Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations
Abstract Background Numerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order meth...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-03-01
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| Series: | Beni-Suef University Journal of Basic and Applied Sciences |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s43088-024-00484-y |
| _version_ | 1797233432736038912 |
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| author | Abdelrady Okasha Elnady Ahmed Newir Mohamed A. Ibrahim |
| author_facet | Abdelrady Okasha Elnady Ahmed Newir Mohamed A. Ibrahim |
| author_sort | Abdelrady Okasha Elnady |
| collection | DOAJ |
| description | Abstract Background Numerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping. Results The results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s. Conclusions The solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately. |
| first_indexed | 2024-04-24T16:16:05Z |
| format | Article |
| id | doaj.art-fc4d0ddc71fc41db90f018b266a854db |
| institution | Directory Open Access Journal |
| issn | 2314-8543 |
| language | English |
| last_indexed | 2024-04-24T16:16:05Z |
| publishDate | 2024-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Beni-Suef University Journal of Basic and Applied Sciences |
| spelling | doaj.art-fc4d0ddc71fc41db90f018b266a854db2024-03-31T11:28:45ZengSpringerOpenBeni-Suef University Journal of Basic and Applied Sciences2314-85432024-03-0113111410.1186/s43088-024-00484-yNovel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equationsAbdelrady Okasha Elnady0Ahmed Newir1Mohamed A. Ibrahim2Mechatronics Department, Faculty of Engineering, October 6 UniversityMechatronics Department, Faculty of Engineering, October 6 UniversityMechatronics Department, Faculty of Engineering, October 6 UniversityAbstract Background Numerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping. Results The results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s. Conclusions The solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately.https://doi.org/10.1186/s43088-024-00484-yVan der PolDuffingVan der Pol–DuffingNonlinear differential equationTaylor expansion |
| spellingShingle | Abdelrady Okasha Elnady Ahmed Newir Mohamed A. Ibrahim Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations Beni-Suef University Journal of Basic and Applied Sciences Van der Pol Duffing Van der Pol–Duffing Nonlinear differential equation Taylor expansion |
| title | Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations |
| title_full | Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations |
| title_fullStr | Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations |
| title_full_unstemmed | Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations |
| title_short | Novel approach for solving higher-order differential equations with applications to the Van der Pol and Van der Pol–Duffing equations |
| title_sort | novel approach for solving higher order differential equations with applications to the van der pol and van der pol duffing equations |
| topic | Van der Pol Duffing Van der Pol–Duffing Nonlinear differential equation Taylor expansion |
| url | https://doi.org/10.1186/s43088-024-00484-y |
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