Research Journal of Mathematics and Statistics 5(4): 32-42, 2013
ISSN: 2042-2024, e-ISSN: 2040-7505
© Maxwell Scientific Organization, 2013
Submitted: August 31, 2013
Accepted: September 10, 2013
Published: November 30, 2013
A Modified Block Adam Moulton (MOBAM) Method for the Solution of Stiff Initial Value
Problems of Ordinary Differential Equations
G.M. Kumleng, J.P. Chollom and S. Longwap
Department of Mathematics, University of Jos, P.M.B. 2084, Plateau State, Nigeria
Abstract: Stiff ordinary differential equations pose computational difficulties as they present severe step size
restrictions on the numerical methods to be used. Construction of numerical methods that possess suitable stability
properties for the solution of such systems has been the target of many researchers. Development of methods
suitable for these systems of equations has been either through the use of derivative of the solution or by introducing
off-step points, additional stages or super future points. These processes have been exploited in Runge-Kutta
methods or linear multistep methods. In this study, an improved class of linear multistep block method has been
constructed based on Adams Moulton block methods. The improved methods are shown to be A-stable, a property
desirable to handle stiff ODEs. Methods of uniform orders 10 and 11 have been constructed. The efficiency of the
new methods tested on stiff systems of ODEs and the results reveal that the MOBAM methods compare favourably
with results obtained using the state of the art Matlab Ode23 solver.
Keywords: Block method, initial value problems, non-linear, stability
INTRODUCTION
In this study, we pursue the path of Adams
Moulton methods by constructing A-stable block linear
multistep methods of uniform order 10 and 11. These
methods are obtained by modifying the Adams Moulton
method of step number 9 by reducing its interpolation
step number from 8 to 3. This approach is similar to
that of Brugnano and Trigiante (1998) where they
developed the Generalized Adams methods.
The modified methods are constructed using the
concept of multistep collocation of Lie and Norsett
(1989) which Onumanyi et al. (1994, 1999) referred to
as block linear multistep methods. The block methods
are arrived at by evaluating the continuous formulation
of the new method at grid and off grid points to yield
the discrete schemes used as block integrators. The
methods are applied simultaneously producing selfstarting methods thus eliminating the issue of overlap of
pieces of solutions.
Stiff initial value problems of ordinary differential
equations were realized in 1952 arising from the study
of chemical kinetics. Since then, many researches have
been involved in the study and development of suitable
numerical methods to handle them. Curtiss and
Hirschfelder (1952), Mitchell and Craggs (1953) and
Gear (1971) developed the Backward Differential
Formulae which were used to solve stiff problems
arising from chemical kinetics. Intensive work done in
this regard yielding several efficient numerical
algorithms has appeared in the literature. Because of the
severe restriction on step size placed on Adams
Moulton methods for the solution of stiff ODEs,
Garfinket et al. (1978) introduced the concept of Astability for linear multistep methods. This property
became the minimum requirement for any linear
multistep method to be used for the solution of stiff
ODEs. Achieving A-stability was difficult, other
stability requirements such as A(α)-stability and stiffly
stability were considered. Chakrararti and Kamel
(1983) developed stiffly stable second derivative
methods with high order and improved stability regions
to handle stiff problems. Several other researchers such
as Samir (2013), Evelyn and Renante (2006), Song
(2010), Vlachos et al. (2009) and Ali and Gholamreza
(2012) developed improved and more friendly
numerical methods with improved stability properties
that could handle stiff ODEs with various degrees of
stiffness.
CONSTRUCTION OF THE NEW METHOD
The k-step general linear multistep method for
numerically solving the differential equation:
y ' =f ( x, y ), y ( x0 ) =y0 , x ∈ [a, b], y ∈ R (1)
is described by the linear difference equation:
k
∑α
j =0
k
j
( x) yn+ j =h∑ β j ( x) f n+ j
j =0
Corresponding Author: G.M. Kumleng, Department of Mathematics, University of Jos, P.M.B. 2084, Plateau State, Nigeria
32
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Res. J. Math. Stat., 5(4): 32-42, 2013
With h being the step size, α j (x) and β j (x) being the continuous coefficients of the method.
The new improved nine step method based on the Adams Moulton method is defined as:
m
yn + v − α v −1 ( x) yn + v −1 =
h∑ β j ( x) f n + j
(3)
j =0
where, v = k2−1 , α v −1 and β j ( x) are the continuous coefficients of the method, m is the number of distinct collocation
points, h is the step size and k = m − 1 = 3,5,7,9,...
Using the method in Onumanyi et al. (1994, 1999) and further studied by Kumleng et al. (2012) and Chollom
et al. (2012) the method (3) for v = k2−1 and j =0, 1… 9 sequence in the matrix form:
DC = I
(4)
where, D = C −1 , C −1 being the elements of the continuous coefficients of the method.
Construction of MOBAM K = 9: This method has its general form as:
k
(5)
y ( x) = α 3 ( x) yn+3 + h∑ β j ( x) f n+ j , j = 0,1,...,9
j =0
Using the procedure in Onumanyi et al. (1999), we obtain the D matrix for (5) as:
1 xn + 3

