2-Point Block BDF Method with Off-Step Points for

Journal of Soft Computing and Applications 2014 (2014) 1-15

Available online at www.ispacs.com/jsca Volume 2014, Year 2014 Article ID jsca-00039, 15 Pages doi:10.5899/2014/jsca-00039 Research Article

2-Point Block BDF Method with Off-Step Points for Solving Stiff ODEs Naghmeh Abasi1∗ , Mohamed Suleiman1 , Neda Abbasi2 , Hamisu Musa3 (1) Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia (2) Department of Mathematics, Shahid Beheshti University, Tehran, Iran. (3) Mathematics Department, Science Faculty, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia c Naghmeh Abasi, Mohamed Suleiman, Neda Abbasi and Hamisu Musa. This is an open access article distributed under the Copyright 2014 ⃝ Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, 2-point block method with two off-step points based on Backward Differentiation Formula (BDF) for solving stiff ODEs is formulated. The strategy of the developed method is to calculate two solution values of the method with two off-step points simultaneously at each iteration. Stability region and convergence of the method are also generated. The numerical results obtained are compared with the fifth order 2-point block BDF method to compare the enhancement of the method in terms of accuracy. Keywords: Off-step, Backward differentiation formula, Block, Stiff.

1 Introduction Consider the first order ordinary differential equations in the form of y′ = f (x, y)

y(x0 ) = y0 ,

a ≤ x ≤ b.

(1.1)

Implicit methods on solving stiff ODEs are known to perform better than explicit ones. The implementation of the BDF methods for solving stiff ODEs was discussed by Gear [3] and he became one of the well-known researchers in the study of stiff ODEs. The order and the accuracy of BDF method for solving stiff ODEs were improved by Cash [1] through adding a future point, the method was called extended BDF. Implicit off-step point method for the integration of stiff differential equations were discussed by Lautsch and can be found in [9]. Block implicit methods first were proposed by Milne [10] and his idea using a Runge-Kutta method later was extended by Rosser [11]. Convergence and stability properties of one-step implicit block method can be followed in [12, 13]. Block methods on solving stiff ODEs via backward differentiation formulae were developed in recent years and can be studied in [6]. Furthermore, Adam-Moulton hybrid block method with two off-step points for solving stiff ordinary differential equations can be seen in [7]. The solution of the 2-point block BDF with off-step points has not developed previously. The motivation of this paper is to discuss the derivation of the two solution values with two off-step points using block BDF method. The ∗ Corresponding

author. Email address: [email protected], Tel:0060173487635

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 2 of 15

implementations and stability region of the method are also presented. The developed method is compared with the fifth order 2-point block BDF method [14]. The advantage of the proposed method is that, the solutions are approximated with off-step points concurrently hopping for better accuracy. 2 Block BDF with off-step points formulation In this section, two solution values, yn+1 and yn+2 with step size h, and two off-step points, yn+ 1 and yn+ 3 which 2 2 are chosen at the points where the step size is halved, are formulated in a block simultaneously. The formulae are computed using two back values with step size h, yn and yn−1 , in the previous block (Figure 1). We have examined several points for choosing the points as off-step. Our investigations indicate that selecting the points where the step is halved are more desired to obtain the optimized point and a zero stable formulae.

Figure 1: 2-point block with off-step points The interpolating polynomial Pk (x) of degree k which interpolates the values of y at the points (xn−1 , yn−1 ), (xn , yn ), (xn+ 1 , yn+ 1 ), (xn+1 , yn+1 ), (xn+ 3 , yn+ 3 ) and (xn+2 , yn+2 ) is defined as 2

2

2

2

k

Pk (x) =

∑ Lk, j (x)y(xn+2− j )

