The Solutions of Second Order Nonlinear Two Point Boundary Value Problems

##plugins.themes.bootstrap3.article.main##

  •   Abdurkadir Edeo Gemeda

Abstract

In this paper, generalized shifted Legendre polynomial approximation on a given arbitrary interval has been designed to find an approximate solution of a given second order nonlinear two point boundary value problems of ordinary differential equations. Here an approach using Tau method based on Legendre operational matrix of differentiation [2] & [5] has been addressed to generate the nonlinear systems of algebraic equations. The unknown Legendre coefficients of these nonlinear systems are the solutions of the system and they have been solved by continuation method. These unknown Legendre coefficients are then used to write the approximate solutions to the second order nonlinear two point boundary value problems. The validity and efficiency of the method has also been illustrated with numerical examples and graphs assisted by MATLAB.


Keywords: Boundary Value Problems (BVPs), Generalized Shifted Legendre Polynomials, Homotopy Continuation Method, Legendre Operational Matrix of Differentiation, Nonlinear Ordinary Differential Equations.

References

Richard L.Burden & J. Douglas Faires.“Numerical Analysis.” Brooks/Cole Learning, 9th Edition, 2010.

Chahn Yong Jung, Zeqing Liu, Arif Rafiq, Faisal Ali,& Shin Min Kang. “Solution of Second Order Linear and Nonlinear Ordinary Differential Equations Using Legendre Operational Matrix of Differentiation.” International Journal of Pure and Applied Mathematics. Volume 93 No. 2 2014.

Md. Shafiqul Islam, Afroza Shirin.“Numerical Solutions of a Class of Second Order Boundary Value Problems on Using Bernoulli Polynomials.” Applied Mathematics, 2, pp.1059-1067, 2011.

Yogesh Gupta.“A Numerical Algorithm for Solution of Boundary Value Problems with Applications.” International Journal of Computer Applications. Volume 40, No.8, February 2012.

Abbas Saadatmandi, Mehdi Dehghan. “A New Operational Matrix for Solving Fractional-Order Differential Equations.” Computers and Mathematics with Applications 59, pp. 1326-1336, 2010.

Nur Nadiah Abd Hamid, Ahmad Abd. Majid & Ahmad Izani Md. Ismail. “Extended Cubic B-Spline Method for Linear Two-Point Boundary Value Problems.” Sains Malaysiana 40(11), pp.1285–1290, 2011.

[J. Rashidinia and Sh. Sharif. “B-Spline Method for Two-Point Boundary Value Problems.” International Journal of Mathematical Modelling & Computations. Vol. 05, No. 02, pp. 111- 125, Spring 2015.

M. M. Rahman, M.A. Hossen, M. Nurul Islam and Md. Shajib Ali. “Numerical Solutions of Second Order Boundary Value Problems by Galerkin Method with Hermite Polynomials.” Annals of Pure and Applied Mathematics. Vol. 1, No. 2, pp.138-148, 2012.

Li-Bin Liu, Huan-Wen Liu, Yanping Chen. “Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions.” Applied Mathematics and Computation 217, pp.6872–6882, 2011.

Abdurkadir Edeo. ‘‘Solution of Second Order Linear and Nonlinear Two Point Boundary Value Problems Using Legendre Operational Matrix of Differentiation.’’ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS), Volume 51, No.1, pp 225-234,2019.

Downloads

Download data is not yet available.

##plugins.themes.bootstrap3.article.details##

How to Cite
[1]
Gemeda, A. 2019. The Solutions of Second Order Nonlinear Two Point Boundary Value Problems. European Journal of Engineering Research and Science. 4, 8 (Aug. 2019), 49-54. DOI:https://doi.org/10.24018/ejers.2019.4.8.1434.