Vol. 3, No. 2 | Jul – Dec 2019 

 

SJCMS | P-ISSN: 2520-0755| E-ISSN: 2522-3003 © 2019 Sukkur IBA University – All Rights Reserved 

37 

 

 

 

A Modified 3-Point Adams Block Method of the Variable 

Step Size Strategy for Solving Neutral Delay Differential 

Equations 
 

Ubaidullah Yashkun1, 2 , Nurul Huda Abdul Aziz1 

Abstract:  
A modified 3-point Adams block method of order six (3ABM6) to solve neutral delay 

differential equations (NDDEs) using variable step size strategy is developed. The 

approximate solution of the retarded ( )y x −  and the neutral terms ( )y x  −  of the neutral 
delay differential equations at the grid points is obtained using the Newton divided difference 

interpolation technique. The proposed method will approximate the solution in each step 

using the three-points concurrently. To determine the performance of the proposed method, 

the maximum errors (MAXERR) and total number of function calls (FNC) will be compared 

with the method of 2-point order six predictor-corrector. The numerical results show that the 

3ABM6 reduces the number of function calls and better accuracy in term of MAXERR. 

 
Keywords: Neutral Delay Differential Equations, Adams Block Method, Newton 

Interpolation Technique, Variable Step Size Technique. 

 

1. Introduction 

In the fields of Science and Technology, the 
ordinary differential equations (ODEs) can 
be used to formulate the real-life problems 
by using initial value problem (IVP). 

( , )
dy

f x y a x b
dx

=  
   (1)                                                                    

Where, the function 
y

 usually represents 
physical quantities that evolve over time. 
Moreover, the delay differential equations 
(DDEs) are applied extensively in 
engineering, particularly the delay 
differential equations of the neutral type that 
usually arise in many scientific areas due to 
their ability to model many real-life 
phenomena. Neutral Delay Differential 
Equations do not depend only on the history 

of the function ( )y x −  formulated but 
also on the history of the function derivative

( )y x  −
.                        

                                                 
1 Institute of Engineering Mathematics, Universiti Malaysia Perlis, Kampus Pauh Putra, 02600, Arau, 
Perlis,Malaysia 
2 Sukkur IBA University, Airport Road, Sukkur, 65200, Sindh, Pakistan. 

Corresponding Author: ubaidullah@iba-suk.edu.pk 

*

0

0

( ) ( , ( ), ( ( )), ( ), ( ( ))) ,

( ) ( ) ,

y x f x y x y x x y x y x x x x

y x x x x

 

 

  = − − 

=  
                                                                  

(2) 

Where, 
*

( ) 0 and ( ) 0x x  
 are the 

delay arguments, and  ( )x  is the initial 
value function. 
Several algorithms are envisaged in the 
literature to approximate the results of 
Neutral Delay Differential Equation 

(NDDEs) of type (2) which is ( )y x  

having f  as an independent function. 
Neves [1], Bellen and Zennaro[2] and Al-
Mutib[3] discussed algorithms based on 
one-step methods, while Jackiewicz and 
Lo[4] and Tavernini[5] discussed 
algorithms based on liner multistep 
methods. Fabiano and Payne[6] extended 
the spline approximation technique for 
Neutral Delay Differential Equations. The 
continuous Galerkin finite element method 

mailto:asnali189@gmail.com


Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

38 

 

is used by Qin et al. [7] to solve the Linear 
Delay Differential Equations. 
Wen et al. [8] discussed the numerical 
stability of NDDEs and Baker et al. [9] 
discussed the role of NDDEs in cell growth 
phenomena. The Neutral Functional 
Differential Equations (NFDEs) numerical 
solution is studied by Hu et al. [10] using 
linear multistep methods (LMM). The 
analytical and numerical stability of the 
nonlinear neutral delay-integral differential 
equations was studied by P. Hu and C. 
Huang [11]. An approximate solution for 
NFNDEs was also obtained by Wang and Li 
[12] using numerical methods, namely the 
one-leg θ methods. 
The Adams Block Methods (ABM) are 
efficient and computational cost-effective 
methods compared to the other numerical 
methods. Seong and Majid[13] studied 
direct two point order four and five 
multistep block method to solve the second 
order delay differential equations (DDEs). 
In their study the formulation and stability 
of the block method is discussed. Aziz et al. 
[14] presented the delay differential 
equations (DDEs) numerical solution using 
a predictor-corrector scheme in modified 

block method. ABM will compute the 
y

approximate values in the block 
corresponding to the independent values of 
x . The one-block r-point technique for the 

