An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

Nathaniel Mahwash Kamoh; Bwebum Cleofas Dang; Comfort Soomiyol Mrumun

doi:doi:10.11648/j.acm.20241304.15

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An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

Nathaniel Mahwash Kamoh^*

, Bwebum Cleofas Dang, Comfort Soomiyol Mrumun

Published in Applied and Computational Mathematics (Volume 13, Issue 4)

Received: 9 July 2024 Accepted: 29 July 2024 Published: 15 August 2024

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Abstract

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Published in	Applied and Computational Mathematics (Volume 13, Issue 4)
DOI	10.11648/j.acm.20241304.15
Page(s)	111-117
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), 2024. Published by Science Publishing Group

Keywords

Legendre Polynomials, Matrix Calculus, Differential Equations

References

[1]	N. Z. Mukhtar, Z. Abdul Majid, F. Ismail and M. Suleiman (2012): Numerical Solution for Solving Second Order Ordinary Differential Equations Using Block Method. International Journal of Modern Physics Conference, Vol. 9. pp 560–565.
[2]	Awoyemi, D. O. (2001). A New Sixth-Order Algorithm for General Second Order Ordinary Differential Equation. International Journal of Computational Mathematics, 77: 117-124.
[3]	Awoyemi, D. O and Kayode, S. J. (2005). An Implicit Collocation Method for Direct Solution of Second Order ODEs. Journal of Nigeria Association of Mathematical Physics, 24: 70-78.
[4]	Fatunla, S. O. (1995). A Class of Block Method for Second Order Initial Value Problems. International Journal of Computer Mathematics, 55(1&2): 119-133.
[5]	Adesanya, A. O., Anake, T. A., Bishop, S. A. and Osilagun, J. A. (2009). Two Steps Block Method for the Solution of General Second Order Initial Value Problems of Ordinary Differential Equation. Journal of Natural Sciences, Engineering and Technology, 8(1): 25-33.
[6]	Okunuga, S. A and Onumanyi, P. (1985). An Accurate Collocation Method for Solving Ordinary Differential Equations American Museum of Science and Energy Review, 4(4): 45-48.
[7]	Jator, S. N. and Li, J. (2009). A Self-Starting Linear Multistep Method for a Direct Solution of the General Second-Order Initial Value Problem. International Journal of Computer Mathematics, 86(5): 827-836.
[8]	Adesanya, A. O., Anake, T. A. and Udoh, M. O. (2008). Improved Continuous Method for Direct Solution of General Second Order Ordinary Differential Equations. Journal of the Nigeria Association of Mathematical Physics, 13: 59-62.
[9]	Abdulnastr Isah and Chang Phang (2016): Operational matrix based on Genocchi polynomials for solution of delay differential equations. Ain Shams Engineering journal, 9(4), PP. 2123-2128.
[10]	M. Sezer, (1996): A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, Int. J. Math. Educ. Sci. Technol. 27, PP. 821–834.
[11]	E. Tohidia, A. H. Bhrawy, K. Erfani (2013): A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Applied Mathematical Modeling 37, PP. 4283–4294.
[12]	Yunika Lestaria Ningsih and Anggria Septiani Mulbasari (2019): Exploring Students’ Difficulties in Solving Nonhomogeneous Second Order Ordinary Differential Equations with Initial Value Problems, Al-Jabar Jurnal Pendidikan Matematika, Vol. 10, No. 2, 2019, PP. 233–242.
[13]	A. O. Adesanya, T. A. Anake, S. A. Bishop and J. A. Osilagu (2009): Two Steps Block Method for the Solution of General Second Order Initial Value Problems of Ordinary Differential Equation. Journal of Natural Sciences, Engineering and Technology, 8(1), 25-33.
[14]	Zill, D. G. and Warren, W. S. (2013). Differential Equations with Boundary- Value Problems. Cengage Learning, books/Cole, eight editions. 664pp.
[15]	Higham, N. J. (2004). The Numerical Stability of Barycentric Lagrange Interpolation. SIMA Journal of Numerical Analysis, 24: 547–556.
[16]	W. M. Abd-Elhameed, Y. H. Youssri and E. H. Doha (2015): A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations, Math Sci (2015) 9: 93–102.

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APA Style

Kamoh, N. M., Dang, B. C., Mrumun, C. S. (2024). An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Applied and Computational Mathematics, 13(4), 111-117. https://doi.org/10.11648/j.acm.20241304.15

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ACS Style

Kamoh, N. M.; Dang, B. C.; Mrumun, C. S. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl. Comput. Math. 2024, 13(4), 111-117. doi: 10.11648/j.acm.20241304.15

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AMA Style

Kamoh NM, Dang BC, Mrumun CS. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl Comput Math. 2024;13(4):111-117. doi: 10.11648/j.acm.20241304.15

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@article{10.11648/j.acm.20241304.15,
  author = {Nathaniel Mahwash Kamoh and Bwebum Cleofas Dang and Comfort Soomiyol Mrumun},
  title = {An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations
},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {4},
  pages = {111-117},
  doi = {10.11648/j.acm.20241304.15},
  url = {https://doi.org/10.11648/j.acm.20241304.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241304.15},
  abstract = {In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.
},
 year = {2024}
}

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TY  - JOUR
T1  - An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

AU  - Nathaniel Mahwash Kamoh
AU  - Bwebum Cleofas Dang
AU  - Comfort Soomiyol Mrumun
Y1  - 2024/08/15
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241304.15
DO  - 10.11648/j.acm.20241304.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 111
EP  - 117
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241304.15
AB  - In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

VL  - 13
IS  - 4
ER  -

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Author Information

Nathaniel Mahwash Kamoh

Department of Mathematics, University of Jos, Jos, Nigeria

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http://orcid.org/0000-0001-6333-1913
Bwebum Cleofas Dang

Department of Mathematics, University of Jos, Jos, Nigeria

Contact Email
Comfort Soomiyol Mrumun

Department of Mathematics, Benue State University, Makurdi, Nigeria

Contact Email

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Plain Text BibTeX RIS

APA Style

Kamoh, N. M., Dang, B. C., Mrumun, C. S. (2024). An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Applied and Computational Mathematics, 13(4), 111-117. https://doi.org/10.11648/j.acm.20241304.15

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ACS Style

Kamoh, N. M.; Dang, B. C.; Mrumun, C. S. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl. Comput. Math. 2024, 13(4), 111-117. doi: 10.11648/j.acm.20241304.15

Copy | Download

AMA Style

Kamoh NM, Dang BC, Mrumun CS. An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations. Appl Comput Math. 2024;13(4):111-117. doi: 10.11648/j.acm.20241304.15

Copy | Download

@article{10.11648/j.acm.20241304.15,
  author = {Nathaniel Mahwash Kamoh and Bwebum Cleofas Dang and Comfort Soomiyol Mrumun},
  title = {An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations
},
  journal = {Applied and Computational Mathematics},
  volume = {13},
  number = {4},
  pages = {111-117},
  doi = {10.11648/j.acm.20241304.15},
  url = {https://doi.org/10.11648/j.acm.20241304.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241304.15},
  abstract = {In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

AU  - Nathaniel Mahwash Kamoh
AU  - Bwebum Cleofas Dang
AU  - Comfort Soomiyol Mrumun
Y1  - 2024/08/15
PY  - 2024
N1  - https://doi.org/10.11648/j.acm.20241304.15
DO  - 10.11648/j.acm.20241304.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 111
EP  - 117
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20241304.15
AB  - In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

VL  - 13
IS  - 4
ER  -

Copy | Download