 0 0

1 0 0 0 0 0 0 0 0 0

0 0 0 0

0 1 0 0 0 0 0 0 0 0

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

( ) det{ } det det

0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 0

0 0 0 0 1 0 0 0 0

R R

R

R RA A R

R R

ρ

⎡

⎧ ⎡ ⎤ ⎡ ⎤⎫

⎪ ⎪

⎪ ⎢ ⎥ ⎢ ⎥⎪ ⎢

⎪ _⎢ _⎥ _⎢ _⎥⎪ _⎢

⎪ ⎪

⎪ _⎢ _⎥ _⎢ _⎥⎪ _⎢

⎪ ⎪

⎪ ⎢ ⎥ ⎢ ⎥⎪ ⎢

⎪ ⎪ ₋

= − = _⎪⎨ _⎢⎢ _⎥⎥−⎢ ⎥⎬_⎪ = ⎢_⎢

⎢ ⎥

⎪ _⎢ _⎥ _⎢ _⎥⎪ _⎢

⎪ ⎪

⎪ _⎢ _⎥ _⎢ _⎥⎪ _⎢

⎪ ⎪

⎪ ⎢ ⎥ ⎢ ⎥⎪ ⎢

⎪ _⎢ _⎥ _⎢ _⎥⎪ _⎢

⎪ ⎣ ⎦⎪

⎩ ⎣ ⎦ ⎭ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

5 ₀

R R

= ⇒ =

By (4.0) the block method (12.0) is zero stable and consistent since its order

p



6

≥

1

hence by Henrici (1962), the block method (12.0) is convergent.

ORDER OF THE BLOCK METHOD: Consider the block hybrid Adams Moulton Method for k=3 with one off grid point at





52 as given below:

5 2 5 2

5 5

2 2

1 2 1800 1 2 3

2 225 1 2 3

2 28800 1 2 3

3 2 1800 1 2

[31 755 1635 704 145 ]

[71 320 15 64 20 ]

[37 335 7455 7808 565 ]

[ 5 285 121

h

n n n n n n n

h

n n n n n n n

h

n n n n n

n n

h

n n n n n n

y y f f f f f

y y f f f f

+ + + + + +

+ + + + +

+ + + +

+ +

+ = + +

= + − − + −

= + + + + −

= + − + + −

= + − + + + 5

2 3

(5.3)

295fn+ ]

+

⎫⎪⎪ ⎪⎪ ⎪⎪⎪ ⎬⎪ ⎪⎪ ⎪⎪

+ _⎪⎪⎭

…13.0

Applying the method described for the order of the block LMM gives

₀ ₀ ₁ ₂ ₅ ₃

2

+ CJJJG = αJJJG+αJJJG+αJJJG+αJJJG αJJJG

1

0 1 0 0

0 0

1 0

0 0

0 0 1 1

0

0 0 1 0 1

0

0 0 0 0 0

− 0

⎡ ⎤

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢₋ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ₋ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= ⎢ ⎥+⎢ ⎥+⎢ ⎥+⎢ ⎥+⎢ ⎥ = ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥_{⎢ ⎥} ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ _{⎣ ⎦} ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

5

2

5

1 1 2 3 0 1 2

2 5

( 2 3 ) (

2 3)

α α α α β β β β β

= + + + − + + + +

CJJJG JJJG JJJG JJJG JJJG JJG JJG JJG JJG JJG

31 0

0 1 0 1800

71

0 ₀

1 2

225

0 ₀

0 2 37

5 ₀ 28800

0 0

1 2

0

0 0 ₀ ₁₈₀₀

⎡ ⎢ ⎧ ⎡ ⎤ ⎫ ⎪⎡ ⎤ ⎡− ⎤ ⎡ ⎤⎪ _⎢ ⎪_{⎢ ⎥} _⎢ _⎥ ⎢ ⎥ _{⎢ ⎥}⎪ ⎪ _{⎢ ⎥} ⎪ _⎢ ⎪_{⎢ ⎥} _⎢ _⎥ _{⎢ ⎥}⎪ ⎪ − _{⎢ ⎥} ⎪ ⎢ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ _⎢ ⎪_{⎢ ⎥} _⎢ _⎥ ⎢ ⎥ _{⎢ ⎥}⎪

⎪ ⎪ _⎢

⎪ ⎢ ⎥ ⎪

=⎨_⎪⎢ ⎥+⎢ ⎥+_{⎢ ⎥}+⎢ ⎥⎬_⎪− _⎢ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ _⎢ ⎪_{⎢ ⎥} _⎢ _⎥ ⎢ ⎥ ⎪ ⎪ _{⎢ ⎥} ⎢ ⎥⎪

