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

Author(s)
Khadijah M. Abualnaja

Affiliation(s)

Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia.

Department of Mathematics and Statistics, College of Science, Taif University, Taif, Saudi Arabia.

ABSTRACT

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.

In this article, we derive a block procedure for some K-step linear multi-step methods (for

KEYWORDS

Collocation Methods with Legendre Polynomials, Initial Value Problems, Perturbation Function, Fourth-Order Runge-Kutta Method

Collocation Methods with Legendre Polynomials, Initial Value Problems, Perturbation Function, Fourth-Order Runge-Kutta Method

Cite this paper

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.

Abualnaja, K. (2015) A Block Procedure with Linear Multi-Step Methods Using Legendre Polynomials for Solving ODEs.

References

[1] Hairer, E. and Wanner, G. (1991) Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin.

http://dx.doi.org/10.1007/978-3-662-09947-6

[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.

http://dx.doi.org/10.1016/j.jcp.2013.05.032

[4] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulations, 16, 2535-2542.

http://dx.doi.org/10.1016/j.cnsns.2010.09.007

[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.

http://dx.doi.org/10.1080/00207160.2013.764994

[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.

http://dx.doi.org/10.1016/j.apm.2013.06.010

[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.

http://dx.doi.org/10.1002/mma.2681

[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.

http://dx.doi.org/10.12785/amis/070541

[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.

http://dx.doi.org/10.1017/S1446181110000830

[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.

http://dx.doi.org/10.1016/j.cam.2010.12.002

[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.

http://dx.doi.org/10.1016/S0096-3003(02)00093-0

[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.

[1] Hairer, E. and Wanner, G. (1991) Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin.

http://dx.doi.org/10.1007/978-3-662-09947-6

[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.

http://dx.doi.org/10.1016/j.jcp.2013.05.032

[4] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulations, 16, 2535-2542.

http://dx.doi.org/10.1016/j.cnsns.2010.09.007

[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.

http://dx.doi.org/10.1080/00207160.2013.764994

[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.

http://dx.doi.org/10.1016/j.apm.2013.06.010

[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.

http://dx.doi.org/10.1002/mma.2681

[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.

http://dx.doi.org/10.12785/amis/070541

[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.

http://dx.doi.org/10.1017/S1446181110000830

[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.

http://dx.doi.org/10.1016/j.cam.2010.12.002

[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.

http://dx.doi.org/10.1016/S0096-3003(02)00093-0

[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.