Explicit and implicit 3-point block methods for solving special ... - m-hikari

Int. Journal of Math. Analysis, Vol. 3, 2009, no. 5, 239 - 254

Explicit and Implicit 3-point Block Methods for Solving Special Second Order Ordinary Differential Equations Directly Fudziah Ismail Department of Mathematics, Universiti Putra Malaysia 43400 Serdang, Selangor, Malaysia [email protected] Yap Lee Ken Institute for Mathematical Research, Universiti Putra Malaysia 43400 Serdang, Selangor, Malaysia

Mohamad Othman Faculty of Computer Science and Information Technology, Universiti Putra Malaysia 43400 Serdang, Selangor, Malaysia

Abstract This paper focused mainly on deriving explicit and implicit 3-point block methods of constant step size using linear difference operator for solving special second order ordinary differential equations (ODEs). The methods compute the solutions of the ODEs at three points simultaneously. Regions of stability for both the explicit and implicit block methods are presented. A standard set of problems are solved using the new methods and the numerical results are compared when the same set of problems are solved using existing methods. The results suggest a significant

240

F. Ismail, Yap Lee Ken and M. Othman

improvement in efficiency of the new methods in terms of number of steps and accuracy. Mathematics Subject Classification: 65L06; 65L05 Keywords: initial value problems, linear difference operator, block method

1. Introduction Since the advent of computers, the numerical solution of Initial Value Problems (IVP) for Ordinary Differential Equations (ODEs) has been the subject of research by numerical analysts, especially methods for the numerical solutions of the special second-order ODEs of the form

y ′′ = f ( x, y ) ,

y ( x0 ) = y 0 ,

y ′( x0 ) = y 0′ .

(1)

in which f does not depend on y ′ . In general the special second order equation (1) can be reduced to an equivalent first-order system of twice the dimension and solved using the standard Runge-Kutta method or multistep method. However, it is more efficient if the equation can be solved directly as suggested by Chakravati and Worland [1], Dahlquist [2], Jeltsch [6], Sharp and Fine [10], Dormand et. al [3] and El-Mikkawy and El- Desouky [4]. These methods compute the numerical solution at one point at a time. More efficient codes can be developed if the solutions to the equations can be computed at more than one point simultaneously. This type of method is called block method.

The block method for numerical solutions of first order ODEs has been proposed by Lee [7], while Omar et al. [9] and Majid and Suleiman [8] developed block method for general second order ODEs. In a block method, a set of new values that are obtained by each application of the formula is referred to as “block”. In r-point block method, each application of the formulas generates a block of r new equally spaced solution values simultaneously. The computation which proceeds in blocks is based on the computed values at earlier blocks. The computed values at the previous k-block are used to compute the current block containing r points and the method is called r-point k-block method. Fatunla [5] has developed 2-point block method for the special second order ODEs. In this paper, we are going to derive 3-point block method for the solution of the special second order ODEs.

Explicit and implicit 3-point block methods

241

2. Derivation of 3-Point 1-Block Method According to Fatunla [5], the r-point k-block method for (1) is given by the matrix finite difference equation Ym =

k

k

i =1

i =0

∑ A(i )Ym−i + h 2 ∑ B (i ) Fm−i

(2)

⎡ f n +1 ⎤ ⎡ y n +1 ⎤ ⎢f ⎥ ⎢y ⎥ where Ym = ⎢ n + 2 ⎥, Fm = ⎢ n + 2 ⎥ , (for n = mr , m = 0,1K ), A (i ) , B ( i ) are r by r ⎢M ⎥ ⎢M ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ f n+r ⎦ ⎣ y n+ r ⎦ matrices. The block scheme is explicit if the coefficient matrix B ( 0 ) is a null matrix. The linear difference operator L associated with the linear multistep method is defined by ∞

[

L[Z ( x), h] = ∑ α j Z ( x + jh) − h 2 β j Z " ( x + jh)

]

