Optimized Hybrid Block Adams Method for Solving First Order Ordinary Differential Equations

Abstract

Multistep integration methods are being extensively used in the simulations of high dimensional systems due to their lower computational cost. The block methods were developed with the intent of obtaining numerical results on numerous points at a time and improving computational efficiency. Hybrid block methods for instance are specifically used in numerical integration of initial value problems. In this paper, an optimized hybrid block Adams block method is designed for the solutions of linear and nonlinear first-order initial value problems in ordinary differential equations (ODEs). In deriving the method, the Lagrange interpolation polynomial was employed based on some data points to replace the differential equation function and it was integrated over a specified interval. Furthermore, the convergence properties along with the region of stability of the method were examined. It was concluded that the newly derived method is convergent, consistent, and zero-stable. The method was also found to be A-stable implying that it covers the whole of the left/negative half plane. From the numerical computations of absolute errors carried out using the newly derived method, it was found that the method performed better than the ones with which we compared our results with. The method also showed its superiority over the existing methods in terms of stability and convergence.

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Computers, Materials & Continua
DOI: 10.32604/cmc.2022.025933
Article
Optimized Hybrid Block Adams Method for Solving First Order Ordinary
Differential Equations
Hira Soomro1,*, Nooraini Zainuddin1, Hanita Daud1and Joshua Sunday2
1Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 32610, Seri Iskandar, Perak,
Malaysia
2Department of Mathematics, University of Jos, 930003, Jos, Nigeria
*Corresponding Author: Hira Soomro. Email: soomro_19001048@utp.edu.my
Received: 09 December 2021; Accepted: 24 January 2022
Abstract: Multistep integration methods are being extensively used in the sim-
ulations of high dimensional systems due to their lower computational cost.
The block methods were developed with the intent of obtaining numerical
results on numerous points at a time and improving computational efficiency.
Hybrid block methods for instance are specifically used in numerical inte-
gration of initial value problems. In this paper, an optimized hybrid block
Adams block method is designed for the solutions of linear and nonlinear
first-order initial value problems in ordinary differential equations (ODEs).
In deriving the method, the Lagrange interpolation polynomial was employed
based on some data points to replace the differential equation function and
it was integrated over a specified interval. Furthermore, the convergence
properties along with the region of stability of the method were examined.
It was concluded that the newly derived method is convergent, consistent, and
zero-stable. The method was also found to be A-stable implying that it covers
the whole of the left/negative half plane. From the numerical computations
of absolute errors carried out using the newly derived method, it was found
that the method performed better than the ones with which we compared our
results with. The method also showed its superiority over the existing methods
in terms of stability and convergence.
Keywords: Initial value problem (IVPs); linear multi-step method; block;
interpolation; hybrid; Adams-Moulton method
1Introduction
There are multiple fields of applications where differential equations are found, however, among
those; only a few applications have analytical solutions [1,2]. One of the major reasons why scientists
are inspired by differential equations is that they have the ability to replicate similar dynamics in the
natural world. This paper focuses on solving the first-order Initial Value Problems (IVPs) of the form:
y=f(t, y(t)),y(t0)=y0,t∈[a,b]. (1)