2nd order initial value problem is proposed 
by Fatunla [15]. A block method will 
compute simultaneously the solution values 
at numerous different points on the 

x axis−  of the blocks by Ishak et al. [16] 
and Alkasassbeh and Omar [17]. 
In this paper, we introduce a new modified 
3-point Adams block method of order six for 
first-order NDDEs. Some numerical results 
are shown in the tables for the proposed 
method. Here is the order of this paper. 
Section 2 formulation of the method is 
presented. In section 3, order, consistency 
and zero stability are discussed. In section 4 
implementation of retarded and neutral 
terms are discussed. In section 5, the 
strategy of variable step size is discussed. 
Section 6, presents the algorithm of the 
method. Section 7, presents the numerical 
results, while discussions and conclusion 
are given in section 8.  

2.  The formulation of the 3-point 
Adams block method 

Consider the first-order NDDEs (2) and 

discretize the interval  
,a b

 to 
subintervals 

4 3 4
, ,..., , 0,1, 2,...

n n n k
x x x k

− − − +
=

 

such that 4n
a x

−
=

 and 4n k
b x

− +
=

. From 
Fig. 1, the proposed method is used to 

evaluate 1 2 3
, and

n n n
y y y

+ + +  
corresponding to the grid values 

1 2 3
, and

n n n
x x x

+ + +  having the variable 

step size h  respectively. The four initial 

values 3 2 1
, , and

n n n n
y y y y

− − −  are 
obtained using the Euler method with the 
initial conditions corresponding to the 

values 3 2 1
, , and

n n n n
x x x x

− − −  having step 

ratios rh  and qh . 

 
Source: Author's own created 

1n
y

+  can be determine by integrating (1) 
as;                

1

1
( ) ( ) ( , )

n

n

x

n n

x

y x y x f x y dx
+

+
= + 

                                                       (3) 

In (3), to integrate the function ( , )f x y , a 

Lagrange interpolation is substituted with 

the function. The approximation of 1n
y

+  

by using Maple with the ratio r  is as 
follows: 

1

1 4
( ) ( ) ( )

n

n

x

n n

x

y x y x P x dx
+

+
= +                             

(4) 
By substitution 

3n n
x x x x

s and s
h h

+
− −

= =
 3-point 



Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

39 

 

ABM of order six for the variable step size 
ratios is as follows: 
 

For the Corrector when 1r =  at the limit 

points 3−  to 2− , 2−  to 1−  and 1−  to 
0  

1

2

3

3

2

2

3

2

1

3

2

3

1

3

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

120

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

24

1
( 5)( 4)( 3)( 1)( )

12

1
( 5)( 4)( 2)( 1)( )

12

1
( 5)( 3)( 2)( 1)( )

24

n n n

n

n

n

n

y y h s s s s s dsf

h s s s s s dsf

h s s s s s dsf

h s s s s s dsf

h s s s s s dsf

+

−

+

−

−

+

−

−

+

−

−

−

−

−

= + + + + + +

− + + + +

+ + + + +

− + + + +

+ + + + +








2

2

2

3

1
( 4)( 3)( 2)( 1)( ) *

120
n

h s s s s s ds f

−

−

−

−

− + + + +





(5) 

2

3

1 3 2 1 1 2

1 3 2 1 1 2

2 3 2 1 1 2

(11 93 802 802 93 11 )
1440

( 27 637 1022 258 77 11 )
1440

(475 1427 798 482 173 27 )
1440

n

n

n n n n n n n n

n n n n n n n

n n n n n n n

h
y y f f f f f f

h
y y f f f f f f

h
y y f f f f f f

+

+

+ + + + − −

+ + + + − −

+ + + + − −

= + − + + − +

= + − + + − + −

= + + − + − +
                      

(6) 
 

For the corrector when 2r =  at the limit points 3−  to 2− , 2−  to 1−  and 1−  to 0  
 