0 1
0 1

0 1
0 1

D = 0 1
0 1

0 1

0 1
0 1

0 1
xn2+ 3
2 xn
2 xn +1
2 xn + 2
2 xn + 3
2 xn + 4
2 xn + 5
2 xn + 6
xn3+ 3
3xn2
3xn2+1
3xn2+ 2
3xn2+ 3
3xn2+ 4
3xn2+ 5
3xn2+ 6
xn4+ 3
4 xn3
4 xn3+1
4 xn3+ 2
4 xn3+ 3
4 xn3+ 4
4 xn3+ 5
4 xn3+ 6
xn5+ 3
5 xn4
5 xn4+1
xn6+ 3
6 xn5
6 xn5+1
xn7+ 3 xn8+ 3
7 xn6
8 xn7
6
7 xn +1 8 xn7+1
5 xn4+ 2
5 xn4+ 3
5 xn4+ 4
5 xn4+ 5
5 xn4+ 6
6 xn5+ 2
6 xn5+ 3
6 xn5+ 4
6 xn5+ 5
6 xn5+ 6
7 xn6+ 2
7 xn6+ 3
7 xn6+ 4
7 xn6+ 5
7 xn6+ 6

xn9+ 3
x10
n+3

8
9 xn
10 xn9 
9 xn8+1 10 xn9+1 

9 xn8+ 2 10 xn9+ 2 
9 xn8+ 3 10 xn9+ 3 

9 xn8+ 4 10 xn9+ 4 
9 xn8+ 5 10 xn9+ 5 

9 xn8+ 6 10 xn9+ 6 

9 xn8+ 7 10 xn9+ 7 
8
9
9 xn +8 10 xn +8 

9 xn8+ 9 10 xn9+ 9 
8 xn7+ 2
8 xn7+ 3
8 xn7+ 4
8 xn7+ 5
8 xn7+ 6
2 xn + 7 3xn2+ 7 4 xn3+ 7 5 xn4+ 7 6 xn5+ 7 7 xn6+ 7 8 xn7+ 7
2 xn +8 3xn2+8 4 xn3+8 5 xn4+8 6 xn5+8 7 xn6+8 8 xn7+8
2 xn + 9 3xn2+ 9 4 xn3+ 9 5 xn4+ 9 6 xn5+ 9 7 xn6+ 9 8 xn7+ 9
(6)
Using the Maple software, the inverse of the matrix (6) is obtained and represented as C, the elements of which
yield the continuous coefficients α 3 (x) and β j (x), j = 0,1,…9 of the method (5) as:
α 3 ( x) = 1
µ
µ
µ
6515µ
4523µ
19µ
3013µ
5µ
29µ
3591h
β 0 ( x) = ( − 12800
+ µ − 7129
5040h + 6048h − 9072h + 128h − 103680h + 1344h − 96768h + 72576h − 3628800h )
2
β1 ( x ) = ( −
147429h
89600
9µ 2
2h
+
4609µ 3
840h 2
−
3
4
5
2
3
4
14139µ 4
4480h 3
+
7667µ 5
7200h 4
−
6
7807µ 6
34560h 5
+
7
5
−
8
6
11µ 7
360h 6
+
9
7
59µ 8
23040h 7
−
10
8
11µ 9
90720h8
+
9
µ 10
403200h 9
)
µ
µ
µ
µ
µ
563µ
7µ
43µ
279h
)
β 2 ( x) = ( 1600
− 9 µh + 5869
− 20837
+ 24901
− 6787
+ 5040
− 720
+ 90720
− 100800
h
h
h
h
420h
2240h
7200h
8640h
2
3
4
2
h
β 3 ( x) = ( − 13273
5600 +
β 4 ( x) = (
19107h
8960
−
14µ 2
h
63µ 2
4h
−
3
6289µ 3
270h 2
6499µ 3
240h 2
+
5
+
−
6
4
72569µ 4
4320h 3
6519µ 4
320h 3
+
−
7
5
4013µ 5
600h 4
122249µ 5
14400h 4
8
6
+
13873µ 6
8640h 5
−
36769µ 6
17280h 5
9
7
−
401µ 7
1680h 6
+
3313µ 7
10080h 6
+
8
31µ 8
1440h 7
−
10
−
353µ 8
11520h 7
9
7µ9
6480h8
+
+
41µ 9
25920h8
µ 10
43200h 9
−
)
(7)
µ 10
28800h 9
)
µ
µ
63µ
265µ
1091µ
5273µ
32773µ
305µ
67µ
h
β 5 ( x) = ( − 73809
44800 + 5 h − 12h + 64h − 720h + 17280h − 1008h + 2304h − 648h + 28800h )
2
3
4
2
5
3
6
4
7
5
8
6
9
10
7
8
9
µ
7µ
6709µ
84307µ
10279µ
9823µ
313µ
53µ
13µ
h
β 6 ( x) = ( 5039
5600 − h + 540h − 8640h + 2400h − 8640h + 1680h − 2880h + 12960h − 43200h )
2
3
4
2
3663h
β 7 ( x) = ( − 11200
+
18µ 2
7h
−
5
3
967µ 3
210h 2
+
6
4
4101µ 4
1120h 3
−
2939µ 5
1800h 4
+
7
8
5
6
3817µ 6
8640h 5
373µ 7
5040h 6
−
9
7
+
10
8
151µ 8
20160h 7
−
9
19µ 9
45360h8
+
µ 10
100800h9
)
µ
µ
µ
3407µ
3487µ
347µ
41µ
37 µ
909h
)
β 8 ( x) = ( 12800
− 916µh + 33360
− 1823
+ 10579
− 34560
+ 20160
− 23040
+ 362880
− 403200
2240h
28800h
h
h
h
h
h
h
2
3
4
2
3
5
6
4
5
7
8
6
9
7
10
8
9
µ
µ
761µ
29531µ
89µ
1069µ
13µ
25h
)
β 9 ( x) = (− 3584
+ 18µ h − 7560
+ 362880
− 2400
+ 103680
− 560µ h + 69120
− 90720
+ 3628800
h
h
h
h
h
h
h
2
3
4
2
3
5
6
4
7
5
8
6
33
9
7
10
8
9
Res. J. Math. Stat., 5(4): 32-42, 2013
Substituting the continuous coefficients (7) into (5) yields the continuous form of the MOBAM method (8):
6515 µ
4523 µ
19 µ
3013 µ
5µ
29 µ
µ
µ
µ
3591h
y ( xn + µ ) = y n+3 + (− 12800
+ µ − 7129
5040 h + 6048 h 2 − 9072 h 3 + 128 h 4 − 103680 h 5 + 1344 h 6 − 96768 h 7 + 72576 h8 − 3628800 h 9 ) f n +
2
3
4
5
6
7
8
9
10
9µ
4609 µ
14139 µ
7667 µ
7807 µ
11µ
59 µ
11µ
µ
h
(− 147429
89600 + 2 h − 840 h 2 + 4480 h 3 − 7200 h 4 + 34560 h 5 − 360 h 6 + 23040 h 7 − 90720 h8 + 403200 h 9 ) f n +1 +
2
3
4
5
6
7
8
9
10
563 µ
7µ
43 µ
µ
