Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method
Mathematics and Computer Science
Volume 2, Issue 5, September 2017, Pages: 66-78
Received: Apr. 28, 2017;
Accepted: Jun. 6, 2017;
Published: Sep. 18, 2017
Views 2214 Downloads 195
Akalu Abriham Anulo, Department of Mathematics, Institute of Technology, Dire Dawa University, Dire Dawa, Ethiopia
Alemayehu Shiferaw Kibret, Department of Mathematics, Jimma University, Jimma, Ethiopia
Genanew Gofe Gonfa, Department of Mathematics, Jimma University, Jimma, Ethiopia
Ayana Deressa Negassa, Department of Mathematics, Jimma University, Jimma, Ethiopia
Follow on us
In this paper, the Galerkin method is applied to second order ordinary differential equation with mixed boundary after converting the given linear second order ordinary differential equation into equivalent boundary value problem by considering a valid assumption for the independent variable and also converting mixed boundary condition in to Neumann type by using secant and Runge-Kutta methods. The resulting system of equation is solved by direct method. In order to check to what extent the method approximates the exact solution, a test example with known exact solution is solved and compared with the exact solution graphically as well as numerically.
Second Order Ordinary Differential Equation, Mixed Boundary Conditions, Runge-Kutta, Secant Method, Galerkin Method, Chebyshev Polynomials
To cite this article
Akalu Abriham Anulo,
Alemayehu Shiferaw Kibret,
Genanew Gofe Gonfa,
Ayana Deressa Negassa,
Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method, Mathematics and Computer Science.
Vol. 2, No. 5,
2017, pp. 66-78.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Yattender Rishi Dubey: An approximate solution to buckling of plates by the Galerkin method, (August 2005).
Tai-Ran Hsu: Mechanical Engineering 130 Applied Engineering analysis, San Jose State University, (Sept 2009).
E. Suli: Numerical Solution of Ordinary Differential Equations, (April 2013).
J. N. Reddy: An Introduction to the finite element method, 3rd edition, McGraw-Hill, (Jan 2011) 58-98.
Marcos Cesar Ruggeri: Theory of Galerkin method and explanation of MATLAB code, (2006).
Jalil Rashidinia and Reza Jalilian: Spline solution of two point boundary value problems, Appl. Comput. Math 9 (2010) 258-266.
S. Das, Sunil Kumar and O. P. Singh: Solutions of nonlinear second order multipoint boundary value problems by Homotopy perturbation method, Appl. Appl. Math. 05 (2010) 1592-1600.
M. Idress Bhatti and P. Bracken: Solutions of differential equations in a Bernstein polynomials basis, J. Comput. Appl. Math. 205 (2007) 272-280.
M. M. Rahman. et.al: Numerical Solutions of Second Order Boundary Value Problems by Galerkin Method with Hermite Polynomials, (2012).
Arshad Khan: Parametric cubic spline solution of two point boundary value problems, Appl. Math. Comput. 154 (2004) 175-182.
Yuqiang Feng and Guangjun Li: Exact three positive solutions to a second-order Neumann boundary value problem with singular nonlinearity, Arabian J. Sci. Eng. 35 (2010) 189-195.
P. M. Lima and M. Carpentier: Numerical solution of a singular boundary-value problem in non-Newtonian fluid mechanics, Computer Phys. Communica. 126(2000) 114-120.
K. N. S. Kasi Viswanadham and Sreenivasulu Ballem: Fourth Order Boundary Value Problems by Galerkin Method with Cubic B-splines, (May 2013).
Jahanshahi et al.: A special successive approximations method for solving boundary value problems including ordinary differential equations, (August 2013).
L. Fox and I. B. Parker: Chebyshev Polynomials in Numerical Analysis, Oxford University Press, (1 May, 1967).