



































IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

ADVANCEMENTS IN LINEAR MULTI-STEP METHOD FOR SOLVING 

THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS 

 

Etim, Uduak James  

Department of Mathematics 

 Akwa Ibom State University, Nigeria  

 

 

Eno John 

Department of General Studies 

Akwa Ibom State Polytechnic, Ikot Osurua, Nigeria 

 

 

Dr. Tombotamunoa W. J. Lawson 

Department of Mathematics/Statistics,  

Ignatius Ajuru University of Education, Port Harcourt, Nigeria. 

 

 

Udo Ukemeobong Monday 

Department of Mathematics 

 Akwa Ibom State University, Nigeria  

 

 

Abstract 

This work addresses the development of four step linear multi-step methods for the solution 

of third order ordinary differential equations. The approach requires the construction of a 

truncation error term and expanding it in Taylor’s series. The resulting FOUR step method are 

analysed to show that it is consistent, zero stable and hence convergent with good interval of 

absolute stability.Thus the new method satisfies the minimum condition for a linear multi-step 

method to be acceptable.  The technique of derivation employed in this work is easier and more 

adaptable than those of collocation 
 

Keywords: Four-Step Method, Third-Order Ordinary Differential Equations, Truncation Error, 

Taylor's Series, Consistency, Zero Stability, Convergence, Absolute Stability, Numerical 

Analysis. 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

1. INTRODUCTION 

Ordinary differential equations (ODEs) are fundamental in modeling dynamic systems across 

various scientific disciplines. [1]’s paper discusses the construction of linear multistep methods, 

providing insights into the techniques used for their development and analysis.Burden and Faires' 

[2] textbook is a comprehensive resource for numerical analysis. Chapter discussions on 

multistep methods offer foundational knowledge in the field.Butcher's book [3] is a classic in the 

field, providing a deep understanding of various numerical methods for ordinary differential 

equations, including multistep methods. [4]’s book covers the computational aspects of ordinary 

differential equations, providing valuable insights into the development and analysis of 

numerical methods. The first volume of [5] series delves into the numerical solution of nonstiff 

ordinary differential equations, offering relevant information for the development of multistep 

methods.Lambert's work [6] is a foundational resource on computational methods for ordinary 

differential equations, providing a historical context for the development of numerical 

techniques.Shampine and Gordon's book [7] is a classic in the field, providing practical insights 

into the numerical solution of ordinary differential equations, including the development of 

multistep methods. This paper presents a novel contribution to the field by introducing a four-

step linear multi-step method for solving third-order ODEs. The methodology involves the 

construction of a truncation error term, which is then expanded using Taylor's series. 

The resulting four-step method undergoes a thorough analysis to establish its key properties. We 

demonstrate its consistency, ensuring an accurate representation of the underlying differential 

equation. Moreover, we prove its zero stability, indicating reliable behavior, and establish its 

convergence with a substantial interval of absolute stability. This work represents a significant 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

advancement in numerical methods for ODEs, providing an efficient and acceptable solution to 

the complex challenges posed by third-order equations. 

 

2. METHODOLOGY 

This section describes the development of a foursteplinear multi-step method for the solution of 

initial value problems of ordinary differential equation. 

MethodsOf Derivation Of The New Linear Multi-Step  2.1 

i. The L. H. S is expanded by Taylor’s series about h 

ii. The R. H. S i.e. ���� is expanded in Taylor’s series about h.   

iii. Replace the function ���� with ����
���  and then expand in Taylor’s about h 

iv. Put the system in matrix form 

v. Determine the values of �′� and �′� 

vi. Finally form the new linear multi-step method 

vii. Test the linear multi-step method for convergence. 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

 

From  

���� = � ��

�

���

���� + ℎ� � �� ����

�

���

                           (2.1) 

Implies 

���� = ���� + ������+������ + ������ + ℎ�[����
��� + ������

��� + ������
��� + ������

��� + ������
��� ](2.2) 