(3)

j =0

where Z (x) is an arbitrary function and is assumed to be differentiable. Expand Z ( x + jh ) and Z ′′(x + jh ) about x and collecting terms to obtain L[Z ( x), h] = C 0 Z ( x) + C1 hZ ' ( x) + L + C q h q Z ( q ) ( x) + O(h q +1 )

whose coefficients C q , q = 0,1,K are constants and given as: k

C0 = ∑ α j j =0

k

C1 = ∑ jα j j =1

M

k ⎤ 1⎡ q (4) − − j α q ( q 1 ) j q−2 β j ⎥ ⎢∑ ∑ j q! ⎣ j =1 j =1 ⎦ The associated linear multistep method and the linear difference operator (3) are said to be of order p if C 0 = C1 = K = C p +1 = 0 and C p + 2 ≠ 0 .

Cq =

k

Derivation of Explicit 3-Point 1-Block Method:

242


The explicit 3-point 1-block method (r = 3, k = 1) for second order IVP can be derived from (3) by ensuring that the coefficient matrix B ( 0 ) is null. It can be described by matrix finite difference equation (2)

A(0 ) Ym = A(1) Ym−1 + h 2 B (1) Fm−1

(5)

with the matrix coefficients specified as

Ym

A

(0 )

⎡1 0 = ⎢⎢0 1 ⎢⎣0 0

⎡ y n +1 ⎤ ⎡ y n−2 ⎤ ⎡ f n−2 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ y n + 2 ⎥, Ym −1 = ⎢ y n −1 ⎥, Fm −1 = ⎢⎢ f n −1 ⎥⎥, , ⎢⎣ y n +3 ⎥⎦ ⎢⎣ y n ⎥⎦ ⎢⎣ f n ⎥⎦ 0⎤ ⎡b11 b12 b13 ⎤ ⎡ c11 c12 ⎥ ⎢ ⎥ (1) (1) 0⎥ , A = ⎢b21 b22 b23 ⎥ , B = ⎢⎢c 21 c 22 ⎢⎣b31 b32 b33 ⎥⎦ ⎢⎣c31 c32 1⎥⎦

c13 ⎤ c 23 ⎥⎥ c33 ⎥⎦

We consider the linear difference operator (3) and using (4) three sets of equations are obtained. The following three sets of equations have to be satisfied so that the method is at least of order three. 1st set of equations: b11 = 0 ,

b12 = −1 ,

c11 + c12 + c13 = 1 ,

b13 = 2 ,

c12 + 2 c13 = 2 ,

c12 + 4 c13 =

25 . 6

c 22 + 4 c 23 =

35 . 2

2nd set of equations: b21 = 0 ,

b22 = −2 ,

c 21 + c 22 + c 23 = 3 ,

b23 = 3 ,

c 22 + 2 c 23 = 7 ,

3rd set of equations:

b32 = −3 , b33 = 4 , c32 + 2 c33 = 16 , c31 + c32 + c33 = 6 ,

b31 = 0 ,

Solving the equations, giving

c32 + 4 c33 = 47 .


A

(0 )

⎡1 0 0⎤ = ⎢⎢0 1 0⎥⎥ , ⎢⎣0 0 1⎥⎦

A

(1)

