Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y)
In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y‴=f(x,y)y‴=f(x,y) denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has t...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2014
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| Online Access: | http://psasir.upm.edu.my/id/eprint/51880/ http://psasir.upm.edu.my/id/eprint/51880/1/Embedded%20explicit%20Runge%E2%80%93Kutta%20type%20methods%20for%20directly%20solving%20special%20third%20order%20differential%20equations%20y%E2%80%B4%3Df%28x%2Cy%29.pdf |
| _version_ | 1848851953357946880 |
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| author | Senu, Norazak Mechee, Mohammed Sahib Ismail, Fudziah Siri, Zailan |
| author_facet | Senu, Norazak Mechee, Mohammed Sahib Ismail, Fudziah Siri, Zailan |
| author_sort | Senu, Norazak |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y‴=f(x,y)y‴=f(x,y) denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge–Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs. |
| first_indexed | 2025-11-15T10:30:22Z |
| format | Article |
| id | upm-51880 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T10:30:22Z |
| publishDate | 2014 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-518802017-05-03T04:09:21Z http://psasir.upm.edu.my/id/eprint/51880/ Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) Senu, Norazak Mechee, Mohammed Sahib Ismail, Fudziah Siri, Zailan In this paper three pairs of embedded Runge–Kutta type methods for directly solving special third order ordinary differential equations (ODEs) of the form y‴=f(x,y)y‴=f(x,y) denoted as RKD methods are presented. The first is the RKD4(3) pair which is third order embedded in fourth-order method has the property first same as last (FSAL) whereby the last row of the coefficient matrix is equal to the vector output. The second method is the RKD5(4) pair followed by the RKD6(5) pair. The methods are derived with the strategies such that the higher order methods are very accurate and the lower order methods will give the best error estimates. Variables stepsize codes are developed based on the methods and used to solve a set of special third order problems. Numerical results are compared with the existing embedded Runge–Kutta pairs which require the problems to be reduced into a system of first order ODEs. Numerical results have clearly shown the advantage and the efficiency of the new RKD pairs. Elsevier 2014 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/51880/1/Embedded%20explicit%20Runge%E2%80%93Kutta%20type%20methods%20for%20directly%20solving%20special%20third%20order%20differential%20equations%20y%E2%80%B4%3Df%28x%2Cy%29.pdf Senu, Norazak and Mechee, Mohammed Sahib and Ismail, Fudziah and Siri, Zailan (2014) Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y). Applied Mathematics and Computation, 240. pp. 281-293. ISSN 0096-3003 http://www.sciencedirect.com/science/article/pii/S0096300314006341 10.1016/j.amc.2014.04.094 |
| spellingShingle | Senu, Norazak Mechee, Mohammed Sahib Ismail, Fudziah Siri, Zailan Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| title | Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| title_full | Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| title_fullStr | Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| title_full_unstemmed | Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| title_short | Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| title_sort | embedded explicit runge–kutta type methods for directly solving special third order differential equations y‴=f(x,y) |
| url | http://psasir.upm.edu.my/id/eprint/51880/ http://psasir.upm.edu.my/id/eprint/51880/ http://psasir.upm.edu.my/id/eprint/51880/ http://psasir.upm.edu.my/id/eprint/51880/1/Embedded%20explicit%20Runge%E2%80%93Kutta%20type%20methods%20for%20directly%20solving%20special%20third%20order%20differential%20equations%20y%E2%80%B4%3Df%28x%2Cy%29.pdf |