2948 CMC, 2022, vol.72, no.2
where, fis the continuous function in [a,b]intervals and the assumption of fgratifying Lipchitz
condition ensures the solution to the problem Eq. (1) exists and is unique [3].
ODEs appear in a variety of contexts in mathematics and science. Several approaches have been
adopted by several authors for the numerical solutions of ODEs among which block methods have the
advantages of being more cost-effective [4–7]. In general, with each block has r-point, the followings
are the advantages of the implementation of the block method [8,9]:
i) Each application of the block method generates rsolutions simultaneously.
ii) The computational time reduces as well as the overall number of steps.
iii) Overcoming the overlapping of pieces of solutions.
Reference [10] advocated the use of block implicit techniques as a way of acquiring begin-
ning values for predictor-corrector systems. Similar considerations were made by [11]. Further,
[12] expanded Milne’s suggestions into general-purpose algorithms, based on the Newton-Cotes
integration equations. A method for higher-order ODEs (stiff and non-stiff) was devised by [13]. For
the non-stiff algorithm, a split difference formulation was used, but for the stiff algorithm, a backward
differentiation formulation was employed. As a direct solution to non-stiff higher-order ODEs, [13]
developed a split difference formulation known as Direct Integration (DI). While creating a block
algorithm, [14] created a novel variant of the DI technique. According to [15,16], one-step methods
based on Newton backward difference formulae were used to solve first-order ODEs. An eighth order
seven-step block Adams type method has been proposed and implemented as a self-starting method to
generate the solutions at (tn+1,yn+1), (tn+2,yn+2), (tn+3,yn+3), (tn+4,yn+4), (tn+6,yn+6), (tn+7,yn+7), and (tn+8,yn+8)
by [17] for the solution of ODEs. Through interpolation and collocation procedures, a self-starting
multistep method was proposed by [18] in which the derivation of Adams-type methods was compiled
into block matrix equations for solving IVPs with an obsessive focus on stiff ODEs. Reference [19]
constructed an improved class of linear multistep block technique based on Adams Moulton block
methods in their study. The enhanced approaches were A-stable, which was a beneficial attribute when
dealing with stiff ODEs. Different implementation methods have also been developed, ranging from
predictor-corrector technique to block method, by many researchers [20–23]. A block technique based
on a stability zone was obtained, in which [24] presented one nonlinear and three linear ODEs using
the block technique. Since it has a large range of absolute stability, it could solve both nonlinear and
linear IVPs in ODEs, as well as stiff problems in systems. The key flaw of this approach is that the
accuracy of the predictors decreased with the increasing step length, and the results were presented at
an overlapping interval [25].
Despite having many advantages, block method, also possessed a major setback which pointed
out that the order of interpolation points must not exceed the differential equations. Because of this
setback, hybrid methods were introduced. Hybrid methods are highly efficient and have been reported
to circumvent the “Dahlquist Zero-Stability Barrier” condition by introducing function evaluation at
off-step points which takes some time in its development but provides better approximation than two
conventional methods (Runge-Kutta and linear multistep methods) [26,27].
Recently, many scholars have developed hybrid methods for the numerical solutions of ODEs. A
four-step hybrid block method is formulated by [28] in which the author has discussed about the new
strategy for the selection of hybrid points. A new single-step hybrid block method with fourth-order
has been proposed by [29] in which the increment of three off-step points enhanced the performance
of the developed method comparatively. The main persistence of [9] is to generate a higher-order block
algorithm with excellent stability properties, such as A-stability, for addressing various types of IVPs.
Reference [30] worked on the hybrid block approach with power series expansion which would aid

CMC, 2022, vol.72, no.2 2949
in the development of a more computationally stable integrator capable of solving problems relating
to first-order differential equations of the form Eq. (1). A highly efficient hybrid technique to find
out the approximate solution of first-order quadratic Riccati differential equations is derived by [31].
Insignificant convergence, implementation regions and inefficiency in terms of accuracy were some
of the major drawbacks of these methods. Due to these, we are motivated to formulate an efficient
algorithm that will address these setbacks. Therefore, the objective behind of this study is to develop
a sixth order hybrid block Adams method for finding the solutions of linear and nonlinear first-order
ODEs using Lagrange polynomial as the basis function. The basis on which the new method is built
based on the suggestion that halved step-size helps to acquire the desired stability and optimized
method according to [24]. For choosing the hybrid points, various points have been examined and
it is concluded that by selecting the points where the step-size is halved will lead toward the zero-stable
formulae. The advantage of the proposed hybrid block method is that it is useful in reducing the step
number of the problem and remains zero stable.
This paper is organized as follows: in Section 2, the derivation of the proposed method is discussed.
Section 3 contains an analysis of the basic properties of the derived method. In Section 4, some
numerical examples are presented, and the discussion of results is examined in Section 5. Finally,
Section 6 consists of conclusions and future recommendations.
2Derivation of 3-Points Hybrid Block Adams Moulton Method (AMM)
This section comprises the derivation of the proposed method for finding the solution of Eq. (1).
Derivation of the block method is based on the derivation presented in [15]. As illustrated in Fig. 1,
the approximate solutions are split into block’s series, and every block comprises three points with one
off-step point.
Figure 1: 3-Points hybrid block AMM
In Fig. 1, three solutions of yn+1,yn+2,andyn+3having one off-step point yn+5
2are simultaneously
computed while using two back values yn−1and ynin a block.
Three points will be computed using the previous block with a fixed step size h. The 3-point hybrid
block method equations are obtained by integrating Eq. (1) using the Lagrange interpolation polyno-
mial with interpolating points (xn−1,yn−1), (xn,yn), (xn+1,yn+1), (xn+2,yn+2), (xn+5
2,yn+5
2)and(xn+3,yn+3).