(2.2)

j=0

where x − xn+2−i x n+2− j − xn+2−i i=0

1 f or j = 0, , ..., k 2

Lk, j (x) = ∏

(2.3)

i̸= j

The associated polynomial for (2.2) can be written as

P(x) =

(x − xn−1 )(x − xn )(x − xn+ 1 )(x − xn+1 )(x − xn+ 3 ) 2

2

(xn+2 − xn−1 )(xn+2 − xn )(xn+2 − xn+ 1 )(xn+2 − xn+1 )(xn+2 − xn+ 3 ) 2

+

2

(x − xn−1 )(x − xn )(x − xn+ 1 )(x − xn+1 )(x − xn+2 ) 2

(xn+ 3 − xn−1 )(xn+ 3 − xn )(xn+ 3 − xn+ 1 )(xn+ 3 − xn+1 )(xn+ 3 − xn+2 ) 2

+

2

2

2

2

(x − xn−1 )(x − xn )(x − xn+ 1 )(x − xn+ 3 )(x − xn+2 ) 2

2

(xn+1 − xn−1 )(xn+1 − xn )(xn+1 − xn+ 1 )(xn+1 − xn+ 3 )(xn+1 − xn+2 )

+

2

2

2

2

(x − xn−1 )(x − xn+ 1 )(x − xn+1 )(x − xn+ 3 )(x − xn+2 ) 2

2

(xn − xn−1 )(xn − xn+ 1 )(xn − xn+1 )(xn − xn+ 3 )(xn − xn+2 ) 2

+

yn+1

(xn+ 1 − xn−1 )(xn+ 1 − xn )(xn+ 1 − xn+1 )(xn+ 1 − xn+ 3 )(xn+ 1 − xn+2 ) 2

2

2

(x − xn−1 )(x − xn )(x − xn+1 )(x − xn+ 3 )(x − xn+2 ) 2

yn+ 3

2

2

+

yn+2

yn+ 1 2

2

yn

2

(x − xn )(x − xn+ 1 )(x − xn+1 )(x − xn+ 3 )(x − xn+2 ) 2

2

(xn−1 − xn )(xn−1 − xn+ 1 )(xn−1 − xn+1 )(xn−1 − xn+ 3 )(xn−1 − xn+2 ) 2

yn−1 .

2

(2.4)

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Define s =

Page 3 of 15

x − xn+1 and substitute x = xn+1 + sh in the interpolating polynomial (2.4), hence h p(x) =

p(xn+1 + sh)

=

(s + 2)(s + 1)(s + 12 )(s)(s − 12 )

+

(s + 2)(s + 1)(s + 12 )(s)(s − 1)

18 4

− 15 16

yn+2 yn+ 3 2

+

(s + 2)(s + 1)(s + 12 )(s − 12 )(s − 1)

+

(s + 2)(s + 1)(s)(s − 12 )(s − 1)

2 4 9 − 16

+

(s + 2)(s + 12 )(s)(s − 12 )(s − 1)

+

(s + 1)(s + 12 )(s)(s − 12 )(s − 1)

6 4

− 90 4

yn+1

yn+ 1 2

yn yn−1 .

(2.5)

Differentiating (2.5) with respect to s gives hp′ (xn+1 + sh) = − + − + + Substituting s = − 21 , 0,

1 2

2 21 3 1 (5 s4 + 12 s3 + s2 − s − ) yn+2 9 4 2 2 16 4 3 (5 s + 10 s − 5 s − 1) yn+ 3 2 15 15 1 2 (5 s4 + 8 s3 − s2 − 5 s + ) yn+1 4 4 16 (5 s4 + 6 s3 − 6 s2 − 3 s + 1) yn+ 1 2 9 2 27 1 1 (5 s4 + 4 s3 − s2 − s + ) yn 3 4 2 2 2 4 1 2 1 (− s + s − ) yn−1 9 6 90

(2.6)

and s = 1 in (2.6) respectively, gives

h fn+ 1 2

h fn+1 h fn+ 3 2

h fn+2

1 3 9 3 1 yn−1 − yn − yn+ 1 + yn+1 − yn+ 3 + yn+2 , 2 2 60 4 4 5 12 1 1 1 1 16 16 = − yn−1 + yn − yn+ 1 + yn+1 + yn+ 3 − yn+2 , 2 2 90 3 9 2 15 9 1 15 5 5 5 31 = yn−1 − yn + yn+ 1 − yn+1 + yn+ 3 + yn+2 , 2 2 60 12 3 4 15 12 48 9 1 3 16 = − yn−1 + yn − yn+ 1 + 9yn+1 − yn+ 3 + yn+2 . 2 2 15 2 3 5 2 =

(2.7)