⎪_{⎢ ⎥} _⎢ _⎥ _{⎢ ⎥}⎪ ₋ ⎪ _{⎢ ⎥} ⎪

⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪

⎪ ⎪

⎩ ⎣ ⎦ ⎭ ⎣

1635 704 755

1800 1800

1800

15 64

320

225 225

225

7455 7808

335

28800 28800 28800

5 285 121

1800 1800 1800

− −

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥₊⎢₋ ⎥₊⎢ ⎥₊⎢ ⎥₊ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥_⎦ ⎢_⎣ ⎥_⎦ ⎢_⎣ ⎥_⎦ ⎢_⎣ ⎥_⎦

145

0 1800

20 ₀

225

0 565

28800 ₀

295

0 1800

−

⎧ ⎡

⎪ ⎪

⎪ ⎢ ⎥⎪

⎪ _⎢ _⎥⎪ _{⎛ ⎡ ⎤ ⎞}

⎪ ⎪ _⎜ _⎟

⎪ _⎢ ₋ _⎥⎪ _⎜_{⎢ ⎥ ⎟}

⎪ ⎪ _{⎜ ⎢ ⎥}⎟

⎪ ⎢ ⎥⎪ _⎜ ⎟_⎟ ⎪ _⎢ _⎥⎪ _⎜_{⎢ ⎥}_⎟

⎪ ⎪ _⎜ _⎟

⎪ _⎢ _⎥⎪ _⎜⎢ ⎥_⎟ ⎪ ₋ ⎪₌_⎜_{⎢ ⎥}_⎟

⎨ _⎢ _⎥⎬ _⎜ _⎟

⎪ _⎢ _⎥⎪ ⎜ ⎢ ⎥⎟_⎟

⎪ ⎪ ⎜ ⎟

⎪ ⎢ ⎥⎪ ⎜⎢ ⎥⎟

⎪ ⎪ ⎜⎢ ⎥⎟

⎪ ⎢ ⎥⎪ ⎜ ⎟

⎪ _⎢ _⎥⎪ ⎜ ⎢ ⎥_⎜ ⎟_⎟

⎪ ⎪ _⎝_{⎢ ⎥}_⎠

⎪ _⎢ _⎥⎪ ⎣ ⎦

⎪ ⎪

⎪ ⎢ ⎥⎪

⎪ ⎪

⎩ ⎣

⎤ ⎫

⎦ ⎭

(5)

( )

5 5

2 2

1 5 5

2 2!( 1 2 2 2 3 3) ( 1 2 2 2 3

C = α + α + α + α − β + β + β + β3)

755 1800 320 225 25 335 1 4 28800 2! 5 1800 0 4 0 1 0 0 4 0 0 0 4

0 4 0 9

0 0 0 0 ₀

− − ⎡ − ⎛ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ _{⎡ ⎤ ⎟}⎞ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎟ ⎜ _⎟ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎟ ⎜ _⎢ _⎥ _{⎢ ⎥} ⎟ ⎜ ⎢ ⎥ ⎢ ⎥ ⎟ ⎜ _{⎢ ⎥} _⎢ _⎥ _{⎢ ⎥} _{⎢ ⎥}⎟ ⎜ ₋ ⎟_⎟ ⎜ =⎜_⎜ _{⎢ ⎥}⎢ ⎥+_⎢⎢ ⎥_⎥+⎢ ⎥_{⎢ ⎥}+⎢ ⎥ ⎟_{⎢ ⎥ ⎟}_⎟− ⎜ ⎢ ⎥ ⎢− ⎥ ⎢ ⎥ ⎢ ⎥⎟ ⎜ ⎟ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎟ ⎜ _⎟⎟ ⎜ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ _{⎢ ⎥ ⎟} ⎜ ⎟ ⎜⎝ ⎢ ⎥⎣ ⎦ ⎢⎣ ⎥⎦ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦⎠ _⎣

3270 1760 435 1800 1800 1800 30 160 60 225 225 225 14910 19520 1695 28800 28800 28800 570 3040 885