⎡0 − 1 2 ⎤ = ⎢⎢0 − 2 3⎥⎥ , B (1) ⎢⎣0 − 3 4⎥⎦

243

1 ⎡1 ⎢12 − 6 ⎢5 7 =⎢ − 2 ⎢4 11 ⎢ − 15 ⎢⎣ 2

13 ⎤ 12 ⎥ 21⎥ ⎥ 4⎥ 31 ⎥ 2 ⎥⎦

Hence, the explicit 3-point 1-block method can be written as 1 13 ⎤ ⎡1 − ⎢ 12 6 12 ⎥ ⎡ f n − 2 ⎤ ⎡1 0 0⎤ ⎡ y n +1 ⎤ ⎡0 − 1 2⎤ ⎡ y n − 2 ⎤ ⎢ ⎥ ⎢0 1 0⎥ ⎢ y ⎥ = ⎢0 − 2 3⎥ ⎢ y ⎥ + h 2 ⎢ 5 − 7 21⎥ ⎢ f ⎥ n −1 ⎥ ⎢ ⎥ ⎢ n+2 ⎥ ⎢ ⎥ ⎢ n −1 ⎥ 2 4 ⎥⎢ ⎢4 ⎢ ⎢⎣0 0 1⎥⎦ ⎢⎣ y n +3 ⎥⎦ ⎢⎣0 − 3 4⎥⎦ ⎢⎣ y n ⎥⎦ ⎢ 11 − 15 31 ⎥ ⎣ f n ⎥⎦ ⎢⎣ 2 2 ⎥⎦ Derivation of Implicit 3-Point 1-Block Method: The implicit 3-point 1-block method (r = 3, k = 1) for second order ODEs can be described by the matrix difference equation

[

A ( 0 )Ym = A (1)Ym −1 + h 2 B ( 0) Fm + B (1) Fm −1

]

(6)

with the matrix coefficients specified as

A

(0)

⎡ y n +1 ⎤ Ym = ⎢⎢ y n + 2 ⎥⎥ , Ym −1 ⎢⎣ y n +3 ⎥⎦ ⎡1 0 0 ⎤ ⎡b11 ⎢ ⎥ (1) = ⎢0 1 0⎥ , A = ⎢⎢b21 ⎢⎣0 0 1⎥⎦ ⎢⎣b31

⎡ y n−2 ⎤ ⎡ f n +1 ⎤ ⎢ ⎥ = ⎢ y n −1 ⎥ , Fm = ⎢⎢ f n + 2 ⎥⎥ , ⎢⎣ y n ⎥⎦ ⎢⎣ f n +3 ⎥⎦ b12 b13 ⎤ ⎡ c11 c12 ⎥ (0) b22 b23 ⎥ , B = ⎢⎢c 21 c 22 ⎢⎣c31 c32 b32 b33 ⎥⎦

⎡ f n−2 ⎤ Fm −1 = ⎢⎢ f n −1 ⎥⎥ , ⎢⎣ f n ⎥⎦ c13 ⎤ ⎡ d11 ⎥ (1) c 23 ⎥ , B = ⎢⎢d 21 ⎢⎣ d 31 c33 ⎥⎦

d12 d 22 d 32

Based on (4), we obtained the following three sets of equations, for the method to be of order at least six. 1st set of equations:

d13 ⎤ d 23 ⎥⎥ . d 33 ⎥⎦

244

F. Ismail, Yap Lee Ken and M. Othman b12 = −1 ,

b13 = 2 , c11 + c12 + c13 + d11 + d12 + d13 = 1 , d12 + 2 d13 + 3 c11 + 4 c12 + 5 c13 = 2 , 25 d12 + 4 d13 + 9 c11 + 16 c12 + 25 c13 = , 6 d12 + 8 d13 + 27 c11 + 64 c12 + 125 c13 = 9 ,

b11 = 0 ,

301 , 15 d12 + 32 d13 + 243 c11 + 1024 c12 + 3125 c13 = 46 . d12 + 16 d13 + 81c11 + 256 c12 + 625 c13 =

2nd set of equations: b22 = −2 ,

b23 = 3 , c 21 + c 22 + c 23 + d 21 + d 22 + d 23 = 3 , d 22 + 2 d 23 + 3 c 21 + 4 c 22 + 5 c 23 = 7 , 35 d 22 + 4 d 23 + 9 c 21 + 16 c 22 + 25 c 23 = , 2 93 d 22 + 8 d 23 + 27 c 21 + 64 c 22 + 125 c 23 = , 2 651 d 22 + 16 d 23 + 81 c 21 + 256 c 22 + 625 c 23 = , 5 d 22 + 32 d 23 + 243 c 21 + 1024 c 22 + 3125 c 23 = 381 .

b21 = 0 ,

3rd set of equations: b33 = 4 , d 31 + d 32 + d 33 + c31 + c32 + c33 = 6 , d 32 + 2 d 33 + 3 c31 + 4 c32 + 5 c33 = 16 ,

b31 = 0 ,

b32 = −3 ,

d 32 + 4 d 33 + 9 d 31 + 16 d 32 + 25 d 33 = 47 , d 32 + 8 d 33 + 27 c31 + 64 c32 + 125 c33 = 150 , 2562 , d 32 + 16 d 33 + 81c31 + 256 c32 + 625 c33 = 5 d 32 + 32 d 33 + 243 c31 + 1024 c32 + 3125 c33 = 1848 .