Expanding the L.H.S.by Taylor’s series, about h and when k = 4, we obtain; 

���� =
(4ℎ)�

0!
�� +

(4ℎ)�

1!
��

�
+

(4ℎ)�

2!
��

�� +
(4ℎ)�

3!
��

��� +
(4ℎ)�

4!
��

�� +
(4ℎ)�

5!
��

� +
(4ℎ)�

6!
��

�� +
(4ℎ)�

7!
��

��� +
(4ℎ)�

8!
��

���� + ⋯  

���� = 4�ℎ��� + 4�ℎ���
� +

����

�
��

�� +
����

�
��

��� +
����

��
��

�� +
����

���
��

� +
����

���
��

�� +
����

����
��

��� +
����

�����
��

���� +
������

��

������
+ ⋯    

���� = ℎ��� + 4ℎ��
� +

16ℎ�

2
��

�� +
64ℎ�

6
��

��� +
256ℎ�

24
��

�� +
1024ℎ�

120
��

� +
4096ℎ�

720
��

�� +
16384ℎ�

5040
��

��� +
65536ℎ�

40320
��

���� +
262144ℎ���

��

362880
+                    (2.3) 

 Expanding the coefficients of ��, ��,��,��, ��,��, ��, ����� ��by Taylor’s series about h as in 

equation(2.1), we obtain;  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

����ℎ� + �� ���ℎ� + ��
� ℎ +

��
��

2!
ℎ� +

��
���

3!
ℎ� +

��
��

4!
ℎ� +

��
�

5!
ℎ� +

��
��

6!
ℎ� +

��
���

7!
ℎ� +

��
����

8!
ℎ� +

��
��

9‼
ℎ� +

��
�

10!
ℎ�� … �

+ �� ���ℎ� + 2��
� ℎ +

4��
��

2!
ℎ� +

8��
���

3!
ℎ� +

16��
��

4!
ℎ� +

32��
�

5!
ℎ� +

64��
��

6!
ℎ� +

128��
���

7!
ℎ� +

256��
����ℎ�

8!
+

512��
��ℎ�

9!

+
1024��

�

10!
ℎ�� … �

+ �� �
(3ℎ)���

0!
ℎ� +

(3ℎ)���
�

1!
ℎ� +

(3ℎ)���
��

2!
ℎ� +

(3ℎ)���
���

3!
ℎ� +

(3ℎ)���
��

4!
ℎ� +

(3ℎ)���
�

5!
ℎ� +

(3ℎ)���
��

6!
ℎ� +

(3ℎ)���
���

7!
ℎ�

+
(3ℎ)���

����

8!
ℎ� +

(3ℎ)���
��

9!
ℎ� +

(3ℎ)����
�

10!
ℎ�� + ⋯ �

+ ℎ� �����
���ℎ� + �� ���

���ℎ� + ��
��ℎ +

��
�

2!
ℎ� +

��
��

3!
ℎ� +

��
���

4!
ℎ� +

��
����

5!
ℎ� +

��
��

6!
ℎ� +

��
�

7!
ℎ� + ⋯ �       

+ �� ���
���ℎ� + 2��

��ℎ +
4��

�

2!
ℎ� +

8��
��

3!
ℎ� +

16��
���

4!
ℎ�  +   

128��
�ℎ�

7!
+ ⋯ �

+ �� ���
���ℎ� + 3��

��ℎ +
9��

�

2!
ℎ� +

27��
��

3!
ℎ� +

81��
���

4!
ℎ� +

243��
����

5!
ℎ� +

729��
��

6!
ℎ� +

2187��
�

7!
ℎ� + ⋯ �

+ �� �
(4ℎ)���

���

0!
+

(4ℎ)���
��

1!
+

(4ℎ)���
�

2!
+

(4ℎ)���
��

3!
+

(4ℎ)���
���

4!
+

(4ℎ)���
����

5!
+

(4ℎ)���
��

6!
+

(4ℎ)���
�

7!
+ ⋯ �� 

 

