# MODIFIED LAGUERRE COLLOCATION BLOCK METHOD FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

• Helen Olaronke Edogbanya
Keywords: Linear Multi-step Method, Laguerre Polynomials, Second Order Ordinary Differential Equations, Basis Function, Grid Point.

### Abstract

In this paper, the derivation of block procedure for linear multi-step methods (K=2) using the Laguerre polynomials as the basis functions was considered. Discrete methods was given which were used in block and implemented for solving the initial value problems, being the continuous interpolation derived and collocated at grid points. The derived scheme was used to solve some second order ordinary differential equations (ODEs) in order to show their validity and accuracy. The numerical results obtained shows that the proposed methods are efficient in solving second order ordinary differential equations.

### References

Abualnaja, M. (2015). A block procedure with linear multistep methods using Legendre polynomials for solving ODEs. Applied Mathematics, 6, 717-723.

Adewale, A. J., .Olaide, A., & Sunday, J. (2013). Continuous block method for the solution of second order initial value problems of ordinary differential equation. International Journal of Pure and Applied Mathematics, 83(3), 405-421.

Awoyemi, D. O. (2001). A new sixth-order algorithm for general second order ODE. International Journal of Computational Mathematics, 117-124.

Awoyemi, D. O. (2003). A p-stable linear multi-step method for solving general third order ordinary differential equations. International Journal of Computational Mathematics, 80(8), 987-993.

Awoyemi, D. O. (1999). A class of continuous methods for general second order initial value problem in Ordinary Differential Equations.72:29-37.

Awoyemi, D. O. (1995). A two-step hybrid multistep method with continuous coefficients for initial value problems of general second order differential equations. Spectrum Journal, 2, 56-63.

Chu, M. T., & Hamilton, H. (1987). Parallel solution of ODEs by multi-block methods. SIAM Journal of Scientific and Statistical Computation, 8, 342-553.

El-Ajou, O., Abu, A., & Momani, S. (2015). Approximate analytical solution of the nonlinear fractional equation: A new iterative algorithm. Journal of Computational Physics, 293, 81-95.

El-Ajou, O., Abu, A. O., & Momani, S. (2015). A novel expansion iterative method for solving linear partial differential equations of fractional order. Applied Mathematics and Computation, 257, 119-133.

Fatunla, S. O. (1994). Block methods for second order initial value problems. International Journal of Computer Mathematics, 41, 55-63.

Isamil, F., Ken, Y. L., & Othman, M. (2009). Explicit and implicit 3-point block methods for solving special second order ordinary differential equations directly. International Journal of Mathematics Analysis, 3(5), 239-254.

Jator, S. N. (2007). A sixth order linear multistep method for the direct solution of second order differential equation. International Journal of Pure and Applied Mathematics, 40(4), 457-472.

Jator, S. N., & Li, J. (2009). A self-starting linear multistep method for a direct solution of the general second-order initial value problem. International Journal of Computer Mathematics, 86(5), 827-836.

Kayode, S. J. (2014). An improved numerical method for direct solution of general second order initial value problems of ordinary equations. National Mathematical Centre Proceedings.

Lambert, J. D. (1973). Computational methods in ordinary differential equations. New York: John Wiley and Sons..

Milne, W. E. (1953). Numerical solution of differential equations. New York: Wiley.

Okedayo, T. G., Owolanke, A. O., Amumeji, O. T. & Ogunbamike, O. K. (2018). Modified laguerre collocation block method for solving initial value problems of first order ordinary differential equations. Journal of Advance in Mathematics and Computer Science, 29(2), 1-13.

Omar, Z. B., & Suleiman, M. B. (1999). Solving second order ODEs directly using parallel 2-point explicit block method. Mathematical, 21(1), 15-23.

Omar, Z., & Suleiman, M. (2003). Parallel R-point implicit block method for solving higher order ordinary differential equation directly. Journal of Information Communication Technology, 3(1), 53-66.

Onumanyi, P., Sirisena, U., & Dauda, Y. (2001). Toward uniformly accurate continuous finite difference approximation of ordinary differential equations. Bagale Journal of Pure and Applied Science, 5-8.

Sararfyan, D. (1990). New algorithm for the continuous approximate solutions of ODEs. Journal of Computer and Mathematics, 77-100.

Yahaya, Y. A. (2004). Some theories and applications of continuous LMM for ordinary differential equations (Unpublished PhD Thesis). Nigeria: University of Jos, .

Yahaya, Y. A., & Badmus, A. M. (2009). Class of collocation methods for general second order ordinary differential equations. African Journal of Mathematics and Computer Science Research, 2(4), 69-72.

Published
2020-04-13
How to Cite
EdogbanyaH. O., & AdesuyiP. M. (2020). MODIFIED LAGUERRE COLLOCATION BLOCK METHOD FOR SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS. FUDMA JOURNAL OF SCIENCES, 4(1), 29 -36. Retrieved from https://fjs.fudutsinma.edu.ng/index.php/fjs/article/view/Remote%20URL
Section
Research Articles