Research Article

Year 2020,
Volume: 17 Issue: 2, 80 - 95, 01.11.2020
### Abstract

### Thanks

### References

This paper presents the construction and implementation of a three-step optimized hybrid method for solving stiff system of first order initial value problems of ordinary differential equations. The method contains six implicit formulas which were obtained from a continuous approximation, using shifted chebyshev polynomial as the basis function, via evaluations at six different points on the selected three-step including three optimized intra-step points. The method is consistent, zero-stable and convergent. Numerical experiments are included to show the competitive and superior strength of the proposed method for solving these kinds of problems over similar properties of methods in literature.

The authors appreciate the anonymous reviewers for their work and criticism that greatly improve the manuscript.

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- [3] O. A. Akinfenwa, S. N. Jator, and N. M. Yao, “Continuous Block Backward Differentiation Formula for Solving Stiff Ordinary Differential Equations,” Computers and Mathematics with Applications , vol. 65, pp. 996-1005, 2013.
- [4] H. Musa, M. B. Suleiman, and N. Senu, Fully “Implicit 3-point Block Extended Backward Differentiation Formula for Stiff Initial Value Problems,” Applied Mathematical Sciences, vol. 6, pp. 4211-4228, 2012.
- [5] H. Musa, M. B. Suleiman, F. Ismail, N. Senu, and Z, B. Ibrahim, “An Improved 2-point Block Backward Differentiation Formula for Solving Stiff Initial Value Problems,” AIP Conference Proceedings, 2013, no. 1522, 211.
- [6] J. Sunday, M. R. Odekunle, A. A. James, and A. O. Adesanya, “Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator,” British Journal of Mathematics and Computer Science, vol. 4, no. 17, pp. 2471-2481, 2014.
- [7] H. Ramos, Z. Kalogiratou, Th. Monovasilis, and T. E. Simos, “A Trigonometrically Fitted Optimized Two-step Hybrid Block Method for Solving Initial Value Problems of the form 𝑦′′=𝑓(𝑥,𝑦,𝑦′) with Oscillatory Solutions,” AIP Conference Proceedings, 2015, no. 1648, 810007.
- [8] H. Ramos and P. Popescu, “How Many 𝑘-step Linear Block Methods Exist and which of them is Most Efficient and Simplest One?,” Applied Mathematics and Computation, vol. 36, pp. 296-309, 2018.
- [9] G. Singh, A. Garg, V. Kanwar, and H. Ramos, “An Efficient Optimized Adaptive Step-size Hybrid Block Method for Integrating Differential Systems,” Applied Mathematics and Computation, vol. 362, 124567, 2019.
- [10] M. O. Ogunniran, O. A. Tayo, Y. Haruna, and A. F. Adebisi, “Linear Stability Analysis of Runge-Kutta Methods for Singular Lane-Emden Equations,” Journal of Nigerian Society of Physical Sciences, vol. 2, pp. 134 – 140, 2020.
- [11] J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1991.
- [12] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1962.
- [13] Z. B. Ibrahim, K. I. Othman, and M. Suleiman, “Implicit 𝑟-block Backward Differentiation Formula for Solving First-order Stiff ODEs,” Applied Mathematics Computation, vol. 186, pp. 558-565, 2007.

Year 2020,
Volume: 17 Issue: 2, 80 - 95, 01.11.2020
### Abstract

### References

- [1] W. E. Milne, Numerical Solution of Differential Equations, John Wiley and Sons, New York, 1953.
- [2] H. Ramos, “An Optimized Two-step Hybrid Block Method for Solving First-order Initial-value Problems in ODEs,” Differential Geometry-Dynamic System, vol. 19, pp. 107-118, 2017.
- [3] O. A. Akinfenwa, S. N. Jator, and N. M. Yao, “Continuous Block Backward Differentiation Formula for Solving Stiff Ordinary Differential Equations,” Computers and Mathematics with Applications , vol. 65, pp. 996-1005, 2013.
- [4] H. Musa, M. B. Suleiman, and N. Senu, Fully “Implicit 3-point Block Extended Backward Differentiation Formula for Stiff Initial Value Problems,” Applied Mathematical Sciences, vol. 6, pp. 4211-4228, 2012.
- [5] H. Musa, M. B. Suleiman, F. Ismail, N. Senu, and Z, B. Ibrahim, “An Improved 2-point Block Backward Differentiation Formula for Solving Stiff Initial Value Problems,” AIP Conference Proceedings, 2013, no. 1522, 211.
- [6] J. Sunday, M. R. Odekunle, A. A. James, and A. O. Adesanya, “Numerical Solution of Stiff and Oscillatory Differential Equations Using a Block Integrator,” British Journal of Mathematics and Computer Science, vol. 4, no. 17, pp. 2471-2481, 2014.
- [7] H. Ramos, Z. Kalogiratou, Th. Monovasilis, and T. E. Simos, “A Trigonometrically Fitted Optimized Two-step Hybrid Block Method for Solving Initial Value Problems of the form 𝑦′′=𝑓(𝑥,𝑦,𝑦′) with Oscillatory Solutions,” AIP Conference Proceedings, 2015, no. 1648, 810007.
- [8] H. Ramos and P. Popescu, “How Many 𝑘-step Linear Block Methods Exist and which of them is Most Efficient and Simplest One?,” Applied Mathematics and Computation, vol. 36, pp. 296-309, 2018.
- [9] G. Singh, A. Garg, V. Kanwar, and H. Ramos, “An Efficient Optimized Adaptive Step-size Hybrid Block Method for Integrating Differential Systems,” Applied Mathematics and Computation, vol. 362, 124567, 2019.
- [10] M. O. Ogunniran, O. A. Tayo, Y. Haruna, and A. F. Adebisi, “Linear Stability Analysis of Runge-Kutta Methods for Singular Lane-Emden Equations,” Journal of Nigerian Society of Physical Sciences, vol. 2, pp. 134 – 140, 2020.
- [11] J. D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1991.
- [12] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1962.
- [13] Z. B. Ibrahim, K. I. Othman, and M. Suleiman, “Implicit 𝑟-block Backward Differentiation Formula for Solving First-order Stiff ODEs,” Applied Mathematics Computation, vol. 186, pp. 558-565, 2007.

There are 13 citations in total.

Primary Language | English |
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Subjects | Engineering |

Journal Section | Articles |

Authors | |

Publication Date | November 1, 2020 |

Published in Issue | Year 2020 Volume: 17 Issue: 2 |