A Block Procedure with Linear Multi-Step Methods Using Legendre Polynomials for Solving ODEs


In this article, we derive a block procedure for some K-step linear multi-step methods (for K = 1, 2 and 3), using Legendre polynomials as the basis functions. We give discrete methods used in block and implement it for solving the non-stiff initial value problems, being the continuous interpolant derived and collocated at grid and off-grid points. Numerical examples of ordinary differential equations (ODEs) are solved using the proposed methods to show the validity and the accuracy of the introduced algorithms. A comparison with fourth-order Runge-Kutta method is given. The ob-tained numerical results reveal that the proposed method is efficient.

Share and Cite:

Abualnaja, K. (2015) A Block Procedure with Linear Multi-Step Methods Using Legendre Polynomials for Solving ODEs. Applied Mathematics, 6, 717-723. doi: 10.4236/am.2015.64067.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Hairer, E. and Wanner, G. (1991) Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin.
[2] Atkinson, K.E. (1889) An Introduction to Numerical Analysis. 2nd Edition, John Wiley and Sons, New York.
[3] Khader, M.M. (2013) Numerical Treatment for Solving the Perturbed Fractional PDEs Using Hybrid Techniques. Journal of Computational Physics, 250, 565-573.
[4] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulations, 16, 2535-2542.
[5] Khader, M.M. (2013) The Use of Generalized Laguerre Polynomials in Spectral Methods for Fractional-Order Delay Differential Equations. Journal of Computational and Nonlinear Dynamics, 8, Article ID: 041018.
[6] Khader, M.M. (2013) An Efficient Approximate Method for Solving Linear Fractional Klein-Gordon Equation Based on the Generalized Laguerre Polynomials. International Journal of Computer Mathematics, 90, 1853-1864.
[7] Khader, M.M. and Sweilam, N.H. (2013) On the Approximate Solutions for System of Fractional Integro-Differential Equations Using Chebyshev Pseudo-Spectral Method. Applied Mathematical Modelling, 37, 9819-9828.
[8] Khader, M.M. and Hendy, A.S. (2013) A Numerical Technique for Solving Fractional Variational Problems. Mathematical Methods in Applied Sciences, 36, 1281-1289.
[9] Khader, M.M. and Babatin, M.M. (2013) On Approximate Solutions for Fractional Logistic Differential Equation. Mathematical Problems in Engineering, 2013, Article ID: 391901.
[10] Khader, M.M., Sweilam, N.H. and Mahdy, A.M.S. (2013) Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM. Applied Mathematics and Information Science, 7, 2011-2018.
[11] Lambert, J.D. (1991) Numerical Methods for ODE. John Wiley and Sons, New York.
[12] Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Integro-Differential Equations. ANZIAM Journal, 51, 464-475.
[13] Sweilam, N.H., Khader, M.M. and Nagy, A.M. (2011) Numerical Solution of Two-Sided Space-Fractional Wave Equation Using Finite Difference Method. Journal of Computational and Applied Mathematics, 235, 2832-2841.
[14] Sweilam, N.H., Khader, M.M. and Kota, W.Y. (2013) Numerical and Analytical Study for Fourth-Order Integro-Differential Equations Using a Pseudo-Spectral Method. Mathematical Problems in Engineering, 2013, Article ID: 434753, 7 pages.
[15] Hirayama, H. (2000) Arbitrary Order and A-Stable Numerical Method for Solving Algebraic Ordinary Differential Equation by Power Series. 2nd International Conference on Mathematics and Computers in Physics, Vouliagmeni, Athens, 9-16 July 2000, 1-6.
[16] Çelik, E. and Bayram, M. (2003) On the Numerical Solution of Differential-Algebraic Equations by Padé Series. Applied Mathematics and Computation, 137, 151-160.
[17] Onumanyi, P., Awoyemi, D.O., Jator, S.N. and Sirisena, U.W. (1994) New Linear Multistep Methods with Continuous Coefficients for First Order Initial Value Problems. Journal of the Nigerian Mathematical Society, 13, 7-51.
[18] Fatokun, J.O., Onumanyi, P. and Sirisena, U. (1999) A Multistep Collocation Based on Exponential Basis for Stiff Initial Value Problems. Nigerian Journal of Mathematics and Applications, 12, 207-223.
[19] Fatokun, J.O., Aimufua, G.I.O. and Ajibola, I.K.O. (2010) An Efficient Direct Collocation Method for the Integration of General Second Order Initial Value Problem. Journal of Institute of Mathematics & Computer Sciences, 21, 327-337.
[20] Fatunla, S.O. (1998) Numerical Method for Initial Value Problems in ODEs. Academic Press Inc., New York.
[21] Funaro, D. (1992) Polynomial Approximation of Differential Equations. Springer Verlag, New York.
[22] Bell, W.W. (1968) Special Functions for Scientists and Engineers. Butler and Tanner Ltd, Frome and London.
[23] Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (2006) Spectral Methods. Springer-Verlag, New York.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.