2950 CMC, 2022, vol.72, no.2
Consider the Lagrange interpolation polynomial given as,
Pq(x)=
k

j=0
Lq,j(x)fxn+3−j(2)
Where
Lq,j=
i=0
i= j
x−xn+3−i
xn+3−j−xn+3−i
By expanding Eq. (2) and substituting s=x−xn+3
hand then replace dx =hds, the corrector formula
for 3-point hybrid block Adams Moulton Method (AMM) can be obtained as, (the detailed derivation
can be seen in [32]),
yn+1=yn+h−11fn+3
180 +
88fn+5
2
315 −49fn+2
120 +283fn+1
360 +151fn
360 −13fn−1
840 
yn+2=yn+h−1fn+3
90 +17fn+2
45 +19fn+1
15 +17fn
45 −1fn−1
90 
yn+5
2=yn+h−125fn+3
4608 +
65fn+5
2
252 +125fn+2
192 +2875fn+1
2304 +55fn
144 −125fn−1
10752 (3)
yn+3=yn+h3fn+3
20 +
24fn+5
2
35 +21fn+2
40 +51fn+1
40 +3fn
8−3fn−1
280 
Assemble the predictor for the 3-point hybrid block AMM by adopting the same procedure carried
out above. Therefore, the predictor formulae for 3-point hybrid block AMM are obtained as,
yn+1
p=yn+h
2(−fn−1+3fn)
yn+2
p=yn+h(−2fn−1+4fn)
yn+5
2
p=yn+h
2(−9fn−1+15fn)(4)
yn+3
p=yn+h
8(−49fn−1+77fn)
Thus, Eq. (4) together with Eq. (3) gives the 3-point hybrid predictor-corrector AMM for the
solutions of problems in the form of Eq. (1).
3Analysis of the Basic Properties of the Proposed Method
This section encompasses the essential features of the proposed method, such as order and error
constants, stability analysis, consistency, and convergence. The stability region of the 3-point hybrid
block AMM will also be determined.

CMC, 2022, vol.72, no.2 2951
3.1 Order and Error Constant
Definition 3.1. (Order and Error Constant)
The linear multistep method (LMM)
k

j=0
αjyn+j=h
k

j=0
βjfn+j, (5)
where αjand βjare the coefficients of Eq. (3) and k=4, is said to be of order p if C0=C1= ··· =
Cp=0andC
p+1= 0where;
C0=
k