1800 1800 1800 0 0 0 − − − − ⎛ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎜_⎢ _⎥ _⎢ _⎥ _⎢ _⎥ _⎢ _⎥ ⎜⎜⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎜_⎢ _⎥ _⎢ _⎥ _⎢ _⎥ _⎢ _⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥₊ ₊ ₊ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎝ _⎦ _⎣ _⎦ _⎣ _⎦ _⎣ _⎦ 0 0 0 0 0 ⎞ _{⎡ ⎤} ⎟⎟ ⎢ ⎥ ⎟⎟ ⎢ ⎥ ⎟ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟_⎟ ⎢ ⎥ ⎜ _⎟₌ ⎜ _⎟ ⎢ ⎥ ⎜ _⎟ _{⎢ ⎥} ⎜ _⎟ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟_⎟ ⎢ ⎥ ⎜ _⎟ _{⎢ ⎥} ⎜ _⎟ ⎜ _⎟ _{⎢ ⎥} ⎜ ⎠ ⎣ ⎦ 5 2 3

3 3 2 2

1 1

5

3 3! 1 2 3 2! 1 2 2

2

5

C = ( 2 3 ) ( 2 ( ) 3 )

2

αJJJG+ αJJJG+ αJJJG+ αJJJG − β + β + β + β

JJG JJG JJG ₅ JJG ₂JG

3 J  1 3! 1 0 0 0 0  −8 8 −8 −8 0  0 0 125 8 0 0  0 0 0 27 0

− 2!1

−755 1800 320 225 −335 28800 5 1800 0  −6540 1800 60 225 29820 28800 1140 1800 0  4400 1800 400 225 48800 28800 7600 1800 0  −1305 1800 −180 225 −5085 28800 2655 1800 0  0 0 0 0 0

( )

5 5

2 2

4 4 3 3

1 5 1 5

4 4!( 1 2 2 2 3 3) 3!( 1 2 2 ( )2 3

C = α + α + αJJJJJG+ α − β + β + β + ₃β

3)

JJJG _JJJG _JJJG _JJJG JJG JJG JJG JJG

1 4! 1 0 0 0 0  −16 16 −16 −16 0  0 0 625 16 0 0  0 0 0 81 0 − 1 3! −755 1800 320 225 −335 28800 5 1800 0  −13080 1800 120 225 59640 28800 2280 1800 0  11000 1800 1000 225 122000 28800 19000 1800 0  −3915 1800 −540 225 −15255 28800 79565 1800 0  0 0 0 0 0

( ) 5 5

2 2

5

5 5 4 4

1 5 1 5

5 5 !( 1 2 2 2 3 3) 4 !( 1 2 2 ( )2 3

C = α + α + αJJJJJG+ α − β + β + β + ₄β

3)

JJJG _JJJG _JJJG _JJJG JJG JJG JJG JGJ

1 5! 1 0 0 0 0  −32 32 −32 −32 0  0 0 3125 32 0 0  0 0 0 243 0

− 4!1

−755 1800 320 225 −335 28800 5 1800 0  −26160 1800 240 225 119280 28800 4560 1800 0  27500 1800 2500 225 305000 28800 47500 1800 0  −11745 1800 −1620 225 −45765 28800 23895 1800 0  0 0 0 0 0

( )

5 5

2 2

6

6 6 5 5

1 5 1 5

6 6!( 1 2 2 2 3 3) 5!( 1 2 2 ( )2 3

C = α + α + αJJJJJG+ α − β + β + β + ₅β

3)

JJJG _JJJG _JJJG _JJJG JJG JJG JJG JGJ

1 6! 1 0 0 0 0  −64 64 −64 −64 0  0 0 15625 64 0 0  0 0 0 729 0 − 1 5! −755 1800 320 225 −335 28800 5 1800 0  −52320 1800 480 225 238560 28800 9120 1800 0  68750 1800 6250 225 762500 28800 118750 1800 0  −35235 1800 −4860 225 −137295 28800 71685 1800 0  11 360 7 900 83 230400 −1 3600 0

(6)

Since

C

0



C

1



C

2



C

3



C

4



C

5



0

and

C

6

≠

0

, the block hybrid method (13.0) for

3

k

=

with one off grid point is of order

p

=

5

and its error constant is

11 360

7 900

83 230400

1 3600

0

n

T

−

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

=

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

Absolute stability regions of the block methods: To plot the absolute stability regions of the block hybrid Adams Moulton method for k=3, equation (13.0) is expressed in the form (9.0) to yield the table below:

5 2

755 704

31 1635 145 1800 1800 1800 1800 1800 1

71 320 15 64 20 225 225 225 225 225 2

37 335 7455 28800 28800 2 3

3 2 1

0 0 0 0 0 | 0 0 1

| 1 0 0

| 0 0 1

n n n n n

n n n

y y y y y

y y y

− − −

+

−

+

−

+

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪₌ ⎨ ⎬ ⎪ ⎪ ⎪− − −⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

