A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs

Duromola Monday Kolawole; Momoh Adelegan Lukuman; Akingbodi Oluwagbenga Joseph

doi:doi:10.11648/j.ajam.20251302.11

Research Article |

| Peer-Reviewed

A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs

Duromola Monday Kolawole

, Momoh Adelegan Lukuman^*

, Akingbodi Oluwagbenga Joseph

Published in American Journal of Applied Mathematics (Volume 13, Issue 2)

Received: 13 January 2025 Accepted: 27 January 2025 Published: 27 February 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 2)
DOI	10.11648/j.ajam.20251302.11
Page(s)	103-116
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Linear Multi-step Method, Hybrid Points, Continuous Scheme, Discrete Scheme

References

[1]	Ming, C. Y. (2017). Solution of Differential Equations with Applications to Engineering Problems. Dynamical Systems - Analytical and Computational Techniques. https://doi.org/10.5772/67539
[2]	Henrici, P. (1962) Discrete Variable Methods in Ordinary Differential Equations. John Wiley & Sons, New York.
[3]	Olabode, B., Momoh, A., and Senewo, E. (2024). Derivative block methods for the solution for fourth-order boundary value problems of ordinary differential equations. Proceedings of the Nigerian Society of Physical Sciences, 80. https://doi.org/10.61298/pnspsc.2024.1.80
[4]	Mereta, S. (n.d.). A general framework for tropical differential equations. https://doi.org/10.23889/suthesis.61189
[5]	Lambert, J. D. (1973) Computational Methods in ODEs. John Wiley & Sons, New York.
[6]	Brujnano, L., and Trigiante, D. (1998). Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam.
[7]	Ramos, H., and Momoh, A. L. (2023). A Tenth-Order Sixth-Derivative Block Method for Directly Solving Fifth-Order Initial Value Problems. International Journal of Computational Methods, 20(09). https://doi.org/10.1142/s0219876223500111
[8]	Hussain, K., Ismail, F., and Senu, N. (2015) Two Embedded Pairs of Runge-Kutta Type Methods for Direct Solution of Special Fourth-Order Ordinary Differential Equations. Mathematical Problems in Engineering, 2015, Article ID: 196595. https://doi.org/10.1155/2015/196595
[9]	Duromola, M. K., Momoh, A. L., and Akingbodi, O. J. (2024). A Chebyshev-generated Generated Block Method for Directly Solving Nonlinear and Ill-posed Fourth-Order ODEs. Earthline Journal of Mathematical Sciences, 1267?1292. https://doi.org/10.34198/ejms.14624.12671292
[10]	Fatunla, S. O. (1991) Block Method for Second-Order IVPs. International Journal of Computer Mathematics, 41, 55-63. https://doi.org/10.1080/00207169108804026
[11]	Awoyemi, D. O. (1996) A Efficient Two-Step Numerical Integrator for General Second-Order Ordinary Differential Equations. Abacus Journal of the Mathematical Association of Nigeria, 24, 31-43.
[12]	Awoyemi, D. O., Kayode, S. J. and Adoghe, O. (2015) A Six-Step Continuous Multistep Method for the Solution of General Fourth-Order Initial Value Problems of Ordinary Differential Equations. Journal of Natural Sciences Research, 5, 131-138.
[13]	Duromola, M. K. (2016) An Accurate Five Off-Step Points Implicit Block Method for Direct Solution of Fourth-Order Differential Equations. Open Access Library Journal, 3, e2667. https://doi.org/10.4236/oalib.1102667
[14]	Cortell, R. (1993) Application of the Fourth-Order Runge-Kutta Method for the Solution of High-Order General Initial Value Problems. Computers & Structures, 49, 897-900. https://doi.org/10.1016/0045-7949(93)90036-D
[15]	Duromola, M. K., and Momoh, A. L. (2019) Hybrid Numerical Method with Block Extension for Direct Solution of Third Order Ordinary Differential Equations. American Journal of Computational Mathematics, 9, 68- 80. http://www.scirp.org/journal/ajcm
[16]	Yap, L., and Ismail, F. (2015) Block Hybrid Collocation Method with Application to Fourth-Order Differential Equations. Mathematical Problems in Engineering, 2015, Article ID: 561489. https://doi.org/10.1155/2015/561489
[17]	Abdelrahim, R., and Omar, Z. (2017) A Four-Step Implicit Block Method with Three Generalized Off-Step Points for Solving Fourth-Order Initial Value Problems Directly. Journal of King Saud University?Science, 29, 401-412. https://doi.org/10.1016/j.jksus.2017.06.003
[18]	Alkasassbeh, M., and Omar, Z. (2017) Generalized Hybrid One-Step Block Method Involving Fifth Derivative for Solving Fourth-Order Ordinary Differential Equation Directly. Journal of Applied Mathematics, 2017, Article ID: 7637651. https://doi.org/10.1155/2017/7637651
[19]	Badmus, A. M. and Yahaya, Y. A. (2014) New Algorithm of Obtaining Order and Error Constants of Third Order Linear Multistep Method (LMM). Asian Journal of Fuzzy and Applied Mathematics, 2, 190-194.
[20]	Duromola, M. K., Momoh, A. L., and Akinmoladun, O. M. (2022). Block Extension of a Single-Step Hybrid Multistep Method for Directly Solving Fourth- Order Initial Value Problems. American Journal of Computational Mathematics, 12(4), 355-371.
[21]	Hussain K., Fudziah I., and Senu N. (2016) Solving directly special fourth-order ordinary differential equations using Runge-Kutta type method. Journal of Computational and Applied Mathematics, 2016.
[22]	Shampine, L. F., and Watts, H. A. (1969) Block Implicit One-Step Methods. Mathematics of Computation, 23, 731-740. https://doi.org/10.1090/S0025-5718-1969-0264854-5
[23]	Kayode, S. J. (2008) A Zero Stable Method for Direct Solution of Fourth Order Ordinary Differential Equation. American Journal of Applied Sciences, 5, 1461-1466. http://dx.doi.org/10.3844/ajassp.2008.1461.1

