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 |
Legendre Polynomials, Matrix Calculus, Differential Equations
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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
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
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
@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} }
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 -