Our focus is the
development and implementation
of a new two-step hybrid method for the direct solution of general second order
ordinary differential equation. Power
series is adopted as the basis function in the development of the method and
the arising differential system of equations is collocated at all grid and off-grid points. The
resulting equation is interpolated at selected points. We then analyzed the resulting scheme for its basic
properties. Numerical examples were taken to illustrate the efficiency of the
method. The
results obtained converge closely with the exact solutions.
Cite this paper
Owolanke, A. O. , Uwaheren, O. and Obarhua, F. O. (2017). An Eight Order Two-Step Taylor Series Algorithm for the Numerical Solutions of Initial Value Problems of Second Order Ordinary Differential Equations. Open Access Library Journal, 4, e3486. doi: http://dx.doi.org/10.4236/oalib.1103486.
Chakravati, P.C. and Worland, P.B. (1971) A Class of Self Starting Methods for the Nu-merical Solution of Ordinary Differential Equation. BIT, 11, 368-383.
https://doi.org/10.1007/BF01939405
Dahlquist, G. (1959) Convergence and Stability in the Numerical Integration of Ordinary Differential Equation. Mathematics Scandinavia, 4, 33-53.
https://doi.org/10.7146/math.scand.a-10454
Sharp and Fine (1992) Some Nystrom Pairs for the General Second Order Value Problem. Journal of Computation and Applied Mathematics, 42, 279-291.
https://doi.org/10.1016/0377-0427(92)90081-8
Bun, R.A. and Vasilsyer, Y.D. (1990) A Numerical Methods for Solving Differential Equations of Any Order. Computational Mathematics and Mathematical Physics, 32, 317-330.
Lambert, J.D. and Wastson, A. (1976) Symmetric Multistep Method for Periodic Initial Value Problem. Journal of the Institute of Mathematics Application, 18, 189- 202.
https://doi.org/10.1093/imamat/18.2.189
El-Mikkawy, M.E. and El-Desouky, R. (2003) A New Optimized Non FSAL Embedded Runge-Kutta-Nystrom Algorithms of Order 6 and 4 in Six Stages. Applied Mathematics and Computation, 145, 33-43.
https://doi.org/10.1016/S0096-3003(02)00436-8
Awoyemi, D.O. and Kayode, S.J. (2005) A Maxima Order Collocation Method for Initial Value Problems of General Second Order Ordinary Differential Equation. Proceeding of the Conference Organized by the National Mathematical Center, Abuja.
Awoyemi, D.O. (1999) A Class of Continuous Method for General Second Order Initial Value Problems in Ordinary Differential Equations. International Journal of Computer Mathematics, 72, 29-37. https://doi.org/10.1080/00207169908804832
Awoyemi, D.O. (2001) A New Sixth Order Algorithms for General Second Order Ordinary Differential Equation. International Journal of Computer Mathematics, 77, 117-124.
https://doi.org/10.1080/00207160108805054
Awoyemi, D.O. (2005) An Algorithm Collocation Methods for Direct Solution of Special and General Fourth Order Initial Value Problems of Ordinary Differential Equation. Journal of Nigerian Association of Mathematical Physics, 6, 271-238.
Kayode, S.J. (2009) An Order Zero Stable Method for Direct Solution of Fourth Order Ordinary Differential Equation. American Journal of Applied Sciences, 5, 1461-1466.
Onumanyi, P., Sirisons, U.W. and Dauda, Y. (2001) Toward Uniformly Accurate Continuous Finite Difference Approximation of Ordinary Differential Equation. Bayero Journal of Pure and Applied Sciences, 1, 5-8.
Adesanya, A.O., Anake, T.A. and Udoh, M.O. (2008) Improved Continuous Method for Direct Solution of General Second Order Ordinary Differential Equation. Journal of Nigeria Association of Mathematical Physics, 13, 59-62.
Jator, S.N. and Li, J. (2009) A Self Stationary Linear Multistep Method for a Direct Solution of General Second Order Initial Value Problem. International Journal of Computer Mathematics, 86, 817-1165.
Adesanya, A.O. (2011) Block Methods for Direct Solution of General Higher Order Initial Value Problem of Ordinary Differential Equations. PhD Thesis, Federal University of Technology, Akure.
Badmus, A.M. and Yahaya, Y.A. (2009) An Accurate Uniform Order 6 Block Method for Direct Solution of General Second Order Ordinary Differential Equations. The Pacific Journal of Science and Technology, 10, 273-277.
