This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2.
| Published in | International Journal of Applied Mathematics and Theoretical Physics (Volume 8, Issue 1) |
| DOI | 10.11648/j.ijamtp.20220801.11 |
| Page(s) | 1-23 |
| 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), 2022. Published by Science Publishing Group |
Block Method, Special Second Order Delay Differential Equations, Five-step, Off-grid Points, Power Series, Exponential Function
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APA Style
Familua A. B. (2022). Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points. International Journal of Applied Mathematics and Theoretical Physics, 8(1), 1-23. https://doi.org/10.11648/j.ijamtp.20220801.11
ACS Style
Familua A. B. Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points. Int. J. Appl. Math. Theor. Phys. 2022, 8(1), 1-23. doi: 10.11648/j.ijamtp.20220801.11
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Download@article{10.11648/j.ijamtp.20220801.11,
author = {Familua A. B.},
title = {Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {8},
number = {1},
pages = {1-23},
doi = {10.11648/j.ijamtp.20220801.11},
url = {https://doi.org/10.11648/j.ijamtp.20220801.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20220801.11},
abstract = {This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2.},
year = {2022}
}
TY - JOUR T1 - Direct Approach for Solving Second Order Delay Differential Equations Through a Five-Step with Several Off-grid Points AU - Familua A. B. Y1 - 2022/02/05 PY - 2022 N1 - https://doi.org/10.11648/j.ijamtp.20220801.11 DO - 10.11648/j.ijamtp.20220801.11 T2 - International Journal of Applied Mathematics and Theoretical Physics JF - International Journal of Applied Mathematics and Theoretical Physics JO - International Journal of Applied Mathematics and Theoretical Physics SP - 1 EP - 23 PB - Science Publishing Group SN - 2575-5927 UR - https://doi.org/10.11648/j.ijamtp.20220801.11 AB - This paper presents a direct approach for numerical solution of special second order delay differential equations (DDEs) directly without reduction to systems of low orders. The methods were generated using collocation approach via a combination of power series and exponential function. The approximate basis functions are interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve second order Delay Differential Equations (DDEs) directly without the rigor of developing separate predictors. The required methods were obtained for step lengths of five with generalized number of hybrid points (3k). The basic properties of the methods were examined, the methods were found to have high order of accuracy of 21, low error constant, gives large interval of absolute stability, zero stable, consistence and convergent. The developed methods were applied to solve some special second order Delay Differential Equations. The methods also solve an engineering problem namely Matheiu’s equation in order to test for the efficiency and accuracy of the new methods. The results obtained were compared with existing methods in the literature. The results obtained showed better performance than some existing methods. The stability domain of the method is showed in figure 1 whereas the efficiency curve of the application problem for linear and nonlinear is presented in figure 2. VL - 8 IS - 1 ER -
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