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 4 of 15

Solving equations in (2.7) for yn+ 1 , yn+1 , yn+ 3 and yn+2 respectively, gives the following solution values 2

yn+ 1

2

yn+1 yn+ 3 2

yn+2

2

1 3 9 3 1 yn−1 − yn + yn+1 − yn+ 3 + yn+2 − h fn+ 1 , 2 2 60 4 4 5 12 32 2 1 2 32 = yn−1 − yn + yn+ 1 − yn+ 3 + yn+2 + 2h fn+1 , 2 2 45 3 9 15 9 25 25 25 15 1 225 yn−1 + yn − yn+ 1 + yn+1 − yn+2 + h fn+ 3 , = − 2 2 124 124 31 124 124 31 1 32 2 32 2 yn−1 − yn + yn+ 1 − 2yn+1 + yn+ 3 + h fn+2 . = 2 2 135 3 27 15 9 =

(2.8) 2.1 Order of the method This section derives the order of the method corresponding to the equations in (2.8). It can be written in the following form 1 3 9 3 1 yn−1 + yn + yn+ 1 − yn+1 + yn+ 3 − yn+2 2 2 60 4 4 5 12 1 2 32 32 2 − yn−1 + yn − yn+ 1 + yn+1 + yn+ 3 − yn+2 2 2 45 3 9 15 9 1 25 25 225 25 yn−1 − yn + yn+ 1 − yn+1 + yn+ 3 + yn+2 2 2 124 124 31 124 124 1 32 32 2 yn−1 + yn − yn+ 1 + 2yn+1 − yn+ 3 + yn+2 − 2 2 135 3 27 15 −

= −h fn+ 1 , 2

= 2h fn+1 , 15 hf 3, 31 n+ 2 2 h fn+2 . 9

= =

(2.9) The matrix form of (2.9) is associated with   3 1 yn− 3 0 0 − 60 4 2      2  0 −1 0  yn−1 45 3        1 25   y  0 0 − 124 n− 12   124    2 1 0 − 135 0 yn 3 





0





1

    32   −  9    +     25   31    − 32 27

−1

   0  =    0   0 Let



1 − 60

    0   −1 45    α0 =   , α1 =   0   1    124 2 0 − 135

0

0

2

0

0

15 31

0

0



− 94

3 5

1 − 12

1

32 15

− 29

− 225 124

1

25 124

2

− 32 15

1

 fn+ 1 2    0  fn+1     fn+ 3 0   2   2 fn+2 9 0



0



 yn+ 1 2      yn+1     y 3   n+ 2  yn+2

          

          



(2.10)

3 4

   2   0       3   , α2 =   , α3 =   0   − 25     124  1 0 3

      

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/



Page 5 of 15





1

 32  − 9  α4 =   25  31 − 32 27

   1     , α5 =    − 225  124  2



−1









− 94



3 5

       , α6 =     

32 15

1 − 32 15



0



0





1 − 12

    −2   9  , α7 =    25   124 1





0

     



         0   2   0   0          β4 =  , β =  , β =  , β =    0  5  0  6  15  7  0     31      2 0 0 0 9 The linear difference operator L is defined by 7

h

h

∑ [ α j y(x + j 2 ) − h β j y′ (x + j 2 ) ]

L[y(x), h] =

(2.11)

j=0

Expanding the function y(x + j h2 ) and its derivative y′ (x + j h2 ) as Taylor series around x gives y(x + j 2h ) = y(x) + j y′ (x +

j

h ′ 2 ) = y (x) +

h 2

y′ (x) +

j

h 2

( j h2 )2 2!

y′′ (x) +

( j h )2 y′′ (x) + 2!2

( j h2 )3 3!

y(3) (x) + · · · ,

( j h )3 y(3) (x) + 3!2

(2.12) y(4) (x) + · · ·

,

Substituting (2.12) into (2.11) represents 7

L[ y(x), h ] =

∑

α j [ y(x) + j

j=0

−

7

∑

h ′ 1 2 h2 ′′ 1 3 h3 (3) y (x) + j y (x) + j y (x) + · · · ] 2 2! 4 3! 8

h β j [ y′ (x) + j

j=0 7

=

1

+

7

∑ [ α j y(x) ] + 2 ∑ [ j α j − 2β j ] h y′ (x) j=0

j=0

+

h ′′ 1 2 h2 (3) 1 3 h3 (4) y (x) + j y (x) + j y (x) + · · · ] 2 2! 4 3! 8

1 4 1 8

7

1

j2 α j − 2 j β j ] h2 y′′ (x)