2

3

1 3 2 1 1 2

1 3 2 1 1 2

2 3 2 1

(1328 9765 63952 46375 1183 93 )
100800

( 2352 47285 65072 9975 847 77 )
100800

(34288 93835 41328 15575 1743
100800

n

n

n n n n n n n n

n n n n n n n

n n n n n n

h
y y f f f f f f

h
y y f f f f f f

h
y y f f f f f

+

+

+ + + + − −

+ + + + − −

+ + + +

= + − + + − +

= + − + + − + −

= + + − + −
1 2

173 )
n

f
− −
+

 
(7) 

For the corrector when 

1

2
r =

 at the limit points 3−  to 2− , 2−  to 1−  and 1−  to 0  
   

2

3

1 3 2 1 1 2

1 3 2 1 1 2

2 3 2 1 1 2

(11 105 1211 1981 704 126 )
2520

( 39 1057 1981 1029 704 154 )
2520

(8011 2639 1869 2261 1728 406 )
2520

n

n

n n n n n n n n

n n n n n n n

n n n n n n n

h
y y f f f f f f

h
y y f f f f f f

h
y y f f f f f f

+

+

+ + + + − −

+ + + + − −

+ + + + − −

= + − + + − +

= + − + + − + −

= + + − + − +
 

   (8)  
 
The similar procedure can be used to get the one less order of predictor formulas. 



Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

40 

 

3. Order Consistency and Zero 

Stability 
In this study, the order of corrector is one 
order higher than the predictor. J. D. 
Lambert [18] gives the following definitions 
of order, consistency and zero-stability for 
the linear multistep methods. The order of 
this new developed 3ABM6 method is 
determined based on [18] and [13]. 

 
Theorem 1 (Lambert [18]). The general 
multistep method; 
                              

0 0

( , )
s s

m n m m n m n m

m m

y h f x y 
+ + +

= =

= 
    (9)                             

The order 
1p 

 if and only if                       

0

0

1

0 0

1

1

0 0

0,

0, 1, 2,..., ,
! ( 1)!

0.
( 1)! !

s

m

m

k ks s

P m m

m m

p ps s

p m m

m m

C

m m
C k p

k k

m m
C

p p



 

 

=

−

= =

+

+

= =

= =

= − = =
−

= − 
+



 

 
     

                     (10) 
The three-point block method can be written 
as matrix difference equation. 

m n m m n m
Y h f 

+ +
=

,  
 (11)                                                                  

 
Where, 

0 0 1 1 0 0

0 0 0 1 1 0

0 0 0 0 1 1

m


− 
 

= −
 
 − 

, 

11 93 802 802 93 11
1

11 77 258 1022 637 27
1440

27 173 482 798 1427 475

m


− − 
 

= − − −
 
 − −  , 

 2 1 1 2 3
T

n m n n n n n n
Y y y y y y y

+ − − + + +
=

 
 and  

 2 1 1 2 3
T

n m n n n n n n
f f f f f f f

+ − − + + +
=

 
From Theorem 1, the order of the corrector formulae of Equation (6) is   

0

1 1 2 3 4 5 6 0 1 2 3 4 5 6

1

0 0 1 1 0 0 0

0 0 0 1 1 0 0

0 0 0 0 1 1 0

2 3 4 5 6 ( )

0 1 1

0 2 0 3 1

0 0 0

C

C

C

            

−             
             

= + + + − + + =
             
             −             

= + + + + + − + + + + + +

−     
     

= + + − +
     
          

1

0 0 0

4 1 5 0 6 0

1 1 0

11 93 802 802 93 11 0
1

11 77 258 1022 637 27 0
1440

27 173 482 798 1427 475 0

0

0C

     
     

+ +
     
     −     

 − −              
              

− − + + − + + + − +              
              − −              

=

2 2 2 2 2

2 1 2 3 4 5 6 1 2 3 4 5 6

2

0

1 2 3 4 5 6
( 2 3 4 5 6 )

2! 2! 2! 2! 2! 2!

0 1 1 0 0 0
1 4 9 16 25 36

0 0 1 1 0 0
2! 2! 2! 2! 2! 2!