Solving the three sets of equations, we obtain the method which can be written as


245

⎡1 0 0⎤ ⎡ y n +1 ⎤ ⎡0 − 1 2⎤ ⎡ y n − 2 ⎤ ⎢ 0 1 0 ⎥ ⎢ y ⎥ = ⎢0 − 2 3 ⎥ ⎢ y ⎥ ⎥ ⎢ n+2 ⎥ ⎢ ⎢ ⎥ ⎢ n −1 ⎥ ⎢⎣0 0 1⎥⎦ ⎢⎣ y n +3 ⎥⎦ ⎢⎣0 − 3 4⎥⎦ ⎢⎣ y n ⎥⎦ ⎛⎡ 1 1 − ⎜⎢ 240 ⎜ ⎢ 10 11 2 ⎜ 121 +h ⎢ ⎜ ⎢120 120 ⎜ ⎢ 29 127 ⎜⎢ 120 ⎝ ⎣ 15

⎤ ⎡ 1 − 0 ⎥ f n +1 ⎤ ⎢ 240 ⎡ ⎢ 1 1 ⎥⎢ ⎥ ⎢ f n + 2 ⎥⎥ + ⎢ − − 240 ⎥ ⎢ 120 1 ⎥ ⎢⎣ f n +3 ⎥⎦ ⎢ 1 ⎢⎣ − 120 15 ⎥⎦

⎞ 1 97 ⎤ ⎟ ⎥ 10 120 ⎡ f n − 2 ⎤ ⎟ 47 103 ⎥ ⎢ ⎥ ⎢ f n −1 ⎥⎥ ⎟ ⎟ 240 60 ⎥ 4 161 ⎥ ⎢⎣ f n ⎥⎦ ⎟ ⎟ 15 60 ⎥⎦ ⎠

3. Stability of 3-Point 1-Block Method One of the practical criteria for a good method to be useful is that it must have region of absolute stability. The most convenient method for finding region of absolute stability is the boundary locus technique. The region R of the complex h - plane is defined by the requirement that for all h ∈ R , where h = h 2 λ , all the roots of the stability polynomial π (r , h ) have modulus less than one. When the explicit 3-point 1-block method is applied to the test equation y ′′ = λy as follows: A (0 ) Ym = A (1) + B (1) h 2 λ Ym−1

(

⎡1 2 ⎢ h λ ⎡1 0 0⎤ ⎡ y n +1 ⎤ ⎢12 ⎢0 1 0 ⎥ ⎢ y ⎥ = ⎢ 5 h 2 λ ⎥ ⎢ n+2 ⎥ ⎢ 4 ⎢ ⎢⎣0 0 1⎥⎦ ⎢⎣ y n +3 ⎥⎦ ⎢ 11 2 h λ ⎢⎣ 2

)

13 2 ⎤ h λ⎥ 12 ⎡ y n−2 ⎤ 21 2 ⎥ ⎢ 3 + h λ ⎥ ⎢ y n −1 ⎥⎥ . 4 ⎥ 31 2 ⎥ ⎢⎣ y n ⎥⎦ 2 − 3 − 15h λ 4 + h λ ⎥⎦ 2 1 − 1 − h2λ 6 7 − 2 − h 2λ 2

2+

We have the stability polynomial of the method by 2 (0) (1) (1) taking, det tA − ( A + B h ) = 0 , where h = h λ . The polynomial can be written as 37 5 37 ⎞ ⎞ ⎛ ⎞ ⎛ 79 ⎛ 145 (7) t3 + t2⎜− h − 2 ⎟ + t ⎜ h 2 + h + 1⎟ + ⎜ − h 3 + h 2 − h ⎟ = 0 6 4 12 ⎠ ⎠ ⎝ ⎠ ⎝ 4 ⎝ 12

[

]

246


Solving (7) for h = h 2 λ which gives t ≤ 1 we obtained the stability region of the method. This is done by using MATHEMATICA Programming. The stability region is shown in Figure 1 where the region of absolute stability lies inside the boundary.