µ
µ
µ
µ
279 h
− 9 µh + 5869
− 20837
+ 24901
− 6787
+ 5040
− 720
+ 90720
− 100800
( 1600
) f n+2 +
h6
h7
h8
h9
420 h 2
2240 h 3
7200 h 4
8640 h 5
2
3
4
5
6
7
8
9
10
14 µ
6289 µ
72569 µ
4013 µ
13873 µ
401µ
31µ
7µ
µ
h
(− 13273
5600 + h − 270 h 2 + 4320 h 3 − 600 h 4 + 8640 h5 − 1680 h 6 + 1440 h 7 − 6480 h8 + 43200 h 9 ) f n +3 +
2
3
4
5
6
7
8
9
10
63 µ
6499 µ
6519 µ
122249 µ
36769 µ
3313 µ
353 µ
41µ
µ
h
( 19107
8960 − 4 h + 240 h 2 − 320 h3 + 14400 h 4 − 17280 h5 + 10080 h 6 − 11520 h 7 + 25920 h8 − 28800 h9 ) f n + 4 +
2
3
4
5
6
7
8
9
10
(8)
63 µ
265 µ
1091µ
5273 µ
32773 µ
305 µ
67 µ
µ
µ
h
(− 73809
44800 + 5 h − 12 h 2 + 64 h3 − 720 h 4 + 17280 h5 − 1008 h 6 + 2304 h 7 − 648 h8 + 28800 h9 ) f n + 5 +
2
3
4
5
6
7
8
9
10
7µ
6709 µ
84307 µ
10279 µ
9823 µ
313 µ
53 µ
13 µ
µ
h
( 5039
5600 − h + 540 h 2 − 8640 h3 + 2400 h 4 − 8640 h5 + 1680 h 6 − 2880 h 7 + 12960 h8 − 43200 h9 ) f n + 6 +
2
3
4
5
6
7
8
9
10
373 µ
151µ
19 µ
µ
µ
µ
µ
µ
3663 h
+ 187µh − 967
+ 4101
− 2939
+ 3817
− 5040
+ 20160
− 45360
+ 100800
(− 11200
) f n+7 +
h6
h7
h8
h9
210 h 2
1120 h3
1800 h 4
8640 h5
2
3
4
5
6
7
8
9
10
3407 µ
3487 µ
347 µ
41µ
37 µ
µ
µ
µ
909 h
− 916µh + 33360
− 1823
+ 10579
− 34560
+ 20160
− 23040
+ 362880
− 403200
( 12800
) f n +8 +
h2
h5
h6
h7
h8
h9
2240 h3
28800 h 4
2
3
4
5
6
7
8
9
10
761µ
29531µ
89 µ
1069 µ
13 µ
µ
µ
25 h
+ 18µ h − 7560
+ 362880
− 2400
+ 103680
− 560µ h6 + 69120
− 90720
+ 3628800
(− 3584
) f n +9
h2
h3
h4
h5
h7
h8
h9
2
3
4
5
6
7
8
9
10
where, µ = x − x n and µ ∈ [0, 9h ] . Evaluating the continuous scheme (8) at the following points
µ = 0, h, 2h, 4h, 5h, 6h, 7 h, 8h, 9h yields the MOBAM method (9) for k = 9 used in block for the solution of ODEs:
y n +3 − y n =
h
89600
(25137 f n + 147429 f n+1 − 15624 f n+ 2 + 212368 f n+3 − 191070 f n+ 4 +
147618 f n+5 − 80624 f n+6 + 29304 f n+7 − 6363 f n+8 + 625 f n+9 )
h
yn+1 − yn+3 = 113400
(729 f n − 38938 f n+1 − 156340 f n+ 2 − 18742 f n+3 − 28234 f n+ 4 +
24266 f n+5 − 13492 f n+6 + 4910 f n+7 − 1063 f n+8 + 104 f n+9 )
y n + 2 − y n +3 =
h
7257600
y n + 4 − y n +3 =
h
7257600
(−10625 f n + 163531 f n+1 − 3133688 f n+ 2 − 5597072 f n+3 + 2166334 f n+ 4
− 1295810 f n+5 + 617584 f n+6 − 206072 f n+7 + 42187 f n+8 − 3969 f n+9 )
(−3969 f n + 50315 f n+1 − 342136 f n+ 2 + 3609968 f n+3 + 4763582 f n+ 4 −
1166146 f n+5 + 462320 f n+6 − 141304 f n+7 + 27467 f n+8 − 2497 f n+9 )
h
yn+5 − yn+3 = 113400
(−23 f n + 334 f n+1 − 2804 f n+ 2 + 46378 f n+3 + 139030 f n+ 4 + 46378 f n+5 −
2804 f n+6 + 334 f n+7 − 23 f n+8 )
y n+ 6 − y n+3 =
h
89600
(−49 f n + 603 f n+1 − 3960 f n+ 2 + 42352 f n+3 + 95454 f n+ 4 + 95454 f n+5 +
42352 f n+6 − 3960 f n+7 + 603 f n+8 − 49 f n+9 )
h
yn+7 − yn+3 = 14175
(13 f n+1 − 224 f n+ 2 + 5494 f n+3 + 17632 f n+ 4 + 10870 f n+5 + 17632 f n+6 +
5494 f n+7 − 224 f n+8 + 13 f n+9 )
y n +8 − y n + 3 =
h
290304
y n+9 − y n+3 =
h
1400
(−425 f n + 4675 f n+1 − 25400 f n+ 2 + 171760 f n+3 + 247150 f n+ 4 +
381550 f n+5 + 185200 f n+6 + 387400 f n+7 + 101635 f n+8 − 2025 f n+9 )
(9)
(9 f n − 90 f n+1 + 396 f n+ 2 − 598 f n+3 + 3798 f n+ 4 − 1494 f n+5 +
3980 f n+6 − 306 f n+7 + 2313 f n+8 + 392 f n+9 )
Construction of Hybrid MOBAM K = 9, µ = 172 : This is the hybrid form of the MOBAM family for k=9. An off
step point is inserted in (5) to give its general form as:
k
y ( x) = α 3 ( x) yn+3 + h∑ β j ( x) f n+ j + hβ u ( x) f n+u , j = 0,1,...,9, µ = 172
j =0
Following a similar procedure as in above section gives the matrix:
34
(10)
Res. J. Math. Stat., 5(4): 32-42, 2013
1 xn+3