 

 

 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

By clearing the bracket, we have; 

����ℎ� + ����ℎ� + ����
� ℎ + ��

��
��

2
ℎ� + ��

��
���

6
ℎ� + ��

��
��

24
ℎ� + ��

��
�

120
ℎ� + ��

��
��

720
ℎ�

+ ��

��
���

5040
ℎ� + ⋯ + ����ℎ� + 2����

� ℎ + 2����
��ℎ� +

8����
���

6
ℎ� +

16����
��

24
ℎ�

+
32����

�

120
ℎ� +

64����
��

720
ℎ� +

128����
���

5040
ℎ� + ��ℎ��� + 3��ℎ��

� +
9

2
��ℎ���

��

+
27

6
��ℎ���

��� +
81

24
��ℎ���

�� +
243

120
��ℎ���

� +
729

720
��ℎ���

�� +
2187

5040
��ℎ���

���

+
6561

40320
��ℎ���

���� +
19683

362880
��ℎ���

�� +
59049

3628800
��ℎ����

� … + ����
���ℎ�

+ ����
��ℎ� +

����
�

2
ℎ� +

����
��

6
ℎ� +

����
���

24
ℎ� +

����
����

120
ℎ� +

����
��

720
ℎ�

+
����

�

5040
ℎ�� +

ℎ������
��

40320
+

ℎ������
���

362880
+

ℎ������
����

3628800
+ ⋯ + ����

���ℎ� + 2����
��ℎ�

+ 2����
�ℎ� +

8��
��

6
ℎ� +

16����
���

24
ℎ� +

32����
����

120
ℎ� +

64����
��

720
ℎ�

+
128����

�

5040
ℎ�� + ⋯ + ����

���ℎ� + 3����
��ℎ� +

9����

�

2
ℎ� +

27����
��

6
ℎ�

+
81����

���

24
ℎ� +

243����
����

120
ℎ� +

729��
����

270
ℎ� +

2187����
�

5040
ℎ�� + ⋯ + ℎ�����

+ 4ℎ�����
� +

16

2
ℎ�����

�� +
64

6
ℎ�����

��� +
256ℎ�����

��

24
+

1024ℎ�����
�

120

+
4096ℎ�����

��

720
+       (2.4) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

Equating equation (2.3) and equation (2.4), we have; 