0 0 0 1 1 0

93
1

77
1440

C

C

           

 
 
 
  

= + + + + + − + + + + +

−           
           

= + + − + + +
           
           −           

−

−

       

2

3 4 5 6

802 802 93 11 0

2 258 3 1022 4 637 5 27 6 0

173 482 798 1427 475 0

0

0

0

0 0 0 , 0 0 0 , 0 0 0 , 0 0 0
T T T T

C

C C C C

 −            
            

+ − + + + − +            
            − −            

 
 

=
 
  

= = = =

 



Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

41 

 

7 7 7 7 7

7 1 2 3 4 5 6

6 6 6 6 6

1 2 3 4 5 6

7 7 7 7 7

7

1 2 3 4 5 6

7! 7! 7! 7! 7! 7!

1 2 3 4 5 6
( )
6! 6! 6! 6! 6! 6!

0 1 1 0 0 0
1 2 3 4 5 6

0 0 1 1 0 0
7! 7! 7! 7! 7! 7!

0 0 0 1 1 0

1

C

C

     

     

= + + + + +

− + + + + +

−           
           

= + + − + + +
           
           −           

−
6 6 6 6 6

7

93 802 802 93 11 0
1 2 3 4 5 6

77 258 1022 637 27 0
1440 6! 6! 6! 6! 6! 6!

173 482 798 1427 475 0

191

60480

271

60480

863

60480

C

 − −            
            

+ − + + + − +            
            − −            

− 
 
 
 =
 
 
−

 
  

0.

         (12) 

Equation (12) implies that 
7

0C   

therefore, the order of the proposed method 

is order six. 

Definition 2. The LMM in (9) is referred to 

as consistent such that order 1p  . The 
characteristics polynomials of LMM (9) are

       

*

0

*

0

( ) ,

( ) .

k
i

i

i

k
i

i

i

A

B

  

  

=

=

=

=





        (13)                                                             

The LMM is consistent iff                                                 

(1) 0 and (1) (1)  = =            (14)                                                                                                                                

Definition 3. Let the root 1 =  for the 

polynomial ( )  , then the LMM (13) is 
referred to as zero-stable. 

4. Implementation of Retarded 
and Neutral Term 

The first order NDDEs 

approximate solution depends on the 

retarded ( )y t −  and neutral ( )y t  −  
terms. The delay of these terms may be 

constant ( t − ), time dependent ( ( )t ) 

or state dependent ( ( , ( ))t X t ). In this 
paper, the Newton divided difference 

technique is used to approximate these 

terms. For implementation, the required 

points are: 

4 4 3 3 2 2 1 1

1 1 2 2 3 3

( , ), ( , ), ( , ), ( , ),

( , ), ( , ), ( , ) and ( , )

n n n n n n n n

n n n n n n n n

x y x y x y x y

x y x y x y x y

− − − − − − − −

+ + + + + +

 

The following recursion relation will be 

used for Newton divided differences. 

3 2 4 4 3 5
4 3 4

4 4

[ , ,..., ] [ , ,..., ]
[ , ,..., ] ,

0,1, 2...

n n n k n n n k
n n n k

n k n

f x x x f x x x
f x x x

x x

k

− − − + − − − +
− − − +

− + −

−
=

−

=

(15)

5. The Strategy of Variable Step-
Size 

The variable step size strategy is 

recommended for this study. Step-size h  

control strategy will be like, the next step 

size is limited to double, as the previous 

step size when the computation proceeds as 

studied by [19]. The variable step size 

strategy may reduce the formulae that need 



Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

42 

 

to be stored in the code, such that the 

storage capacity will increase. The 

evaluation technique is 

( ) or ( )
s s

P EC P EC E , such that P  

represents the predictor, E    represents 
evaluation, C  represents the corrector and 

s  is the number of iterations. The  

( )
s

P EC E  mode is used if we need to 

calculate the function value of 
( ) ( )

( , )
s s

n i n i n i
f f x y

+ + +
=  otherwise we use 

( )
s

P EC  mode. Modified 3-point Adams 

block method of order six varies the step 

size h ,  , to obtain three values during an 

integration step. Therefore, the efficiency 

of the method is acquired by taking the most 

optimal step size while reaching the desired 

accuracy. 