0 1
0 1

0 1
0 1

0 1
D=
0 1

0 1

0 1
0 1

0 1
0 1

xn2+3
2 xn
2 xn+1
2 xn+ 2
2 x n +3
2 xn+ 4
2 xn +5
2 xn+6
2 xn + 7
2 x n +8
2 xn+ 17
2
xn3+3
3xn2
3xn2+1
3xn2+ 2
3xn2+3
3xn2+ 4
3xn2+5
3xn2+6
3xn2+7
3 xn2+8
3 xn2+ 17
2
xn4+3
4 xn3
4 xn3+1
4 xn3+ 2
4 xn3+3
4 xn3+ 4
4 xn3+5
4 xn3+6
4 xn3+7
4 xn3+8
4 xn3+ 17
2
xn5+3
5 xn4
5 xn4+1
5 xn4+ 2
5 xn4+3
5 xn4+ 4
5 xn4+5
5 xn4+6
5 xn4+7
5 xn4+8
5 xn4+ 17
2
xn6+3
6 xn5
6 xn5+1
6 xn5+ 2
6 xn5+3
6 xn5+ 4
6 xn5+5
6 xn5+6
6 xn5+7
6 xn5+8
6 xn5+ 17
2
xn7+3
7 xn6
7 xn6+1
7 xn6+ 2
7 xn6+3
7 xn6+ 4
7 xn6+5
7 xn6+6
7 xn6+7
7 xn6+8
7 xn6+ 17
2
xn8+3
8 xn7
8 xn7+1
8 xn7+ 2
8 xn7+3
8 xn7+ 4
8 xn7+5
8 xn7+6
8 xn7+7
8 xn7+8
8 xn7+ 17
2
xn9+3
9 xn8
9 xn8+1
9 xn8+ 2
9 xn8+3
9 xn8+ 4
9 xn8+5
9 xn8+6
9 xn8+7
9 xn8+8
9 xn8+ 17
2
x10
n +3
10 xn9
10 xn9+1
10 xn9+ 2
10 xn9+3
10 xn9+ 4
10 xn9+5
10 xn9+6
10 xn9+7
10 xn9+8
10 xn9+ 17
2 xn +9
3 xn2+9
4 xn3+9
5 xn4+9
6 xn5+9
7 xn6+9
8 xn7+9
9 xn8+9
10 xn9+9
2

x11
n +3
10 
11xn 

11x10
n +1

11x10
n+ 2 
10 
11xn+3

11x10
n+ 4 

11x10
n +5 
10
11xn+6 

11x10
n+7 

11x10
n +8
10 
11xn+ 17 
2

11x10
n +9 
(11)
The continuous coefficients of the method (10) are similarly obtain from the inverse of (11) as:
α 3 ( x) = 1

β 0 ( x) =  −

4581629h
126233µ 2
610807µ 3 366199µ 4 1205177µ 5
76871µ 6
+µ−
+
−
+
−
+
16755200
85680h
514080h 2
616896h 3
6168960h 4 1762560h 5
3419µ 7
1123µ 8
167µ 9
107µ 10
µ 11
−
+
−
+
514080h 6 1645056h 7
3701376h 8
61689600h 9
33929280h 10

β 1 ( x) =  −




1693113h 51µ 2 83393µ 3
7921µ 4
9335µ 5 12937µ 6
+
−
+
−
+
985600
10h
12600h 2
1920h 3
6048h 4
34560h 5
−

677µ 7
149µ 8
µ9
µ 10
µ 11

+
−
+
−
3326400
11200h 6
23040h 7
2268h 8
57600h 9
 407367h 153µ 2 104813µ 3
401181h 4
3713351µ 5 165181µ 6
−
+
−
+
−
+
13h
112320h 5
5460h 2
29120h 3
655200h 4
 8008000
β 2 ( x) = 

8179µ 7
1039µ 8
2299µ 9
103µ 10
µ 11

−
+
−
+
32760h 6
37440h 7 1179360h 8 1310400h 9
720720h 10 

β 3 ( x) =  −

759127µ 5
332153µ 6
h 238µ 2 111953µ 3 1384609h 4
4461461
+
−
+
−
+
11h
1355200
2970h 2
47520h 3
59400h 4
95040h 5
−
8581µ 7
2257µ 8
367µ 9
101µ 10
µ 11
+
−
+
−
13860h 6
31680h 7
71280h 8
475200h 9
261360h 10
 376763h 119µ
−
4h
 98560
β 4 ( x) = 
2
+
115523µ 3 123821h 4
2547601µ 5
96619µ 6
−
+
−
+
2160h 2
2880h 3
129600h 4
17280h 5
3103µ 7
1403µ 8
703µ 9
11µ 10
µ 11
−
+
−
+
3024h 6
11520h 7
77760h 8
28800h 9 142560h 10