���� = ℎ��� + 4ℎ��
� +

16ℎ�

2
��

�� +
64ℎ�

6
��

��� +
256ℎ�

24
��

�� +
1024ℎ�

120
��

� +
4096ℎ�

720
��

�� +
16384ℎ�

5040
��

��� +
65536ℎ�

40320
��

���� +
262144ℎ���

��

362880
+ ⋯    

= ����ℎ� + ����ℎ� + ����
� ℎ + ��

��
��

2
ℎ� + ��

��
���

6
ℎ� + ��

��
��

24
ℎ� + ��

��
�

120
ℎ� + ��

��
��

720
ℎ� + ��

��
���

5040
ℎ� + ⋯ + ����ℎ�

+ 2����
� ℎ + 2����

��ℎ� +
8����

���

6
ℎ� +

16����
��

24
ℎ� +

32����
�

120
ℎ� +

64����
��

720
ℎ� +

128����
���

5040
ℎ� + ⋯ + ��ℎ��� + 3��ℎ��

�

+
9

2
��ℎ���

�� +
27

6
��ℎ���

��� +
81

24
��ℎ���

�� +
243

120
��ℎ���

� +
729

720
��ℎ���

�� +
2187

5040
��ℎ���

��� +
6561

40320
��ℎ���

����

+
19683

362880
��ℎ���

�� +
59049

3628800
��ℎ����

�  + ����
���ℎ� + ����

���ℎ� + ����
��ℎ�  +

����
�

2
ℎ� +

����
��

6
ℎ� +

����
���

24
ℎ�

+
����

����

120
ℎ� +

����
��

720
ℎ� +

����
�

5040
ℎ�� + ⋯ + ����

���ℎ� + 2����
��ℎ� + 2����

�ℎ� +
8��

��

6
ℎ� +

16����
���

24
ℎ� +

32����
����

120
ℎ�

+
64����

��

720
ℎ� +

128����
�

5040
ℎ�� + ⋯ + ����

���ℎ� + 3����
��ℎ� +

9����

�

2
ℎ� +

27����
��

6
ℎ� +

81����
���

24
ℎ� +

243����
����

120
ℎ�

+
729��

��

270
ℎ� +

2187����
�

5040
ℎ�� + ⋯ + ℎ����� + 4ℎ�����

� +
16

2
ℎ�����

�� +
64

6
ℎ�����

��� +
256ℎ�����

��

24
+

1024ℎ�����
�

120

+
4096ℎ�����

��

720
+ ⋯   

By comparing the coefficient in powers of h, we obtain; 

i. ℎ� ⇒ �� + �� + �� + ��  = 1 

ii. ℎ� ⇒ �� + 2�� + 3��  = 4 

iii. ℎ� ⇒
��

�
+ 2�� +

�

�
��  =  

��

�
 

iv. ℎ� ⇒
��

�
+ ��

�

�
+ 

��

�
��+ �� +    �� + �� + �� + �� =  

��

�
 

v. ℎ� ⇒
��

��
+ ��

��

��
+

��

��
�� + �� + 2�� + 3�� + 4�� =  

���

��
 

vi. ℎ� ⇒
��

���
+ ��

��

���
+

���

���
�� +

��

�
+ 2�� + ��

�

�
+

��

�
�� =  

����

���
 

vii. ℎ� ⇒
��

���
+ ��

��

���
+

���

���
�� +

��

�
+ ��

�

�
+ ��

��

�
+

��

�
�� =  

����

���
 

viii. ℎ� ⇒
��

����
+

�����

����
+

������

����
+

��

��
+

����

��
+

����

��
+

�����

��
=

�����

����
 

ix. ℎ� ⇒
��

�����
+

�����

�����
+

������

�����
+

��

���
+

����

���
+

�����

���
+

������

���
=

�����

�����
 

(� − ��)(2.5) 

 

 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

https://ijojournals.com/index.php/index            Volume 07 Issue 01 || January., 2024 || 

 

 

 

Transforming the above into matrix form, yields 

⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡

1 1 1
0 1 2

0
1

2

4

2

1 0 0
3 0 0
9

2
0 0

0 0 0
0 0 0
0 0 0

  0
1

6

8

2

0
1

24

16

24

0
1

120

32

120

27

6
1 1

81

24
0 1

243

120
0

1

2

1 1 1
2 3 4
4

2

9

2

16

2

0
1

170

64

720

0
1

5040

128

5040

0
1

40320

256

40320

729

720
0

1

6
2187

5040
0

1

24
6561

40320
0

1

120

8

6

27

6

64

6
16

24

81

24

256

24
32

120

243

120

1024

120 ⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤

⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
��

��

��

��

��

��

��

��

��⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤

=

⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡

1
4

16

2
64

6
256

24
1024

120
4096

720
16384

5040
65536

40320⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤

                                                        … (2.6) 

  

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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3. ANALYSIS OF BASIC PROPERTIES OF THE FOUR STEP METHOD 

This section seek to establish the basic properties of the linear multistep method as stated in 

chapter one. 

Properties Of The Four Step Method 3.1 

Order and Error Constant 

From equation  (2.5), we obtain the follow: 

�� = 1 − �� − �� − �� −  �� 

Substituting 1, -2, 0,and 2 for  ��, ��, ��  ��� �� above we have 

�� = 1 − 1 − (−2) − 0 − 2   

�� = 1 − 1 + 2 − 2   

�� = 0.  

