In this communication, the homotopy perturbation method is modified and extended to obtain the analytical solutions of some nonlinear differential equations. Differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc. Fluid mechanics, heat and mass transfer, and electromagnetic theory are all modeled by partial differential equations (PDE) and all have plenty of real life applications. dynamic meteorology and numerical weather forecasting: the weather report you see every night on TV has been obtained from the numerical solution of a complex set of nonlinear PDEs. The numerical solution of nonlinear differential equations is extremely difficult. Here, the proposed technique is implemented to obtain the analytical solutions of the initial-value ordinary and partial differential equations. In the current study, some problems are solved using a newly modified method that outperforms all other known methods, with approximate results in the form of power series. The method's algorithm is described and illustrated using some well-known problems. The obtained results demonstrate the method's efficiency. Furthermore, those results implies that this new method is simpler to implement. The approach is powerful, effective, and promising in analyzing some classes of differential equations for heat conduction problems and other dynamical systems. To crystallize the new approach, some illustrated examples are introduced.
| Published in | American Journal of Mathematical and Computer Modelling (Volume 7, Issue 2) |
| DOI | 10.11648/j.ajmcm.20220702.11 |
| Page(s) | 20-30 |
| 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 |
Homotopy, Perturbation, Klein-Gordon, Duffing Equation, Schrödinger Equation, Picard Method
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APA Style
Galal Mahrous Moatimid, Mohamed Abdel-Latif Ramadan, Mahmoud Hamed Taha, Elsayed Eladdad. (2022). A Reliable Method for Obtaining Analytical Solutions to Some Ordinary and Partial Differential Equations with Initial Values. American Journal of Mathematical and Computer Modelling, 7(2), 20-30. https://doi.org/10.11648/j.ajmcm.20220702.11
ACS Style
Galal Mahrous Moatimid; Mohamed Abdel-Latif Ramadan; Mahmoud Hamed Taha; Elsayed Eladdad. A Reliable Method for Obtaining Analytical Solutions to Some Ordinary and Partial Differential Equations with Initial Values. Am. J. Math. Comput. Model. 2022, 7(2), 20-30. doi: 10.11648/j.ajmcm.20220702.11
AMA Style
Galal Mahrous Moatimid, Mohamed Abdel-Latif Ramadan, Mahmoud Hamed Taha, Elsayed Eladdad. A Reliable Method for Obtaining Analytical Solutions to Some Ordinary and Partial Differential Equations with Initial Values. Am J Math Comput Model. 2022;7(2):20-30. doi: 10.11648/j.ajmcm.20220702.11
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Download@article{10.11648/j.ajmcm.20220702.11,
author = {Galal Mahrous Moatimid and Mohamed Abdel-Latif Ramadan and Mahmoud Hamed Taha and Elsayed Eladdad},
title = {A Reliable Method for Obtaining Analytical Solutions to Some Ordinary and Partial Differential Equations with Initial Values},
journal = {American Journal of Mathematical and Computer Modelling},
volume = {7},
number = {2},
pages = {20-30},
doi = {10.11648/j.ajmcm.20220702.11},
url = {https://doi.org/10.11648/j.ajmcm.20220702.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20220702.11},
abstract = {In this communication, the homotopy perturbation method is modified and extended to obtain the analytical solutions of some nonlinear differential equations. Differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc. Fluid mechanics, heat and mass transfer, and electromagnetic theory are all modeled by partial differential equations (PDE) and all have plenty of real life applications. dynamic meteorology and numerical weather forecasting: the weather report you see every night on TV has been obtained from the numerical solution of a complex set of nonlinear PDEs. The numerical solution of nonlinear differential equations is extremely difficult. Here, the proposed technique is implemented to obtain the analytical solutions of the initial-value ordinary and partial differential equations. In the current study, some problems are solved using a newly modified method that outperforms all other known methods, with approximate results in the form of power series. The method's algorithm is described and illustrated using some well-known problems. The obtained results demonstrate the method's efficiency. Furthermore, those results implies that this new method is simpler to implement. The approach is powerful, effective, and promising in analyzing some classes of differential equations for heat conduction problems and other dynamical systems. To crystallize the new approach, some illustrated examples are introduced.},
year = {2022}
}
TY - JOUR T1 - A Reliable Method for Obtaining Analytical Solutions to Some Ordinary and Partial Differential Equations with Initial Values AU - Galal Mahrous Moatimid AU - Mohamed Abdel-Latif Ramadan AU - Mahmoud Hamed Taha AU - Elsayed Eladdad Y1 - 2022/04/20 PY - 2022 N1 - https://doi.org/10.11648/j.ajmcm.20220702.11 DO - 10.11648/j.ajmcm.20220702.11 T2 - American Journal of Mathematical and Computer Modelling JF - American Journal of Mathematical and Computer Modelling JO - American Journal of Mathematical and Computer Modelling SP - 20 EP - 30 PB - Science Publishing Group SN - 2578-8280 UR - https://doi.org/10.11648/j.ajmcm.20220702.11 AB - In this communication, the homotopy perturbation method is modified and extended to obtain the analytical solutions of some nonlinear differential equations. Differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc. Fluid mechanics, heat and mass transfer, and electromagnetic theory are all modeled by partial differential equations (PDE) and all have plenty of real life applications. dynamic meteorology and numerical weather forecasting: the weather report you see every night on TV has been obtained from the numerical solution of a complex set of nonlinear PDEs. The numerical solution of nonlinear differential equations is extremely difficult. Here, the proposed technique is implemented to obtain the analytical solutions of the initial-value ordinary and partial differential equations. In the current study, some problems are solved using a newly modified method that outperforms all other known methods, with approximate results in the form of power series. The method's algorithm is described and illustrated using some well-known problems. The obtained results demonstrate the method's efficiency. Furthermore, those results implies that this new method is simpler to implement. The approach is powerful, effective, and promising in analyzing some classes of differential equations for heat conduction problems and other dynamical systems. To crystallize the new approach, some illustrated examples are introduced. VL - 7 IS - 2 ER -
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