1

j3 α j − 2

∑ [ 2!

j=0 7

∑ [ 3!

j=0

+ ···

1 2 j β j ] h3 y(3) (x) 2! (2.13)

The difference operator (2.11) and the associated method (2.9) is considered of order p if c0 = c1 = · · · = c p = 0 and c p+1 ̸= 0. In this case

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 6 of 15

7

c0

=

∑ α j = 0,

j=0 7

c1

=

7

∑ ( jα j ) − 2 ∑ β j = 0

j=0 7

c2

=

∑

j=0

j=0

( j2 α 2!

j)

7

− 2 ∑ ( jβ j ) = 0 j=0

.. . 7 ( j5 α j ) ( j4 β j ) −2 ∑ =0 5! j=0 j=0 4!  7

c5

c6

=

=

∑

1 − 20

 4 7 7 ( j6 α j ) ( j5 β j )   − 45 ∑ 6! − 2 ∑ 5! =  5 j=0 j=0  124 4 − 45



  0       0   ̸=   0   0 (2.14)

Therefore, the order of the method is 5 and the error constant is determined by  1  − 20    −4   45  c6 =    5   124  4 − 45

2.2 Stability of the 2-point block BDF with off-step points In this section, the condition for stability of the method is provided. We start by the definition of zero stability and A-stability from [8]. Definition 2.1. The method is said to be zero stable if all the roots of first characteristic polynomial have modulus less than or equal to unity and those of modulus unity are simple. Definition 2.2. The method is said to be A-stable if the stability region covers the entire negative left half plane. The stability region of the method is determined by applying linear test problem y′ = λ y to the obtained solutions as follows

yn+ 1 2

yn+1 yn+ 3 2

yn+2

3 9 3 1 1 yn−1 − yn + yn+1 − yn+ 3 + yn+2 − hλ yn+ 1 , 2 2 60 4 4 5 12 2 32 32 2 1 yn−1 − yn + yn+ 1 − yn+ 3 + yn+2 + 2hλ yn+1 , = 2 2 45 3 9 15 9 1 25 25 225 25 15 = − yn−1 + yn − yn+ 1 + yn+1 − yn+2 + hλ yn+ 3 , 2 2 124 124 31 124 124 31 2 1 32 32 2 = yn−1 − yn + yn+ 1 − 2yn+1 + yn+ 3 + hλ yn+2 . 2 2 135 3 27 15 9 =

(2.15)

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 7 of 15

These equations can be written in matrix form as   3 1 − 94 − 1 + hλ 5 12     32 2   − 32 1 − 2h λ − 9 15 9       25 25 225 15   λ − 1 − h  31  124 31 124    32 32 2 − 27 2 − 15 1 − 9 hλ

yn+ 1 2

yn+1 yn+ 3 2

yn+2





0 − 34

1 60

0

     1  0  45     =    0 − 1   124    2 0 135

0

− 23

0

25 124

0 − 13

 yn− 3 2    yn−1      y 1   n− 2  yn

           (2.16)

If m is number of block and r is the number of points in the block, then n = mr. Here, r = 2 and n = 2m. It follows from [2] as 

y2m+ 1

2     y2m+1  Ym =    y2m+ 3 2   y2m+2



y2(m−1)+ 1

    y2(m−1)+1  Ym−1 =    y2(m−1)+ 3 2   y2(m−1)+2 2









yn+ 1

2         yn+1   =     yn+ 3 2     yn+2



y2m− 3

        y2m−1   =     y2m− 1 2     y2m 2

         





yn− 3

        yn−1   =     yn− 1 2     yn 2

          

Equation (2.16) can be written in the following form AYm = BYm−1 Which the matrix coefficients can be specified as  3 1 − 94 − 12 1 + hλ 5   32  − 32 1 − 2hλ − 29 9 15   A= 225 25  25 − 124 1 − 15  31 31 hλ 124  − 32 27