j=0
αj,C1=
k

j=0
jαj+
k

j=0
βj,C
2=
k

j=0
j2αj
2! +
k

j=0
jβj,C
3=
k

j=0
j3αj
3! +
k

j=0
j2βj
2! ,
Cq=
k

j=0
jpαj
p! +
k

j=0
jp−1βj
(p−1)!,q=4, 5, 6, ... (6)
The term Cp+1is called the error constant of the method [26]. This, therefore, means that the local
truncation error is calculated as in Eq. (7).
tn+k=Cp+2hp+2y(p+2)(tn)+O(hp+3)(7)
Reshaping Eq. (3) inamatrixformgives,
⎡
⎢
⎢
⎣
1000
0100
0010
0001
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
yn+1
yn+2
yn+5
2
yn+3
⎤
⎥
⎥
⎦
=⎡
⎢
⎢
⎣
0001
0001
0001
0001
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
yn−2
yn−1
yn−1
2
yn
⎤
⎥
⎥
⎦
+h
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
283
360
−49
120
88
315
−11
180
19
15
17
45 0−1
90
2875
2304
125
192
65
252
−125
4608
51
40
21
40
24
35
3
20
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎣
fn+1
fn+2
fn+5
2
fn+3
⎤
⎥
⎥
⎦
+h
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0−13
840 0151
360
0−1
90 017
45
0−125
10752 055
144
0−3
280 03
8
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎣
fn−2
fn−1
fn−1
2
fn
⎤
⎥
⎥
⎦
(8)

2952 CMC, 2022, vol.72, no.2
By applying the formulae Eqs. (6) to (8) we obtained, C1=C2=,...,=C6=[0,0,0,0]
Tand
C7= 0. According to Definition 3.1, the order of the 3-point hybrid block AMM is proven to be 6 as
Cp+1= 0(p=6) with the error constant as shown in Eq. (9),
C7=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
311
120960
1
756
1175
774144
1
896
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
. (9)
3.2 Stability Analysis
In this section, we will discuss the stability analysis of the 3-point hybrid block AMM which is
obtained by applying the linear test problem
y=f=λy(10)
where λrepresents the complex constant with Re(λ)<0.
For a technique to be useful in practice, it must have a zone of stability that ensures the approach
can solve at least slightly stiff problems. The technique must be zero-stable as well.
3.2.1 Zero-stability
Definition 3.2. (Zero-Stability)
The linear multistep method is said to be zero-stable if the characteristic polynomial R(t),hasno
root larger than one, and if all modular roots are simple [33–35].
To determine the zero-stability of the 3-point hybrid block AMM for Eqs. (3),(10) is substituted
in Eq. (3) which gives,
yn+1=yn−11hλyn+3
180 +
88hλyn+5
2
315 −49hλyn+2
120 +283hλyn+1
360 +151hλyn
360 −13hλyn−1
840 ,
yn+2=yn−1hλyn+3
90 +17hλyn+2
45 +19hλyn+1
15 +17hλyn
45 −1hλyn−1
90 ,
yn+5
2=yn−125hλyn+3
4608 +
65hλyn+5
2
252 +125hλyn+2
192 +2875hλyn+1
2304 +55hλyn
144 −125hλyn−1
10752 , (11)
yn+3=yn+3hλyn+3
20 +
24hλyn+5
2
35 +21hλyn+2
40 +51hλyn+1
40 +3hλyn
8−3hλyn−1
280

CMC, 2022, vol.72, no.2 2953
Eq. (11) can be inscribed in the matrix form as
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1−283hλ
360
49hλ
120 −88hλ
315
11hλ
180
−19hλ
15 1−17hλ
45 01hλ
90
−2875hλ
2304 −125hλ
192 1−65hλ
252
125hλ
4608
−51hλ
40 −21hλ
40 −24hλ
35 1−3hλ
20
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎣
yn+1
yn+2
yn+5
2
yn+3
⎤
⎥
⎥
⎦
=⎡
⎢
⎢
⎣
0001
0001
0001
0001
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
yn−2
yn−1
yn−1
2
yn
⎤
⎥
⎥
⎦
+h
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0−13λ
840 0151
360λ
0−1
90 λ017
45λ
0−125
10752λ055
144λ
0−3
280λ03
8λ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎣
yn−2
yn−1
yn−1
2
yn
⎤
⎥
⎥
⎦
,
(12)
and Eq. (12) is equivalent to
AYm−(B+Ch)Ym−1=0, (13)
where,
A=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1−283hλ
360
49hλ
120 −88hλ
315
11hλ
180
−19hλ
15 1−17hλ
45 01hλ
90
−2875hλ
2304 −125hλ
192 1−65hλ
252
125hλ
4608
−51hλ
40 −21hλ
40 −24hλ
35 1−3hλ
20
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,B=⎡
⎢
⎢
⎣
0001
0001
0001
0001
⎤
⎥
⎥
⎦
,
C=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0−13λ
840 0151
360λ
0−1
90 λ017
45λ
0−125
10752λ055
144λ
0−3
280λ03
8λ
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,Ym=⎡
⎢
⎢
⎣
yn+1
yn+2
yn+5
2
yn+3
⎤
⎥
⎥
⎦
and Ym−1=⎡
⎢
⎢
⎣
yn−2
yn−1
yn−1
2
yn
⎤
⎥
⎥
⎦

