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## Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations

#### Muideen OGUNNİRAN [1] , Yahaya HARUNA [2] , Rapheal ADENİYİ [3] , Morufu Oyedunsi OLAYİWOLA [4]

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.
Optimized Hybrid Methods, Singularity, Stiff Problems, Intra-step Points
• [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.
Primary Language en Engineering guz Articles Author: Muideen OGUNNİRAN (Primary Author)Institution: Osun State University, Osogbo NigeriaCountry: Nigeria Author: Yahaya HARUNA Institution: University of Ilorin, IlorinCountry: Nigeria Author: Rapheal ADENİYİ Institution: University of Ilorin, IlorinCountry: Nigeria Author: Morufu Oyedunsi OLAYİWOLA Institution: Osun State University, OsogboCountry: Nigeria The authors appreciate the anonymous reviewers for their work and criticism that greatly improve the manuscript. Publication Date : November 1, 2020
 Bibtex @research article { cankujse733487, journal = {Cankaya University Journal of Science and Engineering}, issn = {}, eissn = {2564-7954}, address = {}, publisher = {Cankaya University}, year = {2020}, volume = {17}, pages = {80 - 95}, doi = {}, title = {Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations}, key = {cite}, author = {Ogunni̇ran, Muideen and Haruna, Yahaya and Adeni̇yi̇, Rapheal and Olayi̇wola, Morufu Oyedunsi} } APA Ogunni̇ran, M , Haruna, Y , Adeni̇yi̇, R , Olayi̇wola, M . (2020). Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations . Cankaya University Journal of Science and Engineering , 17 (2) , 80-95 . Retrieved from https://dergipark.org.tr/en/pub/cankujse/issue/57636/733487 MLA Ogunni̇ran, M , Haruna, Y , Adeni̇yi̇, R , Olayi̇wola, M . "Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations" . Cankaya University Journal of Science and Engineering 17 (2020 ): 80-95 Chicago Ogunni̇ran, M , Haruna, Y , Adeni̇yi̇, R , Olayi̇wola, M . "Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations". Cankaya University Journal of Science and Engineering 17 (2020 ): 80-95 RIS TY - JOUR T1 - Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations AU - Muideen Ogunni̇ran , Yahaya Haruna , Rapheal Adeni̇yi̇ , Morufu Oyedunsi Olayi̇wola Y1 - 2020 PY - 2020 N1 - DO - T2 - Cankaya University Journal of Science and Engineering JF - Journal JO - JOR SP - 80 EP - 95 VL - 17 IS - 2 SN - -2564-7954 M3 - UR - Y2 - 2020 ER - EndNote %0 Cankaya University Journal of Science and Engineering Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations %A Muideen Ogunni̇ran , Yahaya Haruna , Rapheal Adeni̇yi̇ , Morufu Oyedunsi Olayi̇wola %T Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations %D 2020 %J Cankaya University Journal of Science and Engineering %P -2564-7954 %V 17 %N 2 %R %U ISNAD Ogunni̇ran, Muideen , Haruna, Yahaya , Adeni̇yi̇, Rapheal , Olayi̇wola, Morufu Oyedunsi . "Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations". Cankaya University Journal of Science and Engineering 17 / 2 (November 2020): 80-95 . AMA Ogunni̇ran M , Haruna Y , Adeni̇yi̇ R , Olayi̇wola M . Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations. CUJSE. 2020; 17(2): 80-95. Vancouver Ogunni̇ran M , Haruna Y , Adeni̇yi̇ R , Olayi̇wola M . Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations. Cankaya University Journal of Science and Engineering. 2020; 17(2): 80-95. IEEE M. Ogunni̇ran , Y. Haruna , R. Adeni̇yi̇ and M. Olayi̇wola , "Optimized Three-step Hybrid Block Method for Stiff Problems in Ordinary Differential Equations", Cankaya University Journal of Science and Engineering, vol. 17, no. 2, pp. 80-95, Nov. 2020

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