− 32 15

2

1 − 92 hλ



(2.17) 

0

1 60

    1   0 45   , B =      0 − 1   124   2 0 − 135

0

− 43

0

− 32

0

25 124

0

− 31

         

ˆ where hˆ = hλ is evaluated by det(At − B) as The stability polynomial R(t, h) ˆ = R(t, h) + +

16 2 17248 3 17264 4 16 2 ˆ 1 2 ˆ2 t + t − t − t h− t h 2511 2511 2511 2511 837 4768 3 ˆ 4702 3 ˆ 2 268 3 ˆ 3 20272 4 ˆ 3593 4 ˆ 2 t h+ t h + t h + t h− t h 837 2511 837 2511 837 364 4 ˆ 3 20 4 ˆ 4 t h − t h . 279 93

(2.18)

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 8 of 15

The absolute stability region of the method is determined by solving det(At − B) = 0 or 16 2 17248 3 17264 4 16 2 ˆ 1 2 ˆ2 t + t − t − t h− t h 2511 2511 2511 2511 837 4768 3 ˆ 4702 3 ˆ 2 268 3 ˆ 3 20272 4 ˆ 3593 4 ˆ 2 + t h+ t h + t h + t h− t h 837 2511 837 2511 837 364 4 ˆ 3 20 4 ˆ 4 + t h − t h 279 93 = 0.

ˆ = R(t, h)

(2.19)

For zero stability, we set hˆ = 0 in (2.19) to obtain the first characteristic polynomial R(t, hλ ) = −

16 2 17264 4 17248 3 t + t + t 2511 2511 2511

= 0

(2.20)

Solving (2.20) for t, gives t = −0.0009267840593, t = 0, t = 0, and t = 1. So, the method is zero stable. The stability region of the method is given in Figure 2. The method is considered as A-stable since the stability polynomial covers the entire negative left half plane.

Figure 2: Stability region of the 2-point block BDF with off-step points 2.3 Convergence of the method This section discusses the convergence of the method (2.8) via given theorem in [8]. The necessary and sufficient conditions for any linear multistep method to be convergent are that it be consistent and zero stable. ”Consistency controls the magnitude of the local truncation error while zero stability controls the manner in which the error is propagated at each step of calculation” [8]. In section 2.1, we proved that the order of the method is 5 that shows the method satisfies the consistency condition. Also, in section 2.2, the zero stability of the developed method has been analyzed. Having these two conditions, consistency and zero stability, the 2-point block BDF with off-step points is convergent. 3 Implementation of the method Newton’s iteration is applied for the implementation of the method. First, we define the error in the (i)th iteration to show the accuracy of the method. Let (i)

(i)

error(i) = |yexact − yapproximate |

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 9 of 15

and the maximum error is given by MAXE = max (error(i) ) 1≤i≤NS

The abbreviation NS gives the total number of steps. (i+1) Let yn+ j , j = 12 , 1, 32 , 2 denote the (i + 1)th iterative values of yn+ j , define (i+1)

en+ j

(i+1)

3 1 j = , 1, , 2. 2 2

(i)

= yn+ j − yn+ j ,

(3.21)

Equations in (2.8) can be written as 1 yn+2 + h fn+ 1 + η 1 , F1 = yn+ 1 − 94 yn+1 + 35 yn+ 3 − 12 2

2

2

2

2

32 2 F1 = − 32 9 yn+ 1 + yn+1 + 15 yn+ 3 − 9 yn+2 − 2h f n+1 + η1 , 2

2

25 yn+ 1 F3 = + 31 2 2

−

F2 = − 32 27 yn+ 21

+ 2yn+1 −

225 124 yn+1 + yn+ 32 32 15 yn+ 32

(3.22)

25 + 124 yn+2 − 15 31 h f n+ 3 + η 3 , 2

+ yn+2 −

2

2 9 h f n+2 + η2 .

where η 1 , η1 , η 3 and η2 represent the back values as 2

2

η1 =

1 3 60 yn−1 − 4 yn ,

2

1 25 η1 = − 124 yn−1 + 124 yn ,

η3 = 2

1 2 45 yn−1 − 3 yn ,

η2 =

2 1 135 yn−1 − 3 yn .