2954 CMC, 2022, vol.72, no.2
The following stability polynomial of 3-point hybrid block AMM is obtained by using
|tA −(B+Ch)|=0
R(t,H)=t41−3961
2520H+13081
12096H2−431
1080H3+283
4032H4
+t3−1−257
180H−6557
7560H2−583
2160H3−29
672H4
+t2−1
2520H+11
60480H2+1
4320H3+1
20160H4, (14)
assuming H=hλ, we obtain R(t,H)at H=0as
R(t,H)=−t3+t4=0, (15)
Resolving Eq. (15) for t, implies t=0, 0, 0, 1. In conclusion, according to Definition 3.2, if all the
major roots are on or in the unit circle, the method is zero-stable.
3.2.2 Stability Region
The collection of points found by substituting t=eiθ=sinθ+icosθ,0≤θ≤2πin the stability
polynomial Eq. (14) defined the stability area. Fig. 2 depicts the stability region which were obtained
using Mathematica software.
Figure 2: The stability region of the 3-point Hybrid Block AMM
3.3 Consistency
Definition 3.3. (Consistency)
The linear multistep method is said to be consistent if it has order p greater than or equal to one,
i.e., p≥1. The 3-point hybrid block AMM is a technique of order six, p=6≥1; thus, it is consistent.
3.4 Convergence
To determine the convergence of the method, we analyze its consistency and zero-stability
according to the following theorem.

CMC, 2022, vol.72, no.2 2955
Theorem 3.1.(Convergence)
The necessary and sufficient conditions for the linear multistep method to be convergent are
consistent and zero-stable.
Therefore, the 3-point hybrid block AMM is convergent since it is both consistent and zero-stable.
4Numerical Examples
We have gone through a few case studies to show the competence of the 3-point hybrid block AMM
. Specified numerical examples have been taken from [36–40]. For computational purpose, C++ code
was used.
Problem 1: Susceptible, Infected, and Recovered (SIR) Model
In the SIR model, the number of individuals infected with an infectious illness in a closed
population, overtime is calculated. In this class of models, the number of susceptible person S(t), the
number of infected people (t), and the recovery rate (t) are all related by coupled equations. This is an
excellent and straightforward model for several infectious illnesses, like measles, rubella, and mumps
[12,41,42]. This problem is also considered by Sunday et al. [36] and given by the three associated
equations as shown below,
dS
dt =μ(1−S)−βIS, (17)
dI
dt =μI−γI+βIS, (18)
dR
dt =−μR+γI, (19)
where μ,γand βare positive parameters. Define yto be
y=S+I+R
by adding and simplifying Eq’s. (17)–(19),weget
y=μ(1−y),
setting μ=1
2and considering the initial condition y(0)=1
2(for a specific closed population), the
following first-order ODE is obtained,
y(t)=1
2(1−y(t)),y(0)=1
2,t[0, 1],
and the exact solution is given by
y(t)=1−1
2e−1
2t
Problem 2: Consider the quadratic Riccati differential equation from [37]
y(t)=− 1
1+t+y(t)−y2(t),y(0)=1, t[0, 1]
and the exact solution is given by
y(t)=1
1+t