β 5 ( x) =  −






1882809h 153µ 2
23533µ 3
20667h 4 109279µ 5 109723µ 6
+
−
+
−
+
−
492800
5h
420h 2
448h 3
5040h 4
17280h 5
(12)

1727µ 7
337µ 8
101µ 9
97µ 10
µ 11

+
−
+
−
1440h 6
2304h 7
9072h 8
201600h 9 110880h 10 
 360737h 119µ 2 119093µ 3 318847µ 4
381983µ 5
45733µ 6
−
+
−
+
−
+
5h
2700h 2
8640h 3
21600h 4
8640h 5
 123200
β 6 ( x) = 
12893µ 7
23µ 8
43µ 9
19µ 10
µ 11
−
+
−
+
12600h 6
180h 7
4320h 8
43200h 9 118800h 10

β 7 ( x) =  −




15627h 102µ 2 17159µ 3 77453µ 4 423559µ 5
29467µ 6
+
−
+
−
+
−
8800
7h
630h 2
3360h 3
37800h 4
8640h 5

1693µ 7
863µ 8
103µ 9
31µ 10
µ 11

+
−
+
−
2520h 6 10080h 7 15120h 8 100800h 9 166320h10 
 1140903h 153µ 2 60439µ 3 2457µ 4 1521413µ 5 80437µ 6
−
+
−
+
−
+
16h
3360h 2
160h 3
201600h 4
3456oh 5
 985600
β 8 ( x) = 
4693µ 7
1391µ 8
1777µ 9
13µ 10
µ 11
−−
+
−
+
6
7
8
9
10080h
23040h
362880h
57600h
221760h10
35



Res. J. Math. Stat., 5(4): 32-42, 2013

β ( x ) =  −
17
2

3046144h 65536µ 2 116801536µ 3 1334272µ 4 148209664µ 5
+
−
+
−
+
4679675
12155
11486475h 2
153153h 3
34459425h 4
9728µ 6
3085312µ 7
256µ 8
59392µ 9
512µ 10
1024µ 11
−
+
−
+
−
7293h 5 11486475h 6 7293h 7
20675655h 8 3828825h 9 379053675h10



 22423h 17 µ 2
4499µ 3 556819µ 4 345019µ 5
24581µ 6
−
+
−
+
−
+
18h
2520h 2
362880h 3
453600h 4 103680h 5
 197120
β 9 ( x ) = 
83µ 7
437 µ 8
71µ 9
89 µ 10
µ 11
−
+
−
+
1728h 6 69120h 7 136080h 8 3628800h 9 1995840h10

 f n +9

Substituting the continuous coefficients (12) into (10) produces the continuous form of the new MOBAM
method for k = 9, µ = 172 as:
 4581629h
126233µ 2
610807 µ 3
366199 µ 4
1205177 µ 5
76871µ 6
−
+µ −
+
−
+
−
+
y(µ + x n ) = y n+3 + 
85680h
514080h 2
616896h 3
6168960h 4
1762560h 5
 16755200
7
8
9
10
11

µ
3419 µ
1123µ
167 µ
107 µ
 fn +
−
+
−
+
514080h 6
1645056h 7
3701376h 8
61689600h 9
33929280h 10 

 1693113h
51µ 2
83393µ 3
7921µ 4
9335µ 5
12937 µ 6

 − 985600 + 10h − 12600h 2 + 1920h 3 − 6048h 4 + 34560h 5

−

µ9
µ 10
µ 11
677 µ 7
149 µ 8
 f n +1 +
+
−
+
−
3326400 
11200h 6
23040h 7
2268h 8
57600h 9

 407367 h
153µ 2
104813µ 3
401181h 4
3713351µ 5
165181µ 6

 8008000 − 13h + 5460h 2 − 29120h 3 + 655200h 4 − 112320h 5 +

µ 11
8179 µ 7
1039 µ 8
2299 µ 9
103µ 10
−
+
−
+
32760h 6
37440h 7
1179360h 8
1310400h 9
720720h 10


 f n+2 +

 4461461h
238µ 2
111953µ 3
1384609h 4
759127 µ 5
332153µ 6

 − 1355200 + 11h − 2970h 2 + 47520h 3 − 59400h 4 + 95040h 5


µ 11
8581µ 7
2257 µ 8
367 µ 9
101µ 10
 f n+3 +
−
+
−
+
−
13860h 6
31680h 7
71280h 8
475200h 9
261360h 10 

119 µ
 376763h
−

4h
 98560
2
+
115523µ 3
123821h 4
2547601µ 5
96619 µ 6
−
+
−
+
2160h 2
2880h 3
129600h 4
17280h 5

µ 11
3103µ 7
1403µ 8
703µ 9
11µ 10
 f n+4 +
−
+
−
+
3024h 6
11520h 7
77760h 8
28800h 9
142560h 10 

153µ 2
23533µ 3
20667 h 4
109279 µ 5
109723µ 6
 1882809h
+
−
+
−
+
−
−
492800
5h
420h 2
448h 3
5040h 4
17280h 5


µ 11
1727 µ 7
337 µ 8
101µ 9
97 µ 10
 f n +5 +
+
−
+
−
1440h 6
2304h 7
9072h 8
201600h 9
110880h 10 

 360737 h
119 µ 2
119093µ 3
318847 µ 4
381983µ 5
45733µ 6

+
−
+
−
+
 123200 −
5h
2700h 2
8640h 3
21600h 4
8640h 5


µ 11
12893µ 7
23µ 8
43µ 9
19 µ 10
 f n+6 +
−
+
−
+
12600h 6
180h 7
4320h 8
43200h 9
118800h 10 

 15627 h 102 µ 2 17159µ 3 77453µ 4 423559µ 5 29467 µ 6
 −
−
+
−
+
−
+
8800
7h
630h 2
3360h3
37800h 4
8640h5