Cite This Article

Plain Text BibTeX RIS

APA Style

Kolawole, D. M., Lukuman, M. A., Joseph, A. O. (2025). A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. American Journal of Applied Mathematics, 13(2), 103-116. https://doi.org/10.11648/j.ajam.20251302.11

Copy | Download

ACS Style

Kolawole, D. M.; Lukuman, M. A.; Joseph, A. O. A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. Am. J. Appl. Math. 2025, 13(2), 103-116. doi: 10.11648/j.ajam.20251302.11

Copy | Download

AMA Style

Kolawole DM, Lukuman MA, Joseph AO. A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. Am J Appl Math. 2025;13(2):103-116. doi: 10.11648/j.ajam.20251302.11

Copy | Download

@article{10.11648/j.ajam.20251302.11,
  author = {Duromola Monday Kolawole and Momoh Adelegan Lukuman and Akingbodi Oluwagbenga Joseph},
  title = {A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {2},
  pages = {103-116},
  doi = {10.11648/j.ajam.20251302.11},
  url = {https://doi.org/10.11648/j.ajam.20251302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251302.11},
  abstract = {This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs
AU  - Duromola Monday Kolawole
AU  - Momoh Adelegan Lukuman
AU  - Akingbodi Oluwagbenga Joseph
Y1  - 2025/02/27
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251302.11
DO  - 10.11648/j.ajam.20251302.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 103
EP  - 116
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251302.11
AB  - This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.
VL  - 13
IS  - 2
ER  -

Copy | Download

Author Information

Duromola Monday Kolawole

Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria; Department of Mathematical Sciences, Adekunle Ajasin University, Akungba-Akoko, Nigeria

Contact Email

http://orcid.org/0000-0002-7117-8418
Momoh Adelegan Lukuman

Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

Contact Email

http://orcid.org/0000-0003-4352-1079
Akingbodi Oluwagbenga Joseph

Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria

Contact Email

http://orcid.org/0009-0001-5062-8950

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Kolawole, D. M., Lukuman, M. A., Joseph, A. O. (2025). A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. American Journal of Applied Mathematics, 13(2), 103-116. https://doi.org/10.11648/j.ajam.20251302.11

Copy | Download

ACS Style

Kolawole, D. M.; Lukuman, M. A.; Joseph, A. O. A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. Am. J. Appl. Math. 2025, 13(2), 103-116. doi: 10.11648/j.ajam.20251302.11

Copy | Download

AMA Style

Kolawole DM, Lukuman MA, Joseph AO. A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs. Am J Appl Math. 2025;13(2):103-116. doi: 10.11648/j.ajam.20251302.11

Copy | Download

@article{10.11648/j.ajam.20251302.11,
  author = {Duromola Monday Kolawole and Momoh Adelegan Lukuman and Akingbodi Oluwagbenga Joseph},
  title = {A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {2},
  pages = {103-116},
  doi = {10.11648/j.ajam.20251302.11},
  url = {https://doi.org/10.11648/j.ajam.20251302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251302.11},
  abstract = {This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Block-based Linear Multistep Formula for Directly Solving Nonlinear Fourth-order Initial Value Problems of ODEs
AU  - Duromola Monday Kolawole
AU  - Momoh Adelegan Lukuman
AU  - Akingbodi Oluwagbenga Joseph
Y1  - 2025/02/27
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251302.11
DO  - 10.11648/j.ajam.20251302.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 103
EP  - 116
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251302.11
AB  - This paper suggested a block-based linear multistep formula for directly solving nonlinear fourth-order initial value problems of ordinary differential equations (ODEs). The method was achieved by applying collocation and interpolation techniques to a first-kind Chebyshev polynomial. A continuous scheme was constructed through this procedure from where the proposed discrete formula was extracted. The extracted discrete formula was then implemented in block mode using the block matrix formulation and written explicitly as block equations. The proposed method is zero-stable, consistent, convergent, and p-stable, as demonstrated by the analysis of the basic properties of the derived scheme, with theoretical order eight. Six numerical examples were solved with the derived method to test its accuracy and effectiveness, all showing minimal error. A comparison with existing methods in the cited literature revealed that the proposed method offers good performance with minor errors.
VL  - 13
IS  - 2
ER  -

Copy | Download