2956 CMC, 2022, vol.72, no.2
Problem 3: Consider the vastly oscillating ODE presented in [38]
y(t)=−sin t−200(y(t)−cos t),y(0)=0, t[0, 0.01]
The exact solution is given by
y(t)=cos t−e−200t
Problem 4: Consider another Riccati differential equation from [37]
y(t)=1−y(t)2,y(0)=2, t[0, 1]
the exact solution is given by
y(t)=e2t−1
e2t+1
Problem 5:We consider a mildly stiff system problem given in [39]
y
1(t)
y
2(t)=998 1998
−999 −1999y1(t)
y2(t),y1(0)
y2(0)=1
1
The exact solution of the system of equations above is given by the sum of two decaying
exponential components as below
y1(t)
y2(t)=4e−t−3e−1000t
−2e−t+3e−1000t
It is important to state that the eigenvalues of the Jacobian matrix which are λ1=−1, λ2=−1000
with the stiffness ratio 1:1000. The problem is solved within the interval [0,70].
5Results and Discussion
From Tab s. 1–5, the solution values are calculated at the various points of the given interval which
is represented by “t”. On the contrary, the efficiency of 3-point hybrid block AMM is proven when
smaller step sizes are used as it is capable of outperforming at a step size 10−6for each problem.
Table 1: Comparison of absolute error for Problem 1
t Error in 3-point hybrid block AMM Error in [30]Errorin[43]
0.1 6.10623 ×10−14 1.21802 ×10−13 6.78013 ×10−13
0.2 3.19744 ×10−14 1.39999 ×10−13 6.35936 ×10−13
0.3 1.19016 ×10−13 1.18494 ×10−12 6.38045 ×10−13
0.4 2.77001 ×10−13 1.53899 ×10−12 1.18994 ×10−12
0.5 4.30989 ×10−13 1.11000 ×10−12 1.12410 ×10−12
0.6 5.58997 ×10−13 5.27022 ×10−12 1.09901 ×10−12
0.7 6.77902 ×10−13 2.10898 ×10−12 1.54798 ×10−12
0.8 7.80931 ×10−13 1.29789 ×10−11 1.46805 ×10−12
0.9 8.68972 ×10−13 3.08229 ×10−11 1.41909 ×10−12
1.0 9.59011 ×10−13 4.12192 ×10−11 1.78202 ×10−12

CMC, 2022, vol.72, no.2 2957
Tab. 1 depicts the comparison of the numerical outcomes by 3-point hybrid block AMM with the
two-step block hybrid method by Ajileye et al. [30] and 3-step hybrid Adams type methods by Yahaya
et al. [43] for the SIR model. Absolute error was computed by finding the difference between the exact
solution and proposed method’s solution at distinctive values of t. The solution of the 3-point hybrid
block AMM performs better than [30,43].
In Tab. 2, the comparison of the numerical outcomes by 3-point hybrid block AMM with the
quarter-step method for the solution of Riccati differential equations by [37] has been done based
on absolute error. It is obvious from the above results that the proposed method is computationally
reliable in handling the Riccati differential equations also.
Table 2: Comparison of absolute error for Problem 2
t Error in 3-point hybrid block AMM Error in [37]
0.1 2.886579 ×10−15 2.1491 ×10−10
0.2 4.440892 ×10−16 4.7505 ×10−10
0.3 3.330669 ×10−16 7.8751 ×10−10
0.4 2.220446 ×10−16 1.1604 ×10−9
0.5 6.661338 ×10−16 1.6031 ×10−9
0.6 1.110223 ×10−16 2.1260 ×10−9
0.7 2.220446 ×10−16 2.7412 ×10−9
0.8 3.663735 ×10−15 5.9894 ×10−9
0.9 4.218847 ×10−15 4.3048 ×10−9
1.0 4.142744 ×10−15 4.3370 ×10−9
Tab. 3 displays the results from the 3-point hybrid block AMM for solving problem 3. It can be
seen that the proposed method exhibits better accuracy compared with the results obtained by the
two-step hybrid block method [38].
Table 3: Comparison of absolute error for Problem 3
t Error in 3-point hybrid block AMM Error in [38]
0.001 1.821626 ×10−12 8.818301 ×10−09
0.002 5.080380 ×10−13 1.785094 ×10−08
0.003 5.773159 ×10−13 2.694481 ×10−08
0.004 9.997558 ×10−13 3.596013 ×10−08
0.005 2.787770 ×10−13 3.595560 ×10−08
0.006 3.168576 ×10−13 5.400622 ×10−08
0.007 5.485611 ×10−13 6.304263 ×10−08
0.008 1.529887 ×10−13 7.208465 ×10−08
0.009 1.738609 ×10−13 8.113123 ×10−08
0.010 3.009814 ×10−13 9.018154 ×10−08