µ 11
1693µ 7
863µ 8
103µ 9
31µ 10
 fn+7 +
+
−
+
−
2520h 6 10080h 7 15120h8 100800h9 166320h10 
 1140903h 153µ 2 60439 µ 3 2457 µ 4 1521413µ 5 80437 µ 6
+ 
−
+
−
+
−
+
16h
3360h 2
160h3
201600h 4
3456oh5
 985600

µ 11
4693µ 7
1391µ 8
1777 µ 9
13µ 10
 f n +8 +
−−
+
−
+
10080h 6
23040h 7 362880h8 57600h9 221760h10 
 3046144h 65536 µ 2 116801536 µ 3 1334272 µ 4 148209664 µ 5
+
−
+
+  −
+
−
12155
11486475h 2
153153h3
34459425h 4
 4679675
6
7
8
9
10
11

9728µ
3085312 µ
256 µ
59392 µ
512 µ
1024 µ
 f 17 +
−
+
−
+
−
7293h5 11486475h 6 7293h 7 20675655h8 3828825h9 379053675h10  n + 2
36
(13)
Res. J. Math. Stat., 5(4): 32-42, 2013
 22423 h 17 µ 2 4499 µ 3 556819 µ 4 345019 µ 5
24581 µ 6
83 µ 7

−
+
−
+
−
+
−
2
3
4
5
18 h
2520 h
362880 h
453600 h
103680 h
1728 h 6
 197120

437 µ 8
71 µ 9
89 µ 10
µ 11
f
+
−
+
7
8
9
10  n + 9
69120 h
136080 h
3628800 h
1995840 h 
µ = x − x n and µ ∈ [0, 9h] .
where,
µ = 0, h, 2h, 4h, 5h, 6h, 7 h, 8h,
Evaluating
the continuous
scheme
(13)
at
the following
points
h, 9h yields the discrete members of the Modified Block Hybrid Adams Method
(MOBHAM) used as block integrators as:
17
2
h
y n -y n+3 = - 2395993600
( 655172947 f n+4115957703 f n+1-
1218842064 f n+2+7887863048 f n+3-9159108530 f n+4+9154217358 f n+5 7015613176 f n+6+4254794544 f n+7 -2773535193 f n+8+1559625728 f n+17 -272551565 f n+9 )
2
h
y n+1-y n+3 = 9097288200
( 50461697 f n -3041910300 f n+1-
12919661700 f n+2 -462284706 f n+3-4173983242 f n+4+4401077538 f n+5 3373129188 f n+6+2030152410 f n+7 -1312470159 f n+8+735182848 f n+17 -128011598 f n+9 )
2
h
y n+2 -y n+3= - 5822264448
00 ( 607643465 f n -10622743131 f n+1+
239872909584 f n+2+480783779736 f n+3-232034579062 f n+4+178839894090 f n+5 119438161128 f n+6+66455799696 f n+7 -40827431931 f n+8+22431268864 f n+17 -3841935383 f n+9 )
2
h
y n+4 -y n+3= - 5822264448
00 ( 188536777 f n - 2711763483 f n+1+
21333375888 f n+2 -272743194984 f n+3-413057496566 f n+4+133291433418 f n+5 74179086696 f n+6+37828966032 f n+7 -22073336571 f n+8+11903565824 f n+17 -2007444439 f n+9 )
2
h
y n+5 -y n+3= - 9097288200
( 1469039 f n -22958364 f n+1+
207240132 f n+2 -3671758974 f n+3-11242913110 f n+4 -3605498754 f n+5+
117533988 f n+6+49927878 f n+7 -55696641 f n+8+34471936 f n+17 -6393530 f n+9 )
2
h
y n+6 -y n+3 = - 2395993600
( 676819 f n -9663225 f n+1+
76071600 f n+2 -1050296312 f n+3-2703306034 f n+4 -2358687474 f n+5 1313459576 f n+6+235126320 f n+7 -113048793 f n+8+58064896 f n+17 -9459021 f n+9 )
2
h
y n+7 -yn+3 = - 1137161025
( 177320 f n - 2851563 f n+1+
26317632 f n+2 - 463764522 f n+3-1372289776 f n+4 -926283930 f n+5 1363849344 f n+6 -476918442 f n+7+45099912 f n+8 -16252928 f n+17 +1971541 f n+9 )
2
5h
y n+8 -yn+3 =- 2328905779
2 ( 1480765 f n - 20558967 f n+1+
156227280 f n+2 -2062826376 f n+3-5235912110 f n+4 -4488330990 f n+5 4496046984 f n+6 -5126687280 f n+7 -2447435991 f n+8+489291776 f n+17 -58258915 f n+9 )
2
11h
y n+17 -yn+3 = - 2463635865
600 (60848359 f n - 868413975 f n+1 +
2
6868940100 f n+2 - 97078557732 f n+3 - 256542831974 f n+4 - 207787036314 f n+5 227009590236 f n+6 - 232587334980 f n+7 - 189017501673 f n+8 - 27073183744 f n+17 - 783270631 f n+9 )
2
h
y n+9 -yn+3= - 37437400
( 15301 f n -204204 f n+1+
(14)
1460844 f n+2 -17238442 f n+3-40641458 f n+4 -38375766 f n+5 33324148 f n+6 -44035134 f n+7 -22688523 f n+8 -23461888 f n+17 -6130982 f n+9 )
2
STABILITY ANALYSIS OF THE NEW BLOCK METHODS
The block method (9) is represented by a matrix finite difference equation in block form as:
A (1) Yw+1 = A (0) Yw + hB (1) Fw
(15)
37
Res. J. Math. Stat., 5(4): 32-42, 2013
where,
Yw+1 = ( y n +1 , y n + 2 , y n + 3 , y n + 4 , y n + 5 , y n + 6 , y n + 7 , y n +8 , y n + 9 ) T
Yw = ( y n −8 , y n − 7 , y n − 6 , y n −5 , y n − 4 , y n −3 , y n − 2 , y n −1 , y n ) T
Fw = ( f n +1 , f n + 2 , f n + 3 , f n + 4 , f n + 5 , f n + 6 , f n + 7 , f n +8 , f n + 9 ) T
for w = 0, … and n = 0, 9, …, N-9 and the matrices A (1) , A (0) , B (1) being 9 by 9 matrices whose entries are given by
the coefficients of (9) and are as defined in equation (16) below:
1 0 0 0 0 0 0
0
0
0 − 1 0 0 0 0 0 0