(3.23)

Newton’s iteration takes the form  

(i+1) n+ 21

y

    (i+1)  yn+1    (i+1)  y 3  n+ 2   (i+2) yn+2



 y

(i) n+ 21

        (i)   yn+1   =   (i)   y 3   n+ 2     (i) yn+2

                 −                

(i)

∂ F1 2 (i)

∂y

n+ 12 (i)

∂ F1

(i)

∂y

n+ 12 (i)

∂ F3 2 (i)

∂y

n+ 12 (i)

∂ F2

(i)

∂y

n+ 12

(i)

∂ F1

2 (i)

∂ yn+1

(i)

∂ F1 2 (i)

∂y

n+ 32

(i)

∂ F1

(i) ∂ yn+1

(i)

∂ F1

(i)

∂ F3

2 (i)

∂ yn+1

(i)

∂y

n+ 32 (i)

∂ F3 2 (i)

∂y

n+ 32

(i)

∂ F2

(i) ∂ yn+1

(i)

∂ F2

(i)

∂y

n+ 32

(i)

∂ F1

2 (i)

∂ yn+2 (i)

∂ F1

(i) ∂ yn+2

(i)

∂ F3

2 (i)

∂ yn+2 (i)

∂ F2

(i) ∂ yn+2

−1                      



(i)

F1

 2    (i)  F1  .  (i)  F3  2   (i) F2

       ,     

(3.24)

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 10 of 15

Equation (3.24) using (3.21) is equivalent to  (i) (i)                       |

∂ F1

∂ F1

2 (i)

2 (i)

∂y

(i)

(i)

(i)

(i)

(i)

(i)

(i)

∂ yn+2

n+ 32 (i)

∂ F3

2 (i)

∂ F1

∂y

(i)

∂ F3

(i)

∂ F1

∂ yn+1

n+ 1 2

∂ yn+2

n+ 32

∂ F1

∂y

2 (i)

∂y

(i)

∂ F1

(i)

∂ F1

2 (i)

∂ yn+1

n+ 1 2

(i)

∂ F1

(i)

∂ F3

2 (i)

∂ F3

2 (i)

2 (i)

∂y

∂ yn+1

∂y

∂ yn+2

(i) ∂ F2 (i) ∂y 1 n+ 2

(i) ∂ F2 (i) ∂ yn+1

(i) ∂ F2 (i) ∂y 3 n+ 2

(i) ∂ F2 (i) ∂ yn+2

n+ 1 2

n+ 32

{z

                   .               

e

(i)

F1

 2       (i)   F1     = −  (i)   F3   2      (i) F2

(i+1)

en+1 e





(i+1) n+ 12

(i+1) n+ 23 (i+1)

en+2

       .     

(3.25)

}

Jacobian matrix

Let J denote the Jacobian matrix, it can written as  (i) ∂f

n+ 1 2 (i) ∂y 1 n+ 2

 1+h      (i)  32 ∂ fn+1  − 9 − 2h (i)  ∂y 1  n+ 2  J=  (i) ∂f   25 − 15 h n+ 32  31 31 (i) ∂y 1  n+ 2    (i)   − 32 − 2 h ∂ fn+2 (i) 27 9

− 94 + h

1 − 2h

∂f

∂y

n+ 12

3 5

(i)

∂ fn+1 (i) ∂ yn+1

15 − 225 124 − 31 h

2 − 29 h

(i)

n+ 12 (i) ∂ yn+1

∂f

32 15

∂f

+h

− 2h

(i)

∂ fn+2 (i) ∂ yn+1

∂f

1 − 12 +h

(i)

∂ fn+1

− 92 − 2h

(i)

∂y

1 − 15 31 h

∂f

(i)

n+ 32 (i) ∂y 3 n+ 2

2 − 32 15 − 9 h

25 124

(i)

∂ fn+2 (i)

∂y

n+ 32

− 15 31 h

1 − 29 h

(i)

n+ 12 (i) ∂ yn+2

n+ 32

(i)

n+ 23 (i) ∂ yn+1

(i)

n+ 12 (i) ∂y 3 n+ 2

(i)

