Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations
We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form y(iv)=f(x,y) denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will ca...
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| Language: | English |
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Hindawi Publishing Corporation
2015
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| Online Access: | http://psasir.upm.edu.my/id/eprint/51877/ http://psasir.upm.edu.my/id/eprint/51877/1/51877.pdf |
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| author | Hussain, Kasim Ismail, Fudziah Senu, Norazak |
| author_facet | Hussain, Kasim Ismail, Fudziah Senu, Norazak |
| author_sort | Hussain, Kasim |
| building | UPM Institutional Repository |
| collection | Online Access |
| description | We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form y(iv)=f(x,y) denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them. |
| first_indexed | 2025-11-15T10:30:21Z |
| format | Article |
| id | upm-51877 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-15T10:30:21Z |
| publishDate | 2015 |
| publisher | Hindawi Publishing Corporation |
| recordtype | eprints |
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| spelling | upm-518772017-05-02T08:06:03Z http://psasir.upm.edu.my/id/eprint/51877/ Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations Hussain, Kasim Ismail, Fudziah Senu, Norazak We present two pairs of embedded Runge-Kutta type methods for direct solution of fourth-order ordinary differential equations (ODEs) of the form y(iv)=f(x,y) denoted as RKFD methods. The first pair, which we will call RKFD5(4), has orders 5 and 4, and the second one has orders 6 and 5 and we will call it RKFD6(5). The techniques used in the derivation of the methods are that the higher order methods are very precise and the lower order methods give the best error estimate. Based on these pairs, we have developed variable step codes and we have used them to solve a set of special fourth-order problems. Numerical results show the robustness and the efficiency of the new RKFD pairs as compared with the well-known embedded Runge-Kutta pairs in the scientific literature after reducing the problems into a system of first-order ordinary differential equations (ODEs) and solving them. Hindawi Publishing Corporation 2015 Article PeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/51877/1/51877.pdf Hussain, Kasim and Ismail, Fudziah and Senu, Norazak (2015) Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations. Mathematical Problems in Engineering, 2015. art. no. 196595. pp. 1-12. ISSN 1024-123X; ESSN: 1563-5147 https://www.hindawi.com/journals/mpe/2015/196595/abs/ 10.1155/2015/196595 |
| spellingShingle | Hussain, Kasim Ismail, Fudziah Senu, Norazak Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations |
| title | Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations |
| title_full | Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations |
| title_fullStr | Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations |
| title_full_unstemmed | Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations |
| title_short | Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations |
| title_sort | two embedded pairs of runge-kutta type methods for direct solution of special fourth-order ordinary differential equations |
| url | http://psasir.upm.edu.my/id/eprint/51877/ http://psasir.upm.edu.my/id/eprint/51877/ http://psasir.upm.edu.my/id/eprint/51877/ http://psasir.upm.edu.my/id/eprint/51877/1/51877.pdf |