A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations
Applied and Computational Mathematics
Volume 8, Issue 3, June 2019, Pages: 50-57
Received: Jan. 30, 2019;
Accepted: Mar. 17, 2019;
Published: Aug. 12, 2019
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Adoghe Lawrence Osa, Department of Mathematics, Ambrose Alli University, Ekpoma, Edo State, Nigeria
Omole Ezekiel Olaoluwa, Department of Mathematics, Federal University of Technology, Akure, Nigeria
In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were investigated and found to be zero stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing method.
Adoghe Lawrence Osa,
Omole Ezekiel Olaoluwa,
A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations, Applied and Computational Mathematics.
Vol. 8, No. 3,
2019, pp. 50-57.
Fatunla, S. O. (1988): Numerical methods for initial value problems in ordinary differential equations, Academic press Inc. Harcourt Brace Jovanovich Publishers, New York.
A. Olaide Adesanya, D. Mfon Udoh and, A. M. Ajileye (2013): A New Hybrid Block Method For The Solution Of General Third Order Initial Value Problems Of OrdinaryDifferential Equations: International Journal of Pure and Applied Mathematics Volume 86 No. 2, 365-375.
Adoghe L. O, Ogunware B. G and Omole E. O (2016): A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations: Journal of Theoretical Mathematics & Applications, 6 (3): 67-84.
Adoghe L. O and Omole E. O (2018): Comprehensive Analysis of 3-Quarter-Step Collocation Method for Direct Integration of Second Order Ordinary Differential Equations Using Taylor Series Function. ABACUS (Mathematics Science Series) Vol. 44, N0 2, pp. 311-321.
Awoyemi D. O (2001): A new Sixth –Order Algorithm for General Second Order Ordinary Differential Equations. International Journal of Computer mathematics, Vol 77, pp. 117-124.
Awoyemi, D. O and Idowu, O. M. (2005): A class of hybrid collocation Method for third ordinary differential equations. International Journal of Computer Math, 82 (10), 1287-1293.
Atkinson K. E (1989): An introduction to Numerical Analysis, 2nd Edition, John Wiley and Sons, New York.
Fatokun, J. O (2007): Continuous Approach for deriving Self –starting multistep methods for initial value problems in ordinary differential Equations. Journal of Engineering and Applied Sciences Vol. 2 (3) pp. 504-508.
Jain. M. K, Iyengar, S. R. K, Jain, R. K (2008): Numerical Methods for scientific and engineering computations (fifth edition). New Age International Publishers Limited.
Henrici, P. (1962): Discrete Variable method in ordinary differential equations, John Wiley and Sons, New York.
Kayode, S. J and A. Adeyeye (2011): A 3-step hybrid method for direct solution of second order initial value problems, Aust. J. of Basic and Applied Sciences, 5, No. 12 (2011), 2121-2126.
Lambert J. D (1973): Computational Methods in ODES. John Wiley & Sons, New York.
Lambert J. D. (1991): Numerical methods for initial value problems in ordinary differential equation, New York, Academics Press Inc.
Mohammed U. and R. B Adeniyi (2014): A Three Step Implicit Hybrid Linear Multistep Method for the Solution of Third Order Ordinary Differential Equations. Gen. Math. Notes, Vol. 25, No. 1, pp. 62-74.
Samuel. N. Jator (2008): On the numerical integration of third order boundary value problems by a linear multistep method; International Journal of Pure and Applied Mathematics, Vol. 46, No 3,375-388.
Skwame, Y., Sabo, J. & Kyagya, T. Y. (2017): The constructions of implicit one-step block hybrid methods with multiple off-grid points for the solution of stiff ODEs. JSRR, 16 (1): 1-7.
Tumba, P., Sabo, J. & Hamadina, M., (2018): Uniformly Order Eight Implicit Second Derivative Method for Solving Second- Order Stiff Ordinary Differential Equations ODEs. Academic Journal of Applied Mathematical Sciences, 4: 43-48.
Ogunware B. G, Adoghe L. O, Awoyemi D. O, Olanegan O. O., and Omole E. O (2018): Numerical Treatment of General Third Order Ordinary Differential Equations Using Taylor Series as Predictor. Physical Science International Journal, 17 (3): 1-8; DOI: 10.9734/PSIJ/2018/22219.
Yakubu D. G., Manjah N. H, Buba. S. S, and Masksha. A. I. (2011): A family of uniform accurate Lobatto–Runge–Kutta collocation methods: Journal of Computational and Applied Mathematics., Vol. 30, N. 2, pp 315-330.
Awoyemi D. O., Kayode S. J. and Adoghe L. O (2014).: A four–point fully implicit method for numerical integration of third-order ordinary differential equations, Int. J. Physical Sciences, 9 (1), 7-12.
T. A. Anake, A. O. Adesanya. G. J. Oghonyon, and M. C. Agarana (2013): Block Algorithm For General Third Order Ordinary Differential, ICASTOR Journal of Mathematical Sciences Vol. 7, No. 2, 127 – 136.
A. O. Adesanya, M. O. Udoh, A. M. Alkali (2012) A new block-predictor corrector algorithm for the solution of y′′′ = f (x, y, y′, y′′), American J. of Computational Mathematics, (2), 341-344.