∂ fn+1 (i) ∂ yn+2

∂f

(i)

n+ 23 (i) ∂ yn+2

(i)

∂ fn+2 (i) ∂ yn+2

                      

(i+1)

Therefore, the values en+ j , j = 12 , 1, 32 , 2 can be approximated and then the solution values yn+ j , j = 21 , 1, 32 , 2 are computed from (i+1)

yn+ j

(i)

(i+1)

= yn+ j + en+ j ,

3 1 j = , 1, , 2. 2 2

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 11 of 15

4 Numerical results In this section, we will compare the numerical results of the 2-point block BDF method with off-step points of order 5 with the fifth order 2-point block BDF of constant step size [14]. To evaluate the performance of the methods, the following stiff problems are tested. Problem 1: [5] y′ = −100(y − x) + 1,

y(0) = 1,

0 ≤ x ≤ 10,

y(0) = 1,

0 ≤ x ≤ 2,

y1 (0) = 2, y1 (0) = 0.

0 ≤ x ≤ 20,

Exact Solution y(x) = e−100x + x. Problem 2: [4] y′ = −20 y + 20 sin x + cos x, Exact Solution y(x) = sin x + e−20x . Problem 3: [5] y′1 = −20y1 − 19y2 , y′2 = −19y1 − 20y2 , Exact Solution y1 = e−39x + e−x , y2 = e−39x − e−x . Problem 4: [6] y′1 = 198y1 + 199y2 , y′2 = −398y1 − 399y2 ,

y1 (0) = 1, y1 (0) = −1.

0 ≤ x ≤ 10,

y1 (0) = 13 , y1 (0) = 13 .

0 ≤ x ≤ 1,

Exact Solution y1 = e−x , y2 = −e−x . Problem 5: [4] y′1 = 32y1 + 66y2 + 23 x + 23 , y′2 = −66y1 − 133y2 − 13 x − 13 , Exact Solution y1 = 23 x + 23 e−x − 13 e−100x , y2 = − 13 x − 13 e−x + 32 e−100x . The following abbreviations are used in the tables: 2BBDF(5) fifth order 2-point block BDF method 2OBBDF 2-point block BDF method with off-step points of order 5 h constant step size MAXE maximum global error TIME the execution time in micro seconds The results of maximum global error and the execution time are given in Tables 1-5.

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 12 of 15

Table 1: Numerical results for problem 1. h 10−2 10−3 10−4 10−5 10−6

METHOD 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF

MAXE 1.96146e-2 1.95754e-2 5.69310e-2 5.59424e-2 7.18212e-3 7.16455e-3 7.33994e-4 7.33813e-4 7.35582e-5 7.35564e-5

TIME 1.24015e-3 2.37103e-3 9.79317e-3 1.69611e-2 6.46887e-2 7.71449e-2 3.48632e-1 5.57695e-1 7.35144e0 9.49209e0

Table 2: Numerical results for problem 2. h 10−2 10−3 10−4 10−5 10−6

METHOD 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF

MAXE 8.85478e-2 8.05923e-2 1.40157e-2 1.39480e-2 1.46428e-3 1.46355e-3 1.47063e-4 1.47055e-4 1.47126e-5 1.47126e-5

TIME 4.25750e-4 6.33001e-4 1.76333e-3 1.56188e-3 1.70041e-2 1.90020e-2 1.68796e-1 1.50976e-1 2.14071e0 3.43859e0

Table 3: Numerical results for problem 3. h 10−2 10−3 10−4 10−5 10−6

METHOD 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF

MAXE 8.81087e-2 7.00088e-2 2.61263e-2 2.58857e-2 2.84765e-3 2.84492e-3 2.87178e-4 2.87150e-4 2.87419e-5 2.87417e-5

TIME 1.96250e-3 3.07107e-3 2.26088e-2 3.00581e-2 1.94097e-1 2.92691e-1 2.80007e0 4.18854e0 3.34886e1 5.30707e1

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 13 of 15

Table 4: Numerical results for problem 4. h 10−2 10−3 10−4 10−5 10−6

METHOD 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF

MAXE 7.13926e-3 7.17251e-3 7.33994e-4 7.33813e-4 7.35582e-5 7.35564e-5 7.35741e-6 7.35740e-6 7.35744e-7 7.35775e-7