Similarly, the stability region of the implicit 3-point 1-block method is obtained by tracing the boundary for which the modulus of roots of the stability polynomial of the method does not exceed one. Substituting the test equation into (6) gives

(A

( 0)

)

(

)

− h 2 λB (0 ) Ym = A (1) + h 2 λB (1) Ym −1

[

]

The stability polynomial is obtained by taking det t (A ( 0) − B (0 ) h ) − ( A (1) + B (1) h ) = 0 which gives the polynomial 59 11 2 31 ⎞ ⎛ 11957 3 2153 2 679 ⎞ ⎛ t3⎜− h3 + h − h + 1⎟ + t 2 ⎜ − h − h − h − 2⎟ 43200 360 120 21600 480 80 ⎠ ⎝ ⎠ ⎝ 1 1 1 ⎛ 89 3 1 2 1 ⎞ + t⎜ h + h − h + 1⎟ − h3 + h2 − h =0 1440 240 15 4 ⎝ 4320 ⎠ 21600 The stability polynomial is solved for h = h 2 λ which gives t ≤ 1 and the stability region is obtained by tracing the values of h . The stability region is shown in Figure 2. Where the stability region lies inside the boundary.


247 Im aginary Part

S ta bility R e gion of E xplic it 3P 1B

0.04

0.02

Real Part - 0.25

- 0.2

- 0.15

- 0.1

- 0.05

- 0.02

- 0.04

Figure 1 S t a b ilit y

R e g io n o f I m p lic it

I m a g in a r y P a r t

3 P 1B

0 .3

0 .2

0 .1

- 5

- 4

- 3

- 2

R ea l P a rt

- 1

- 0 .1

- 0 .2

- 0 .3

Figure 2

248


4. Test Problems To validate the new method, the following problems are solved and the numerical results are compared with existing methods.

Problem 1: y ″ y1 = − 1 r y ″ y2 = − 2 r

y1 (0 ) = 1 ,

′ y1 (0 ) = 0

y 2 (0 ) = 0 ,

′ y 2 (0) = 1

x ∈ [0,15π ] with r = y1 + y 2 , y1 ( x ) = cos( x ) Exact Solution: 2

2

Problem 2: 2y ″ y1 = −4 x 2 y1 − 2 r y 2 ″ y 2 = −4 x 2 y 2 + 1 r with r 2 = y1 + y 2 , 2

2

Exact solution:

y 2 (x ) = sin ( x )

y1 ( x0 ) = 0

′ y1 ( x0 ) = − 2π

y 2 (x0 ) = 1

′ y 2 (x0 ) = 0

⎡ π ⎤ x∈⎢ ,10⎥ ⎣ 2 ⎦ y1 ( x ) = cos(x 2 )

( )

y 2 ( x ) = sin x 2

Problem 3: z ′′ + z = 0.001e ix , z (0 ) = 1, z ′(0 ) = 0.9995i

z ∈C ,

z ( x ) = u ( x ) + iv( x ), u , v ∈ ℜ ,

u ( x ) = cos( x ) + 0.0005 x sin ( x ) v( x ) = sin ( x ) − 0.0005 x cos( x )

We choose to solve the equivalent real problems u ′′ + u = 0.001 cos( x )

u (0) = 1 ,

u ′(0) = 0

v ′′ + v = 0.001 sin ( x )

v(0) = 0 ,

v ′(0 ) = 0.9995

x ∈ [0, 40π ]


249

5. Numerical Results

The tables below show the numerical results when Problems 1 to 3 are solved numerically using Fatunla [5], 2-point 1-block (2P1B) method, 2-point third order block method by Omar et. al [9] and the 3-point 1-block (3P1B) method derived in the previous section. To obtain the starting values, we use the well-known fourthorder Runge-Kutta method whereby the second order equations are reduced to system of first order equation. The following notations are used in the tables:

h

Step size used.