�� = 4 − �� − 2�� − 3�� 

Whichimplies that 

�� = 4 − (−2) − 2(0) − 3(2)   

�� = 0  

�� =
16

2
−

��

2
− 2�� −

9

2
�� 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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 Which implies that 

�� =
16

2
−

1

2
(−2) − 2(0) −  

9

2
(2) 

Taking the L.C.M, we have 

�� =
16 + 2 − 18

2
 

Therefore 

�� = 0. 

�� =
64

6
−

1

6
�� −

8

6
�� −

27

6
�� − �� − �� − �� − �� − �� 

 

 

By Substitution and for 
�

���
,

�

��
,

��

��
,

�

��
, ,

�

���
= ��, ��, ����, ��  we have 

�� =
1280 + 40 − 1080 − 1 − 56 − 126 − 56 − 1

120
 

Therefore 

�� =
−7

15
 

�� =
256

24
− 

��

24
− ��

16

24
−

81

24
�� − �� − 2�� − 3�� − 4�� 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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Substituting thevalue of the coefficient matrix, we have 

�� =
1280 + 10 − 3105 − 56 − 252 − 168 − 4

120
 

Therefore 

�� =
−2295

120
 

�� =
1024

120
−

��

120
− ��

32

120
−

243

120
�� −

��

2
− 2�� − ��

9

2
−

16

2
�� 

Substituting the values of the coefficientmatrix, we have 

�� =
2048 + 4 − 972 − 560 − 504 − 504 − 16

120
 

 

And therefore 

�� =
���

��
 . 

�� =
4096

720
−  

��

720
− ��

64

720
−

729

720
�� −

��

6
− ��

8

6
−

27

6
−

64

6
�� 

By substitution, we have 

�� =
4096

720
+  

2

720
−

1458

720
−

7

90
−

168

120
−

189

90
−

64

720
 

Therefore we have that 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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�� = 0 . 

�� =  
16384

5040
−  

��

5040
−

128��

5040
−

2187��

5040
−

��

24
−

16��

24
−

81��

24
−

256��

24
 

Similarly upon substitution, we obtain that 

�� =  0. 

Finally we have 

�� =
65536

40320
−

��

40320
−

256��

40320
−

6561��

40320
−

��

120
−

32��

120
−

243��

120
−

1024��

120
 

 

 

Substituting the value of the coefficient matrix, we have 

��  =
��

�����
= ���� ≠ 0. 

Hence we can deduce that our four step method is of order p = 5 with error constant 

���� =
��

�����
. 

Stability and Consistency 3.2 

Form 

� ������ =  ℎ� � ������

�

���

���

���

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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, for � = 4, if we substitute the coefficients 1,-2,0, 2,
�

���
,

�

��
,

��

��
,

�

��
,

�

���
 respectively, for 

��, ��, ��, ��,��, ��, ��, ��,��, we have 

���� = ���� + ������ + ������+�� + ℎ�[���� + ������ + ������ + ������ + ��] 

Which implies, that 

���� − 2���� + 2���� − �� = ℎ� �
1

120
���� +

7

15
���� +

21

20
���� +

7

15
���� +

1

120
��� 

Taking the L.C.M of the terms inside the square bracket, we have 

���� − 2���� + 2���� − �� =
1

120
[���� + 56 ���� + 126 ���� + 56 ���� + ��]                                              … (3.1) 

From equation(3.1), the first characteristic polynomial denoted by �(�) is: 

�(�) = �� − 2�� + 2�� − ��                                                                                                                                     … (3.2) 

Whichimplies that 

�(�) = �� − 2�� + 2� − 1                                                                                                                                      (3.3) 

and the second characteristic polynomial denoted by �(�) is: 

�(�) =
1

120
[�� + 56�� + 126�� + 56�� + ��](3.4) 

where ℎ� is ignored, 

Which implies that 

�(�) =
1

120
[�� + 56�� + 126�� + 56� + 1](3.5) 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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Consistency has it that a linear multi-step method must satisfies the following properties: 

(1) �(1) = 0  and 

(2)����(1)  = 3! �(1).  

