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. DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 1 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 DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 2 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. DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 3 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; DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 4 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! + ⋯ �� DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 5 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) DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 6 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) DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 7 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) DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 8 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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 �� DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 9 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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�� DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 10 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 11 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || �� = 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� � ������ � ��� ��� ��� DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 12 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || , 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) DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 13 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 14 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || ���(�) = 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. DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 15 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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 �� = ����� + ������ DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 16 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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����)�� DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 17 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 18 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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: DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 19 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 20 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || �(�) = 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. DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 21 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Etim, Uduak James* https://ijojournals.com/index.php/index Volume 07 Issue 01 || January., 2024 || 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. DOI 10.5281/zenodo.10571505 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | https://ijojournals.com/index.php/m/index 22