0
1 − 1 0 0 0 0 0 0


0 − 1 1 0 0 0 0 0
0
(1)
0 − 1 0 1 0 0 0 0, A = 0


0 − 1 0 0 1 0 0 0
0
0
0 − 1 0 0 0 1 0 0


0 − 1 0 0 0 0 1 0
0
0

0 − 1 0 0 0 0 0 1

A( 0 )
0
1

0

0
= 0

0
0

0
0

B (1)
 147429
 89600
 − 19469

 56700
 163531
 7257600
 10063

 1451520
167
= 
56700
 603

 89600
 13
 14175
 4675

 290304
 −9
 140
0
− 279
1600
− 7817
5670
− 391711
907200
− 42767
907200
− 701
28350
− 99
2240
− 32
2025
− 3175
36288
99
350
13273
5600
− 9371
56700
− 349817
453600
225623
453600
23189
56700
2647
5600
5494
14175
10735
18144
− 299
700
− 19107
8960
− 14117
56700
1083167
3628800
2381791
3628800
13903
11340
47727
44800
17632
14175
123575
145152
1899
700
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
73809
44800
12133
56700
− 129581
725760
− 583073
3628800
23189
56700
47727
44800
2174
2835
190775
145152
− 747
700
− 5039
5600
− 3373
28350
38599
453600
5779
90720
− 701
28350
2647
5600
17632
14175
11575
18144
199
70
3663
11200
491
11340
− 25759
907200
− 17663
907200
167
56700
− 99
2240
5494
14175
48425
36288
− 153
700
− 909
12800
− 1063
113400
42187
7257600
27467
7257600
− 23
113400
603
89600
− 32
2025
101635
290304
2313
1400
25 
3584 
13 