METHOD

Method employed.

TSTEP

Total number of steps taken to obtain the solution.

MAXERR

Maximum error.

E2P1B

Explicit 2-point 1-block method in Fatunla [5].

E3P1B

Explicit 3-point 1-block method.

E2PO

Explicit 2-point block method of order three in Omar et. al [9].

I2P1B

Implicit 2-point 1-block method in Fatunla [5].

I3P1B

Implicit 3-point 1-block method.

I2PO

Implicit 2-point block method of order three in Omar et. al [9].

The maximum error is defined as the absolute value of the true solution minus the computed solution. MAXERR = max ( y ( xi ) − y i ) 1≤ i ≤TSTEP

250


Table 1: Performance Comparison between E2P1B and E3P1B for Solving Problem 1. h 0.01

0.005

0.001

0.0005

METHOD

TSTEP

MAXERR

E2PO E2P1B E3P1B E2PO E2P1B E3P1B E2PO E2P1B E3P1B E2PO E2P1B E3P1B

2358 2357 1573 4714 4713 3143 23563 23563 15710 47125 47125 31418

6.80011(-3) 1.64890(-3) 6.79302(-4) 3.32365(-3) 4.35356(-4) 8.49136(-5) 6.65714(-4) 1.81544(-5) 6.79035(-7) 3.33056(-4) 4.56201(-6) 8.43528(-8)

Table 2: Performance Comparison between E2P1B and E3P1B for Solving Problem 2.

h

0.01

0.005

0.001

0.0005

METHOD E2PO E2P1B E3P1B E2PO E2P1B E3P1B E2PO E2P1B E3P1B E2PO E2P1B E3P1B

TSTEP

MAXERR

439 438 293 876 876 585 4375 4374 2917 8748 8748 5833

1.78118(-1) 5.84193(-1) 2.17171(-1) 2.47420(-2) 1.43272(-1) 3.15329(-2) 1.09983(-3) 5.71844(-3) 2.60846(-4) 5.49964(-4) 1.42972(-3) 3.26213(-5)


251

Table 3: Performance Comparison between E2P1B and E3P1B for Solving Problem 3.

h

0.01

0.005

0.001

0.0005

METHOD

TSTEP

MAXERR

E2PO E2P1B E3P1B E2PO E2P1B E3P1B E2PO E2P1B E3P1B E2PO E2P1B E3P1B

6285 6284 4191 12568 12567 8379 62833 62833 41890 125665 125665 83778

5.00010(-3) 3.65778(-3) 1.14794(-4) 2.49800(-3) 9.14567(-4) 1.43600(-5) 4.99503(-4) 3.65869(-5) 1.14912(-7) 2.49750(-4) 9.14662(-6) 1.42808(-8)

Table 4: Performance Comparison between I2P1B and I3P1B for Solving Problem 1. h

0.01

0.005

0.001

0.0005

METHOD

TSTEP

MAXERR

I2PO I2P1B I3P1B I2PO I2P1B I3P1B I2PO I2P1B I3P1B I2PO I2P1B I3P1B

2358 2357 1573 4714 4713 3143 23563 23563 15710 47125 47125 31418

1.61702(-3) 3.23212(-8) 2.14918(-8) 8.15183(-4) 2.15792(-9) 1.34949(-9) 1.65754(-4) 1.22006(-10) 8.64701(-11) 8.30992(-5) 6.19152(-10) 3.95277(-10)