7808 565 8800 28800 28800 1 5 285 121 295 1800 1800 1800 1800 1800

1 5 285 121 295 1800 1800 1800 1800 1800 71 71 71 71 71 225 225 225 225 225 31 31 31 31 31 1800 1800 1800 1800 1800

| 1 0 0

|

| 1 0 0

| 0 0 1

| 1 0 0

−

⎧⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪

⎨

− − − − − − − − − − − − − − − − − − − − − − − −

5 2

1 2

3

2 1

n n n n n

n n n

hf hf hf hf hf

y y y

+

⎫⎪⎪_⎧ _⎫ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪ ⎪⎪ ⎪

⎪ ⎪⎪ ⎪

⎪ ⎪⎪_⎬⎨ ⎪_⎬

⎪ ⎪

⎪ _{⎪ − − −}

⎪ ⎪

⎪ ⎪⎩ ⎭

⎪ ⎪

⎩ ⎭

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Where

755 704 31 1635 145 1800 1800 1800 1800 1800 71 320 15 64 20 225 225 225 225 225 37 335 7455 7808 565 28800 28800 28800 28800 28800 1 5 285 121 295 1800 1800 1800 1800 1800

0 0 0 0 0 _{0 0 1}

1 0 0 0 0 1

A U

− − −

−

− −

−

⎧ ⎫

⎪ ⎪

=⎨_⎪ ⎬_⎪ =

⎪ ⎪

⎩ ⎭

, 1 0 0 1 0 0

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

,

1 5 285 121 295 1800 1800 1800 1800 1800 71 71 71 71 71 225 225 225 225 225

31 31 31 31 31 1800 1800 1800 1800 1800

1 0 0

, 0 0

1 0 0

B V

−

⎧ ⎫ _⎧ _⎫

⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪

⎪ ⎪ _⎪

⎪ ⎪

=⎨_⎪ ⎬_⎪ =

⎪ ⎪

⎪ ⎪ _⎪_⎩ _⎪_⎭

⎪ ⎪

⎩ ⎭

1⎪⎪

⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Substituting the values of the matrices

A

,

B

,

U

and

V

into the stability matrix (10.0) and the stability function (11.0) and using the maple software produces the stability polynomial of the method. The stability polynomial is used in a matlab environment and produces the region of absolute stability of the block method k=3 as shown in Fig. 1

(7)

FIG1. THE ABSOLUTE STABILITY REGION OF THE BHAM K=3 WITH ONE OFF GRID POINT CONCLUSION

The convergence, order and region of absolute stability of block linear multi-step methods including its hybrid forms have been

determined for the case and These

examples demostrate that the new approaches have resolved the difficulties of determining the convergence, order and absolute stability regions of block linear multi-step methods. The stability region displayed in figure 1 showed that the Block linear multi-step method for k=3 is

A

-Stable and is of order . while the method

k

is convergent. Future work will include implementation issues such as error estimates and step size control and changing strategy.

3

k

=

k

=

4.

5

p

=

4

=

REFERENCES

Burage, K. & Butcher, J. C. 1980. Non Linear Stability for a general class of differential equations methods, BIT 20:185-203

Chollom, J. & Onumanyi, P. 2004. Variable order A-Stable Adams Moulton type block hybrid methods for the solution of stiff first order ODEs Abacus, Journal Mathematical Association of Nigeria 31(2B)177-200

Dahlquist, G. 1959. Convergence and stability in the numerical integration of ordinary differential equations. Mathematica Scandinavica 4:33-53

Fatunla, S.O. 1994 Block methods for second order initial value problems. International Journal of Computer Mathematics 41:55-63

Henrici, P. 1962. Discrete variable methods in Ordinary Differential Equations. John Willey and sons Inc. New York-London Sydney

Lie, I. & Norsett, P. 2004. Super convergence for the multi-step collocation. Mathematics and Computer.52:65-79

Onumanyi, P.; Awoyemi, D.O.; Jatau, S.N & Sirisena,W.U. 1994. New Linear Multi-Step methods with continuous coefficients for the first order Ordinary Initial Value Problems. Journal of Nigerian Mathematical Society 13:37-51

Onumanyi, P.; Sirisena, W.U. & Jatau, S.N. 1999. Continuous finite difference approximations for solving differential equations. International Journal of Computer & Mathematics 72:15-27

Onumanyi, P.; Sirisena, W.U. & Chollom, J. P. 2001, Continuous hybrid methods through multistep collocation. Abacus, Journal Mathematical Association of Nigeria:58-64

Onumanyi, P. Yakubu, D. G. & Chollom, J. 2004. A new family of general linear methods based on block Adams Moulton multi step methods. Science Forum Journal of pure and applied science, 7(1):98-106