 

6. Algorithm of the 3-point Adams 
Block Method of Order Six to 

Solve NDDEs 

Step 1:    Set TOL, initial condition 
0 0
,x y

and step size h  

Step 2:
*

( ) ( , ( ), ( ( )), ( ), ( ( ))).y x f x y x y x x y x y x x   = − −

Step 3: 

( ) ( )y x x= , (0) 1y = . 
Step 4: 

For 1, 2,3 and 4i =  calculate x − for 

y  and its derivative. If 0xx −  , go to 

3 else calculate ( , )f x y from 2, go to 5 
Step 5: 

1 1
*

i i i
y y h f

− −
= +  

Step 6: 

For 5, 6and 7i =  calculate x − for y

and its derivative. If 
0

xx −  , go to 3 

else go to 7 

Step 7: 
6

1 4

1

( ) ( )
1440

n n i n i

i

h
y x y x f

+ − +

=

= +    

Step 8: If 
( 1)

1.0 *
k k

n n
y y TOL

+
−   go to 

9 else go to 4 

Step 9:     Stop 

 

7.  Problems and Numerical 
Results  

In this section, three problems are present 

which demonstrates the implementation of 

the modified 3-point Adams block method 

of order six for first-order neutral delay 

differential equations (2). The 

computational results of the problems 

present confirm the efficiency and accuracy 

of the method. The analysis of the accuracy 

and efficiency of the method are based on 

maximum errors and the number of total 

steps taken. The approximate results are 

compared with the results of the 2-point 

order six predictor-corrector method. The 

maximum errors and number of total steps 

taken are plotted in Figs. 2-4.  

 
Problem 1. (Jackiewicz [20]) 

( )2 2 2

2

( ) 0.75 ( ) / (1 ) / 2 ( (( 1) / 2)) ,

[0,1], ( ) 1 ,

[ 0.5, 0]

y x x y x x y x y x

x y x x

x

 = + + − −

 = +

 −

        

The exact solution 
2

( ) 1 , [ 0.5,1]y x x x= +  −        

Problem 2. (Jackiewicz [20]) 
2 2 1

( ) exp(1 2 ) ( )( ( 1 / (1 ))) , [0,1],

( ) exp( ), [ 1, 0].

x
y x x y x y x x x

y x x x

+ = − − + 

=  −

       

The exact solution is 

( ) exp( ), [ 1,1].y x x x=  −  
Problem 3. (Jackiewicz [20]) 

2
( ) 1 ( ) 2 ( / 2) ( ), [0, ],

( ) cos( ), [ , 0].

y x y x y x y x x

y x x x

 



 = + − − − 

=  −
       

The exact solution 

( ) cos( ), [ , ].y x x x  =  −       
 

Following abbreviations are used in tables 

1-3 

TOL: Tolerance Defined 

MTD: Employed Method  

TS: Total Number of steps  

FS: Number of failing steps 



Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

43 

 

MAXERR: Maximum Error  

FCN: Number of function calls 

3ABM6: Modified 3-Point Adams block 

method 

N2PC6: Neutral 2-point 6-order 

Table I: Approximate results for Problem 1 

TOL MTD TS FS FNC MAXERR  AVERR 

2
2
−

 

 

3ABM6 

N2PC6 

10 

11 

0 

0 

32 

39 

4.48E(-07) 

1.42E(-05) 

1.86E(-07) 

2.49E(-06) 

4
2
−

 

 

3ABM6 

N2PC6 

14 

14 

0 40 

51 

4.34E(-09) 

1.24E(-08) 

2.84E(-09) 

3.17E(-09) 

6
2
−

 

 

3ABM6 

N2PC6 

17 

18 

0 

0 

46 

67 

4.34E(-11) 

1.71E(-09) 

2.75E(-11) 

2.06E(-10) 

8
2
−

 

 

3ABM6 

N2PC6 

20 

21 

0 

0 

52 

80 

4.04E(-13) 

3.23E(-13) 

2.68E(-13) 

2.39E(-13) 

10
2
−

 

 

3ABM6 

N2PC6 

24 

24 

0 

0 

60 

92 

2.81E(-15) 