2958 CMC, 2022, vol.72, no.2
In Tab. 4, the representation of absolute errors demonstrates the comparison of the results by 3-
point hybrid block AMM with [37]. Hence, it is obvious that the proposed method performs better
than that of [37].
Table 4: Comparison of absolute error for Problem 4
t Error in 3-point hybrid block AMM Error in [37]
0.1 8.326672 ×10−16 1.149081 ×10−14
0.2 4.163336 ×10−16 6.716849 ×10−14
0.3 5.551115 ×10−17 1.833533 ×10−13
0.4 7.771561 ×10−16 3.386180 ×10−13
0.5 7.771561 ×10−16 4.861112 ×10−13
0.6 1.110223 ×10−16 5.798695 ×10−13
0.7 3.219646 ×10−15 5.948575 ×10−13
0.8 7.771561 ×10−16 5.327960 ×10−13
0.9 1.110223 ×10−16 4.161116 ×10−13
1.0 6.661338 ×10−16 2.745582 ×10−13
In Tab. 5, comparison have been made at points (x=5, x=40, and x=70) with the step
size h=10−6for the 3-point hybrid block AMM. From the comparison of the absolute error of the
proposed methods as shown in Tab. 5, it is obvious that the 3-point hybrid block AMM with order 6
exhibit superiority over the method given in [39] with lesser order 4 in terms of accuracy. Tab. 5 shows
that the proposed method is also well suited for mildly stiff linear problems.
Table 5: Comparison of absolute error for Problem 5
xy
iError in 3-point hybrid block AMM Error in [39]
5y12.9144 ×10−12 1.3920 ×10−11
y21.4572 ×10−12 6.9700 ×10−12
40 y17.6611 ×10−23 3.3628 ×10−12
y23.8305 ×10−23 1.6818 ×10−13
70 y11.3829 ×10−35 2.9325 ×10−13
y26.9144 ×10−36 1.4664×10−13
As a result, the 3-point hybrid block AMM, which developed a block method of order six using
Lagrange interpolation as an approximation solution, performs better, and the error analysis reveals
that the proposed method is giving more accurate results in comparisons to the other approaches. In
Tab. 1, it is observed that the proposed method reduces the error, approximately by the average of
33% and 41% compared to [30,43] respectively. The efficiency of the proposed method can also be
checked from Tabs. 2 –5that the 3-point hybrid block AMM decreases the absolute error an average
of approximately 50% compare with [37,39].

CMC, 2022, vol.72, no.2 2959
6Conclusion
In this paper, an optimized 3-point hybrid block Adams method for the solution of first order
ODEs has been derived. The method derived was implemented using C++ language that compute the
solutions to problems of the form in Eq. (1). The basic properties of the method developed were also
analyzed and from the results of the analyses, it is confirmed that the method is zero-stable, consistent,
and convergent. Thus, because of the zero stability of the method, it is suitable for solving stiff systems
of equations (Problem 5) as well as nonlinear equations. Also, from the results presented in Tabs. 1–5,
it is obvious that the new method derived performs better than the existing ones based on the results
produced. We therefore conclude that the proposed method is computationally reliable in solving first-
order problems of the form in Eq. (1).
For future work this method shall be applied to the problems of chemical kinetics to investigate
the efficiency and accuracy of the proposed method which is the main requirement of such type of
problems.
Acknowledgement: The authors are grateful to Universiti Teknologi PETRONAS for providing
facilities for conducting this study.
Funding Statement: This research was funded by Fundamental Research Grant Scheme (FRGS) under
the Ministry of Higher Education Malaysia, grant number with project ref: FRGS/1/2019/STG06/
UTP/03/2.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the
present study.
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