We now test conditions (1) and (2)by using(3.3) as follows: 

1. �(1) = 1� − 2(1�) + 2(1) − 1 

Which implies that  

�(1) = 1 − 2 + 2 − 1 

And therefore  

�(1) = 0 , 

This verifies condition 1. 

Taking the first derivative of  (3.3), we have 

��(�) = 4�� − 6�� + 2                                                                                                                     (3.6) 

For r = 1 we have 

��(1) = 4. 1� − 6(1�) + 2 

 

Which implies that 

��(1) = 4 − 6 + 2 

Therefore, 

��(1) = 0 

2. ����(1) = 3! �(1) 

We need the second and third derivatives of (3.6),thus 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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���(�) = 12�� − 12�  

Implying, 

����(�) = 24� − 12, then 

����(1) = 24 − 12 = 12. 

and 

�(�) =
�

���
[�� + 56�� + 126�� + 56� + 1] 3! 

Which implies that 

�(1) =
�

���
[1� + 561� + 1261� + 56 + 1] 3! 

�(1) =
1

120
[1440] 

Without loss of generality 

�(1) =  12 

����(1) = �(1) = 12. 

This result verifies condition 2. 

It has been established that our fourstep method satisfies the conditions1 ��� 2, hence it is 

convergent. 

The first characteristic polynomial  

�(�) = �� − 2�� + 2� − 1, 

gives the possible values of r when �(�) is zero to be (� − 1)� = 0. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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Which implies that � = 1,1,1. Therefore, the four-step method satisfies the root condition, hence 

it is zero-stable. 

Interval of absolute stability 3.3 

We now seek to obtain the interval of absolute stability; this is done by applying the boundary 

locus method which is define as: 

ℎ(�) =
�(�)

�(�)
(3.7) 

where�(�) is the first characteristic polynomial and �(�) is the second characteristic polynomial. 

From equation(3.3)and  (3.5) 

�(�) = �� − 2�� + 2� − 1  and �(�) =
�

���
[�� + 56�� + 126�� + 56� + 1]. 

Then from (3.7) , We have 

ℎ(�) =
120[�� − 2�� + 2� − 1]

�� + 56�� + 126�� + 56� + 1
(3.8) 

byDeMoiveries’s theorem 

�� = ���� = ����� + ������  where ����� is the real part of  ���� and  �����   is the 

imaginary part of����, 

but here we will replace  �� with  �� .  So 

�� = ����� + ������ 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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For n = 1 we have 

�� = ���� + �����  

Which implies that 

� = ���� + �����                                                                                                                           (3.9) 

For n = 2, we have 

�� = ���2� + ����2�                                                                                                              (3.10) 

For n = 3, we have 

�� = ���3� + ����3�                                                                                                              (3.11) 

and  

For n = 4 we obtain 

�� = ���4� + ����4�                                                                                                                (3.12) 

Substituting equation(3.9)– (3.12)into(3.8), we have  

ℎ(�) =
120[���4� + ����4� − 2(���3� + ����3�) + 2(���� + �����) − 1]

���4� + ����4� + 56(���3� + ����3�) + 126(���2� + ����3�) + 56(���� + �����) + 1
 

which implies that 

ℎ(�) =
120[(���4� − 2���3� + 2���2� − 1) + �(���4� − 2���3� + 2����)]

�����4� + 56���3� + 126���2� + 56���� + 1 + �(���4� + 56���3� + 126���2� + 56����)��
 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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By rationalization, we have 

ℎ(�) =

120[(���4� − 2���3� + 2���� − 1) + �(���4� − 2���3� + 2����)]

[(���4� + 56���3� + 126���2� + 56���� + 1) − �(���4� + 56���3� + 126���2� + 56����)]

