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A Five-Step P-Stable Method for the Numerical Integration of Third Order Ordinary Differential Equations
Department of Mathematics, Landmark University, Umuaru, Nigeria
Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
Department of Mathematics, Ambrose Alli University, Ekpoma, Nigeria
- 1 Department of Mathematics, Landmark University, Umuaru, Nigeria
- 2 Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria
- 3 Department of Mathematics, Ambrose Alli University, Ekpoma, Nigeria
American Journal of Computational Mathematics·Volume 04 (2014)·Pages 119–126·Published 17 April 2014·DOI10.4236/ajcm.2014.43011
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Abstract
In this paper we derived a continuous linear multistep method (LMM) with step number k = 5 through collocation and interpolation techniques using power series as basis function for approximate solution. An order nine p-stable scheme is developed which was used to solve the third order initial value problems in ordinary differential equation without first reducing to a system of first order equations. Taylor’s series algorithm of the same order was developed to implement our method. The result obtained compared favourably with existing methods.
KeywordsContinuous CollocationMultistep MethodsInterpolationThird OrderPower SeriesApproximate Solution
- Lambert, J.D. (1973) Computional Methods in ODEs. John Wiley & Sons, New York.
- Fatunla, S.O. (1988) Numerical Methods for Initial Value Problems in Ordinary Differential Equations. Academic Press Inc., Harcourt Brace, Jovanovich Publishers, New York.
- Butcher, S.C. (2003) Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, New York. http://dx.doi.org/10.1002/0470868279
- Henrici, P. (1962) Discrete Variable Method in Ordinary Differential Equations. John Wiley & Sons, New York. http://dx.doi.org/10.1002/zamm.19660460521
- Bruguano, L. and Trigiante. D. (1998) Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam, 280-299.
- Awoyemi, D.O. (2003) A P-Stable Linear Multistep Method for Solving General Third Order Ordinary Differential Equations. International Journal of computer Mathematics, 80, 987-993. http://dx.doi.org/10.1080/0020716031000079572
- Okunuga, S.A. and Ehijie, J. (2009) New Derivation of Continuous Multistep Methods Using Power Series as Basis Function. Journal of Modern Mathematics and Statistics, 3, 43-55.
- Adesanya, A.O. (2011) Block Methods for Direct Solutions of General Higher Order Initial Value Problems of Ordinary Differential Equations. Ph.D. Thesis, the Department of Mathematical Sciences, Federal University of Technology, Akure.
- Adeniyi, R.B. and Alabi, M.O. (2006) Derivation of Continuous Multistep Methods Using Chebyshev Polynomial Basis Functions. Abacus, 33, 351-361.
- Onumanyi, P., Oladele, J.O., Adeniyi, R.B. and Awoyemi, D.O. (1993) Derivation of Finite Difference Method by Collocation. Abacus, 23, 2-83.
- Olabode, B.T (2007) Some Linear Multistep Methods for Special and General Third Order Initial Value Problems in Ordinary Differential Equations. Ph.D. Thesis, the Department of Mathematical Sciences, The Federal University of Technology, Akure.