252



0.01

0.005

0.001

0.0005

METHOD I2PO I2P1B I3P1B I2PO I2P1B I3P1B I2PO I2P1B I3P1B I2PO I2P1B I3P1B

TSTEP

MAXERR

439 438 293 876 876 585 4375 4374 2917 8748 8748 5833

2.70602(-3) 2.21752(-3) 2.61036(-3) 1.36406(-3) 1.39189(-4) 8.03535(-5) 2.74575(-4) 2.22967(-7) 1.22110(-7) 1.37397(-4) 1.41201(-8) 2.34300(-8)


0.01

0.005

0.001

0.0005

METHOD

TSTEP

MAXERR

I2PO I2P1B I3P1B I2PO I2P1B I3P1B I2PO I2P1B I3P1B I2PO I2P1B I3P1B

6285 6284 4191 12568 12567 8379 62833 62833 41890 125665 125665 83778

1.24874(-3) 5.39496(-8) 4.10154(-9) 6.24373(-4) 3.40287(-9) 2.23512(-9) 1.24875(-4) 1.44249(-10) 5.04009(-11) 6.24376(-5) 2.32200(-10) 2.31953(-10)


253

6. Discussions and conclusion The two measured parameters in the previous section, namely the total number of steps and accuracy will be used to illustrate the effectiveness of the method. The results demonstrate the advantage of using both the explicit and implicit 3-point 1-block method over the explicit and implicit 2-point 1-block methods by Fatunla [5] and Omar et.al [9] in terms of total number of steps taken. In all the problems, it is observed that the number of steps taken by the 3-point 1 block methods are always less than the number of steps taken by the existing methods. These results are expected since the new methods calculate the values of y at three points simultaneously compared to two points at a time for the existing methods. From the tables the new methods reduced the number of steps to almost two third compared to the existing methods. Comparing the explicit and the implicit methods, it is observed that the implicit methods are more accurate than the explicit counterparts this is true for both the new and the existing ones. From the magnitude of the maximum error we can conclude that the new explicit method is more accurate compared to the existing explicit methods. From the numerical results we can say that the new implicit method is more accurate than the I2P1B method and just as accurate as I2PO method. Thus, it can be concluded that the numerical results suggest a significant improvement in the efficiency of the new methods.

References [1]

P.C. Chakravati and P.B. Worland, A class of self-starting methods for the numerical solution of y ′′ = f ( x, y ) . BIT 11 (1971), pp. 368-383.

[2]

G. Dahlquist, On accuracy and unconditional stability of linear multistep methods for second order differential equations. BIT 18 (1978), pp. 133-136.

[3] J. R. Dormand , M. E. A. El-Mikkawy and P. J. Prince, Families of Runge-Kutta Nystrom Formula, IMA Journal of Numerical Analysis, 7(1987): 235-250. [4] M. E. A. El-Mikkawy and R. El- Desouky, A new optimized non-FSAL embedded Runge-Kutta-Nystrom algorithm of orders 6 and 4 in six stages, Applied Mathematics and Computation 145(2003) : 33-43.

254


[5] S.O. Fatunla, Block methods for second order ODEs. Intern. J. Computer Math 41 (1990), pp. 55-63. [6] R. Jeltsch, Complete characterization of multistep methods with an interval of periodicity for solving y ′′ = f ( x, y ) . Math Comp. 32(1978), pp. 1108-1114. [7] L.S. Lee, Two and three-point block methods for solving first order Ordinary Differential Equations in parallel. MSc Thesis (2000), Universiti Putra Malaysia. [8] Z. Majid and M.B. Suleiman, Parallel Block Codes for solving large systems of ordinary differential equations, Int. J. Simulation and Process Modelling, 2 (2006), No 1: 98-112. [9] Z. Omar, M.B. Suleiman, M.Y. Saman, and D.J. Evans, Parallel r-point explicit block method for solving second order Ordinary Differential Equations directly. Intern. J. Computer Math. 79(3) (2002), 289-298. [10] Sharp and Fine, Some Nystrom pairs for the general second order initial-value Problem, Journal of Computational and Applied Mathematics 42(1992): 279-291.

Received: June 8, 2008