[(���4� + 56���3� + 126���2� + 56����) + �(���4� + 56���3� + 126���2� + 56����)]

[(���4� + 56���3� + 126���2� + 56���� + 1) − �(���4� + 56���3� + 126���2� + 56����)]

 

 

 

 

which implies that 

ℎ(�) =

120[(���4� − 2���3� + 2���� − 1 )(���4� + 56���3� + 126���2� + 56���� + 1 )]

−120�[(���4� − 2���3� + 2���� − 1 )(���4� + 56���3� + 126���2� + 56����)]

+120� �
(���4� − 2���3� + 2���� )(���4� + 56���3� + 126���2� + 56���� + 1 )

+120[(���4� − 2���3� + 2����)(���4� + 56���3� + 126���2� + 56���� )]
�

�

(���4� + 56���3� + 126���2� + 56���� + 1 )(���4� + 56���3� + 126���2� + 56���� + 1 )

+�(���4� + 56���3� + 126���2� + 56���� + 1)(���4� + 56���3� + 126���2� + 56���� )

−�(���4� + 56���3� + 126���2� + 56���� + 1)(���4� + 56���3� + 126���2� + 56���� )

+(���4� + 56���3� + 126���2� + 56����)(���4� + 56���3� + 126���2� + 56���� )

�

 

Opening the bracket of the numerator and that of the denominator, we have 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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ℎ(�) =
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
�����������������������������������������������������������������������������

�����������������������������������������������������������������
������������������������������������������������������������������������

�����������
��������������������������������������������������������������

�����������������������������
����������������������������������

����������������
������������������������������

�������������������������
����������������������������������������������

����������������
��������������������������������

�������������������������������� ����������
�����������������

��������������������� �����
������������������������������

����������������
������������������������

���� ���������������������
����������������

����������������������������
�������������������������������������������

�������������
���������������

�������������������������  ⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤

⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎡

���������������������������������������������������
������������� �������������������������������������������������

��������������������������������������������������
��������������������������������������������������������������������������

���������������������������������� ��������������������������������������
�������������������������  ����������������������������

�������������������������������������������������������������������
�������������������������������������� ⎦

⎥
⎥
⎥
⎥
⎥
⎥
⎤

. 

Since we are interested only on the real part, then we collect like terms only on the real part as 

follows: 

  

�(�) =

120 �

����4� + 54���4����3� + 126���4����2� + 58���4�����
−112����3� − 252���3����2� − 58���� + 252�������2 + 1125�����
−54����� − 126���2� − 1 + ����4� + 54���4����3� + 126���4����2�
+58����4� − 112����3� − 252���3����2� + 252���2������ + 56�����

�

⎣
⎢
⎢
⎢
⎢
⎡

����4� + 112���4����3� + 252���4����2� + 112���4�����
+2���4� + 3136����3� + 14112���2����� + 6272���3�����

+112���3� + 15876����2� + 14112���2����� + 112����
+126���2� + 1 + ����4� + 112���4����3� + 252���4����2�

+112���4����� + 3136����3� + 14112���3����2� + 6272���3�����
+15876����2� + 14112���2����� + 3136����� ⎦

⎥
⎥
⎥
⎥
⎤

 

We now evaluate  �(�)for the interval 0 ≤ � ≤ 180 as follows: 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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For � = 0 we have 

�(�) =

120

⎣
⎢
⎢
⎢
⎢
⎡

����0 + 54���0���0 + 126���0���0
+58���0���0 − 112����0 − 252���0���0

−58���0 + 252���0���0 + 1125����0
−54���0 − 126���0 − 1 + ����0

+54���0���0 + 126���0���0 + 58����0 − 112����0
−252���0���0 + 252���0���0 + 56����0 ⎦