14175 
−7 
12800 
− 2497 

7257600 

0

−7 

12800 
13 
14175 
− 25 

3584 
7

25 
(16)
where, w = 0,1,2,...9 and n is the grid index.
Zero-stability: Zero-stability is concerned with the stability of the difference system (15) as h tends to zero. Thus,
as h → 0 , the method (15) tends to the difference system:
A (1) Yw+1 = A (0) Yw
(17)
whose first characteristic polynomial ρ (λ ) is given by:
ρ (λ ) = λA (0) − A(1)
(18)
Substituting A( 0) , A(1) from (16) into the characteristics polynomial ρ (λ ) (18) gives:
ρ (λ ) = λA (0) − A(1) = λ8 (λ − 1)
(19)
By Fatunla (1991), the block method (9) is zero-stable since in (19), ρ (λ ) = 0 satisfies λ j ≤ 1, j = 1,... and for
those roots with λ j = 1 , the multiplicity does not exceed 1. The block method is therefore convergent according to
38
Res. J. Math. Stat., 5(4): 32-42, 2013
2.5
mobam
mobham
2
1.5
1
Im(z)
0.5
0
-0.5
-1
-1.5
-2
-2.5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Re(z)
Fig. 1: Absolute stability regions of the MOBAM methods
4
y1 mobam
y2 mobam
y1 ode23
y2 ode23
3
Solution(y)
2
1
0
-1
-2
-3
1
0
2
3
4
5
Time(t)
6
7
8
9
10
Fig. 2: Solution of example 1 using (10) and ode23
24307
28249
36919
c12 = ( 5420800
, 54885600
,− 188179200
,
Henrici (1962). Since the MOBAM method is both
consistent and zero-stable, it is convergent. Using the
same procedure, the MOBHAM method in (14) is also
zero-stable and consistent, hence convergent.
37987
4997
443
,− 164656800
,− 5420800
,
− 439084800
31
5905
4908607
17 T
,− 52690176
,− 5945425920
,− 96800
)
− 1715175
0
Order of the MOBAM methods: The orders and error
constants of the new block methods are obtained using
Chollom et al. (2007).
The block method (10) is of uniform order
(10,10,10,10,10,10,10,10,10)T with error constants:
Absolute stability region of the MOBAM methods:
Following Chollom et al. (2007) and Butcher (1985),
the block methods (9) and (14) are reformulated as
General linear methods and the region of absolute
stability of the method was plotted using the matlab
program and is shown in Fig. 1.
Figure 1 reveals that the MOBAM methods are Astable.
NUMERICAL EXPERIMENTS
11899
5609
c11 = (− 1971200
,− 7484400
,
171137
479001600
443
1971200
90817
263
, 479001600
, 7484400
,
18665
179 T
62
,− 467775
, 19160064
,− 30800
)
In this section, the MOBAM methods (9) and (14)
are tested on linear and nonlinear stiff initial value
problems of ODEs. The solution curves obtained for the
MOBAM methods are compared with the well-known
MATLAB ODE 23 solver.
while the block method (15) is of uniform order
with
error
(11,11,11,11,11,11,11,11,11,11)T
constants:
39
Res. J. Math. Stat., 5(4): 32-42, 2013
4
y1 mobham
y2 mobham
y1 ode23
y2 ode23
3
Solution(y)
2
1
0
-1
-2
-3
0
1
2
3
4
5
Time(t)
7
6
8
9
10
Fig. 3: Solution of example 1 using (15) and ode23
1
y1 mobam
y2 mobam
y1 ode23
y2 ode23
0.9
0.8
0.7
Solution(y)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time(t)
7
6
8
9
10
Fig. 4: Solution of example 2 using (10) and ode23
1
y1 mobham
y2 mobham
y1 ode23
y2 ode23
0.9
0.8
0.7
Solution(y)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
Time(t)
Fig. 5: Solution of example 2 using (15) and ode23
40
6
7
8
9
10
Res. J. Math. Stat., 5(4): 32-42, 2013
3
y1 mobam
y2 mobam
y1 ode23
y2 ode23
2.5
2
Solution(y)
1.5
1
0.5
0
-0.5
-1
-1.5
0
1
2
3
4
5
Time(t)
6
7
8
9
10
Fig. 6: Solution of example 3 using (10) and ode23
3
y1 mobham
y2 mobham
y1 ode 23
y2 ode 23
2.5
2
Solution(y)
1.5
1
0.5
0
-0.5
-1
-1.5
0
1
2
3
4
5
Time(t)
6
7
8
9
10
Fig. 7: Solution of example 3 using (15) and ode23
y1' = −(ε −1 + 2) y1 + ε −1 y 22 , y1 (0) = 1
Example 1: We consider a well -known classical
system experimented in Baker (1989), Stabrowski
(1997), Vigo-Aguiar and Ramos (2007) and Akinfenwa
et al. (2011), in the range 0 ≤ t ≤ 10 :
y = 998 y1 + 1998 y 2 ,
'
1
y = −999 y1 − 1999 y 2 ,
'
2
y 2' = y1 − y 2 − y 22 , y1 (0) = 1
The smaller is the ε , the more serious the stiffness
of the system. Its exact solution is given by (Fig. 4
and 5):
y1 (0) = 1
y1 (0) = 1
y1 = y 22 , y 2 = e − t , h = 0.1, ε = 10
Its exact solution is given by the sum of two
decaying exponential components:
−t
y1 = 4e − 3e
−1000 t
−t
, y1 = −2e + 3e
−8
Example 3: Consider the stiff problem; see Brugnano
and Trigiante (1998):
−1000 t
y1' = −2 y1 + y 2 + 2 sin x, y1 (0) = 2
The stiffness ratio is 1:1000, h = 0.1 (Fig. 2 and 3)
y 2' = 998 y1 − 999 y 2 + 999(cos x − sin x), y1 (0) = 3
Example 2: Consider the stiffly nonlinear problem
which was proposed by Kaps et al. (1981) in the range
0 ≤ t ≤ 10.
The stiffness ratio of the problem is 1000. h = 0.1,
0≤x≤10. Its exact solution is given by (Fig. 6 and 7):
41
Res. J. Math. Stat., 5(4): 32-42, 2013
y1 ( x) = 2e − x + sin x, y 2 ( x) = 2e − x + cos x
Garfinket, D., C.D. Morbachi and N.Z. Shapiro, 1978.
Stiff differential equations. Ann. Rev. Biophys.
Bioeng., 6: 525-542.
Gear, C.W., 1971. Numerical Initial Value Problems in
Ordinary Differential Equations. Prentice-Hall,
Englewood Cliffs, NJ.
Henrici, P., 1962. Discrete Variable Methods in
Ordinary Differential Equations. John Wiley, New
York, pp: 407.
Kaps, P., G. Dahlguist and R. Jeltsch, 1981.
Rosenbrock-type Methods. In: Berich, G.D. and R.
Jeltsch (Eds.), Vol. 9, Numerical Methods for Stiff
Initial Value Problems. For Geometric and
Practical Mathematic, the RWTH Aachen,
Germany.
Kumleng, G.M., U.W. Sirisena and J.P. Chollom, 2012.
Construction of a class of block hybrid implicit
multistep methods for the solution of stiff ordinary
differential equations. Niger. J. Pure Appl. Sci., 5.
Lie, I. and S.P. Norsett, 1989. Super convergence for
multistep collocation. Math. Comput., 52(185):
65-79.
Mitchell, A.R. and J.W. Craggs, 1953. Stability of
difference relations in the solution of ordinary
differential equations. Math. Comput., 7: 127-129.
Onumanyi, P., D.O. Awoyemi, S.N. Jator and U.W.
Sirisena, 1994. New linear multistep methods with
continuous coefficients for first order initial value
problems. J. Niger. Math. Soc., 13: 37-51.
Onumanyi, P., U.W. Sirisena and S.N. Jator, 1999.
Continuous finite difference approximations for
solving differential equations. Int. J. Comput.
Math., 72: 15-27.
Samir, K., 2013. Improved accuracy of linear multistep
methods. App. Math. Inf. Sci., 7(2): 491-496.
Song, M.H., 2010. The improved linear multistep
methods for DE with piecewise continuous
arguments. Appl. Math. Comput., 217(8):
4002-4009.
Stabrowski, M.M., 1997. An efficient algorithm for
solving stiff ordinary differential equations. Simul.
Pract. Th., 5(4): 333-344.
Vigo-Aguiar, J. and H. Ramos, 2007. A family of astable runge-kutta collocation methods for higher
order for initial value problems. IMA J. Numer.
Anal., 27(4): 798.
Vlachos, D.S., Z.A. Anastassi and T.E. Simos, 2009. A
new family of multistep methods with improved
phase-lag characteristics for the integration of
orbital problems. Astronom. J., 138(86).
CONCLUSION
New MOBAM methods have been proposed and
implemented as self-starting methods for the solution of
stiff ordinary differential equations. The MOBAM
methods are A-stable making them suitable for the
numerical solution of stiff problems. The accuracy and
efficiency of the new block methods have been
demonstrated on both linear and non-linear stiff
problems as can be seen in Fig. 2 to 7. The authors in
the next work will address the theoretical aspect and
implementation strategies of the MOBAM class.
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