This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.
| Published in | Mathematics Letters (Volume 6, Issue 2) |
| DOI | 10.11648/j.ml.20200602.12 |
| Page(s) | 13-31 |
| 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), 2020. Published by Science Publishing Group |
Derivation, Analysis, Fourth-order Ordinary Differential Equations, New numerical Method, Hybrid Techniques, Convergence, Zero Stability, Consistency, Taylor Series, Order 10, Integrators
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APA Style
Ezekiel Olaoluwa Omole, Luke Azeta Ukpebor. (2020). A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Mathematics Letters, 6(2), 13-31. https://doi.org/10.11648/j.ml.20200602.12
ACS Style
Ezekiel Olaoluwa Omole; Luke Azeta Ukpebor. A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Math. Lett. 2020, 6(2), 13-31. doi: 10.11648/j.ml.20200602.12
AMA Style
Ezekiel Olaoluwa Omole, Luke Azeta Ukpebor. A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Math Lett. 2020;6(2):13-31. doi: 10.11648/j.ml.20200602.12
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Download@article{10.11648/j.ml.20200602.12,
author = {Ezekiel Olaoluwa Omole and Luke Azeta Ukpebor},
title = {A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations},
journal = {Mathematics Letters},
volume = {6},
number = {2},
pages = {13-31},
doi = {10.11648/j.ml.20200602.12},
url = {https://doi.org/10.11648/j.ml.20200602.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20200602.12},
abstract = {This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.},
year = {2020}
}
TY - JOUR T1 - A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations AU - Ezekiel Olaoluwa Omole AU - Luke Azeta Ukpebor Y1 - 2020/09/23 PY - 2020 N1 - https://doi.org/10.11648/j.ml.20200602.12 DO - 10.11648/j.ml.20200602.12 T2 - Mathematics Letters JF - Mathematics Letters JO - Mathematics Letters SP - 13 EP - 31 PB - Science Publishing Group SN - 2575-5056 UR - https://doi.org/10.11648/j.ml.20200602.12 AB - This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods. VL - 6 IS - 2 ER -
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