⎥
⎥
⎥
⎥
⎤

⎣
⎢
⎢
⎢
⎢
⎢
⎡

                  ����0 + 112���0���0 + 252���0���0 + 112���0����
+2���0 + 3136����0

 +14112���0���0 + 6272���0���0 + 112���0 + 15876����0
                            +14112���0���0 + 112���0 + 126���0 + 1 + ����0

+112���0���0
+252���0���0 + 112���0���0 + 3136����0 + 14112���0���0

+6272���0���0 + 15876����0 + 14112���0���0 + 3136����0 ⎦
⎥
⎥
⎥
⎥
⎥
⎤

 

�(�) =
120 �

1 + 54 + 126 + 58 − 112 − 252 − 58 + 252
+112 − 54 − 126 − 1 + 0 + 0 + 0 + 0 − 0 − 0 + 0 + 0

�

�
1 + 112 + 252 + 112 + 2 + 3136 + 14112 + 6272 + 1 + 112 + 252

+112 + 2 + 3136 + 14112
+6272 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 15876

�

 

 

Therefore 

�(�) = 0. 

And for  � = 180 we have 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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�(�) =

120

⎣
⎢
⎢
⎢
⎢
⎡

����720 + 54���720���540 + 126���720���360
+58���720���180 − 112����540 − 252���540���360 − 58���180

+252���180���360 + 1125����180 − 54��180 − 126���360
−1125���180 − 1 + ����720 + 54���720���540

+126���720���360 + 58����720 − 112����540 + 252���540���360
+252���360���180 + 56����180 ⎦

⎥
⎥
⎥
⎥
⎤

⎣
⎢
⎢
⎢
⎢
⎢
⎡

����720 + 112���720���540 + 252���720���360
+112���720���180 + 2���720 + 3136����540 + 14112���360���180

+6272���540���180 + 112���540 + 15876����360 + 14112���360���180
+112���180 + 126���360 + 1 + ����720 + 112���720���540

+252���720���360 + 112���720���180 + 3136����540
+14112���540���360 + 6272���540���180 + 15876����360 + 14112���360���180

+3136����180 ⎦
⎥
⎥
⎥
⎥
⎥
⎤

 

Therefore 

�(�) ≅ −5.  

Therefore our four step method has an interval of absolute stability[−5,0]. 

4. CONCLUSION 

The developed four-step linear multi-step method stands as a noteworthy solution for the 

numerical approximation of third-order ordinary differential equations. Its consistency, zero 

stability, and convergence properties validate its reliability and effectiveness. The substantial 

interval of absolute stability further enhances its applicability to a wide range of dynamic 

systems. Notably, the derivation technique employed in this work is characterized by its 

simplicity and adaptability, setting this method apart from traditional collocation-based 

approaches. This research offers a valuable contribution to numerical analysis, providing a 

practical and efficient tool for solving complex ODEs in scientific and engineering applications. 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                             Etim, Uduak James* 

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References 

[1] Ismail, M. S., El-Tawil, M. A., & El-Danaf, T. S. (2014). On the Construction of Linear 

Multistep Methods.Abstract and Applied Analysis, 2014, 1-7. 

[2] Burden, R. L., & Faires, J. D. (2016). Numerical Analysis.Cengage Learning. 

[3] Butcher, J. C. (2008). Numerical Methods for Ordinary Differential Equations.John Wiley & 

Sons. 

[4] Ascher, U. M., & Petzold, L. R. (1998). Computer Methods for Ordinary Differential 

Equations and Differential-Algebraic Equations.SIAM. 

[5] Hairer, E., Nørsett, S. P., &Wanner, G. (1993). Solving Ordinary Differential Equations I: 

Nonstiff Problems.Springer-Verlag. 

[6] Lambert, J. D. (1973). Computational Methods in Ordinary Differential Equations.John 

Wiley & Sons. 

[7] Shampine, L. F., & Gordon, M. K. (1975). Computer Solution of Ordinary Differential 

Equations: The Initial Value Problem.W. H. Freeman and Company. 

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