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Peer-reviewedCC BYTikrit Journal of Pure Science

Efficient Solution of Third-Order ODEs with Diagonally Implicit RK Method Using Tri-Colored Tree and B-series Analysis

  • Esraa R. Darwesh1
  • Firas A. Fawzi1
  1. 1Department of Mathematics, Faculty of Computer Science and Mathematics, Tikrit University, Salahaddin, Iraq.
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Abstract

This study presents a pioneering approach to a new type of diagonally implicit Runge-Kutta-type (DIRKT) method developed specifically for solving third-order ODEs of this type. ѵ'''=f(ȶ,ѵ,ѵ'). These methods are called RKTG and are constructed using B series and Tri-colored tree theories to derive the required order conditions. We derive two methods: DRKTG4 and DRKTG5. The first achieves fourth-order accuracy in three stages, while the second reaches fifth order in four stages. Among other things, numerical experiments show these methods are accurate and efficient for solving these problems compared with existing Runge-Kutta methods, which posed these problems as first-order systems for testing.

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Cite

BibTeXRIS
BibTeX
@article{darwesh2026efficient,
  title = {{Efficient Solution of Third-Order ODEs with Diagonally Implicit RK Method Using Tri-Colored Tree and B-series Analysis}},
  author = {Esraa R. Darwesh and Firas A. Fawzi},
  journal = {Tikrit Journal of Pure Science},
  year = {2026},
  month = oct,
  doi = {10.25130/tjps.v31i5.2014},
  url = {https://doi.org/10.25130/tjps.v31i5.2014},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Efficient Solution of Third-Order ODEs with Diagonally Implicit RK Method Using Tri-Colored Tree and B-series Analysis
AU  - Esraa R. Darwesh
AU  - Firas A. Fawzi
PY  - 2026
DA  - 2026/10/25
JO  - Tikrit Journal of Pure Science
DO  - 10.25130/tjps.v31i5.2014
UR  - https://doi.org/10.25130/tjps.v31i5.2014
AB  - This study presents a pioneering approach to a new type of diagonally implicit Runge-Kutta-type (DIRKT) method developed specifically for solving third-order ODEs of this type. ѵ'''=f(ȶ,ѵ,ѵ'). These methods are called RKTG and are constructed using B series and Tri-colored tree theories to derive the required order conditions. We derive two methods: DRKTG4 and DRKTG5. The first achieves fourth-order accuracy in three stages, while the second reaches fifth order in four stages. Among other things, numerical experiments show these methods are accurate and efficient for solving these problems compared with existing Runge-Kutta methods, which posed these problems as first-order systems for testing.
SN  - 1813-1662
SN  - 2415-1726
ER  -