TIME 1.22776e-3 3.79395e-3 1.66640e-2 2.66598e-2 9.89231e-2 1.38798e-1 1.06522e0 2.46586e0 1.83490e1 2.63500e1

Table 5: Numerical results for problem 5. h 10−2 10−3 10−4 10−5 10−6

METHOD 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF 2BBDF(5) 2OBBDF

MAXE 1.21580e-2 1.20347e-2 3.79477e-2 3.72766e-2 4.78743e-3 4.77571e-3 4.89264e-4 4.89142e-4 4.90322e-5 4.90310e-5

TIME 3.32570-4 5.87940e-4 3.76255e-3 5.66602e-3 1.32177e-2 2.15800e-2 8.46313e-2 1.44442e-1 9.19200e-1 2.37217e0

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 14 of 15

5 Conclusion A 2-point block BDF method with off-step points of order 5 is formulated in this paper. The developed method is on solving stiff ODEs to produce two solution values with off-step points simultaneously at each iteration. The method is shown to be A-stable and convergent. Accuracy and the execution time of the derived method are compared with the existing fifth order 2-point block BDF method. Numerical results obtained indicate that the methods are competitive in terms of accuracy and the execution time. Acknowledgements The authors would like to thank Institute for Mathematical Research, Universiti Putra Malaysia for supporting this research. References [1] J. Cash, On the integration of stiff systems of ODEs using extended backward differentiation formulae, Numerical Mathematik, 34 (3) (1980) 235-246. [2] S. O. Fatunla, Block methods for second order ODEs, International Journal of Computer Mathematics, 41 (1991) 55-63. http://dx.doi.org/10.1080/00207169108804026 [3] C. W. Gear, Numerical initial value problems in ordinary fifferential equations, COMM. ACM, 14 (1971) 185190. [4] Z. B. Ibrahim, Block multistep methods for solving ordinary differential equations, PhD Thesis, Universiti Putra Malysia, (2006). [5] Z. Ibrahim, K. I. Othman, M. Suleiman, Fixed coefficients block backward differentiation formulas for the numerical solution of stiff ordinary differential equations, Europeam Journal of Scientific Research, 21 (3) (2008) 508-520. [6] Z. B. Ibrahim, K. I. Othman, M. Suleiman, Implicit R-point block backward differentiation formula for solving first-order stiff odes, Applied Mathematics and Computation, 186 (2007) 558-565. http://dx.doi.org/10.1016/j.amc.2006.07.116 [7] G. M. Kumleng, S. O. Adee, Y. Skwame, Implicit two step Adam-Moulton hybrid block method with two offstep points for solving stiff ordinary differential equations, Journal of Natural Sciences Research, 3 (9) (2013) 77-82. [8] J. Lambert, Computational methods in ordinary differential equations, New York, John Wiley and Sons, Inc, (1973). [9] M. Lautsch, An implicit off-step point method for the integration of stiff differential equations, Computing, 31 (2) (1983) 177-183. http://dx.doi.org/10.1007/BF02259913 [10] W. E. Milne, Numerical solution of differential equations, John Wiley, New York, (1953). [11] J. B. Rosser, A Runge-Kutta for All Seasons, Siam Review, 9 (3) (1967) 417-452. http://dx.doi.org/10.1137/1009069 [12] L. F. Shampine, H. A. Watts, Block implicit one-step methods, Mathematics of Computation, 23 (1969) 731-740. http://dx.doi.org/10.1090/S0025-5718-1969-0264854-5

International Scientific Publications and Consulting Services

Journal of Soft Computing and Applications http://www.ispacs.com/journals/jsca/2014/jsca-00039/

Page 15 of 15

[13] H. A. Watts, L. F. Shampine, A-stable block implicit one-step methods, BIT, 12 (1972) 252-266. http://dx.doi.org/10.1007/BF01932819 [14] N. A. A. M. Nasir, Z. B. Ibrahim, K. I. Othman, M. B. Suleiman, Numerical Solution of First Order Stiff Ordinary Differential Equations using Fifth Order Block Backward Differentiation Formulas, Sains Malaysiana, 41 (4) (2012) 489-492.

International Scientific Publications and Consulting Services