1.90E(-11) 

2.44E(-15) 

7.37E(-13) 

 

Table II: Approximate results for Problem 2 

TOL MTD TS FS FNC MAXERR AVERR 

2
2
−

 

 

3ABM6 

N2PC6 

10 

11 

0 

0 

32 

43 

3.71E(-07) 

3.94E(-07) 

1.50E(-07) 

1.41E(-07) 

4
2
−

 

 

3ABM6 

N2PC6 

15 

14 

0 

0 

42 

53 

5.54E(-07) 

2.14E(-09) 

7.03E(-08) 

1.04E(-09) 

6
2
−

 

 

3ABM6 

N2PC6 

19 

19 

0 

0 

53 

68 

1.88E(-07) 

3.27E(-09) 

2.34E(-08) 

3.60E(-10) 

8
2
−

 

 

3ABM6 

N2PC6 

30 

27 

0 

0 

72 

92 

3.45E(-10) 

1.91E(-11) 

7.45E(-10) 

3.33E(-12) 

10
2
−

 

 

3ABM6 

N2PC6 

53 

33 

0 

0 

118 

110 

5.60E(-11) 

6.11E(-12) 

1.25E(-11) 

1.10E(-12) 

 

Table III: Approximate results for Problem 3 

TOL MTD TS FS FNC MAXERR AVERR 

2
2
−

 
3ABM6 

N2PC6 

14 

13 

0 

0 

40 

50 

3.24E(-04) 

7.03E(-04) 

2.40E(-05) 

6.41E(-05) 

4
2
−

 
3ABM6 

N2PC6 

22 

17 

0 57 

72 

2.92E(-05) 

2.04E(-05) 

1.76E(-06) 

1.38E(-06) 

6
2
−

 
3ABM6 

N2PC6 

32 

25 

0 

0 

85 

106 

9.27E(-07) 

4.73E(-07) 

1.54E(-07) 

4.14E(-08) 

8
2
−

 
3ABM6 

N2PC6 

50 

41 

0 

0 

122 

155 

2.19E(-07) 

2.69E(-09) 

1.96E(-08) 

3.25E(-10) 

10
2
−

 
3ABM6 

N2PC6 

101 

62 

0 

0 

240 

245 

3.34E(-09) 

9.46E(-10) 

3.74E(-10) 

1.75E(-10) 

 



Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

44 

 

 
Fig 2: Numerical Results Comparison of 

Problem 1. 

 
Fig 3: Numerical Results Comparison of 

Problem 2. 

 

  
Fig 4: Numerical Results Comparison of 

Problem 3. 

8. Discussion and Conclusion 

For problem 1, the total number of 

function calls and the maximum errors 

obtained by the modified 3-point Adams 

block method of order six are less than 

the 2-point order six PC method. For 

problem 2, the total number of function 

calls is less than the N2PC method, 

except for tolerance 
10

10
−

 which is 

slightly high. Also, for problem 3, the 

total number of function calls is less than 

the N2PC method. However, for the 

given tolerances, the maximum errors for 

problems 1-3 are within the acceptable 

range, so both methods achieved the 

desired accuracy. It is clear from the 

approximate results that the proposed 

method is more efficient than the 2-point 

order six predictor-corrector method in 

solving NDDEs to the desired accuracy. 

Figures 2-4 represents the approximate 

solutions graphically of the problems 1-

3. Our new proposed algorithm is 

efficient because during the iteration 

process the total number of function calls 

are less than the other algorithms used in 

literature.  

In this paper, we have developed a 

modified 3-point Adams block method of 

order six based on the variable step size 

strategy for first-order neutral delay 

differential equations. The coding of the 

C programming language has been used 

to test the three numerical problems. The 

results obtained of the three problems 

tested indicate that the proposed method 

is more efficient and accurate in terms of 

the total number of function calls and 

maximum errors than the 2-point order 

six predictor-corrector method. 

 

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Ubaidullah Yashkun et.al.. A Modified 3-Point Adams Block Method of the Variable Step Size Strategy for 
Solving Neutral Delay Differential Equations                                                                                         (pp. 37 - 45)            

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 3 No. 2 July - December 2019 © Sukkur IBA University 

45 

 

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