Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method

Akalu Abriham Anulo; Alemayehu Shiferaw Kibret; Genanew Gofe Gonfa; Ayana Deressa Negassa

doi:doi:10.11648/j.mcs.20170205.12

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Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method

Akalu Abriham Anulo, Alemayehu Shiferaw Kibret, Genanew Gofe Gonfa, Ayana Deressa Negassa

Published in Mathematics and Computer Science (Volume 2, Issue 5)

Received: 28 April 2017 Accepted: 6 June 2017 Published: 18 September 2017

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Abstract

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.

Published in	Mathematics and Computer Science (Volume 2, Issue 5)
DOI	10.11648/j.mcs.20170205.12
Page(s)	66-78
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Second Order Ordinary Differential Equation, Mixed Boundary Conditions, Runge-Kutta, Secant Method, Galerkin Method, Chebyshev Polynomials

References

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[3]	E. Suli: Numerical Solution of Ordinary Differential Equations, (April 2013).
[4]	J. N. Reddy: An Introduction to the finite element method, 3rd edition, McGraw-Hill, (Jan 2011) 58-98.
[5]	Marcos Cesar Ruggeri: Theory of Galerkin method and explanation of MATLAB code, (2006).
[6]	Jalil Rashidinia and Reza Jalilian: Spline solution of two point boundary value problems, Appl. Comput. Math 9 (2010) 258-266.
[7]	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.
[8]	M. Idress Bhatti and P. Bracken: Solutions of differential equations in a Bernstein polynomials basis, J. Comput. Appl. Math. 205 (2007) 272-280.
[9]	M. M. Rahman. et.al: Numerical Solutions of Second Order Boundary Value Problems by Galerkin Method with Hermite Polynomials, (2012).
[10]	Arshad Khan: Parametric cubic spline solution of two point boundary value problems, Appl. Math. Comput. 154 (2004) 175-182.
[11]	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.
[12]	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.
[13]	K. N. S. Kasi Viswanadham and Sreenivasulu Ballem: Fourth Order Boundary Value Problems by Galerkin Method with Cubic B-splines, (May 2013).
[14]	Jahanshahi et al.: A special successive approximations method for solving boundary value problems including ordinary differential equations, (August 2013).
[15]	L. Fox and I. B. Parker: Chebyshev Polynomials in Numerical Analysis, Oxford University Press, (1 May, 1967).

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APA Style

Akalu Abriham Anulo, Alemayehu Shiferaw Kibret, Genanew Gofe Gonfa, Ayana Deressa Negassa. (2017). Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method. Mathematics and Computer Science, 2(5), 66-78. https://doi.org/10.11648/j.mcs.20170205.12

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ACS Style

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. Math. Comput. Sci. 2017, 2(5), 66-78. doi: 10.11648/j.mcs.20170205.12

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AMA Style

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. Math Comput Sci. 2017;2(5):66-78. doi: 10.11648/j.mcs.20170205.12

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@article{10.11648/j.mcs.20170205.12,
  author = {Akalu Abriham Anulo and Alemayehu Shiferaw Kibret and Genanew Gofe Gonfa and Ayana Deressa Negassa},
  title = {Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method},
  journal = {Mathematics and Computer Science},
  volume = {2},
  number = {5},
  pages = {66-78},
  doi = {10.11648/j.mcs.20170205.12},
  url = {https://doi.org/10.11648/j.mcs.20170205.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170205.12},
  abstract = {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.},
 year = {2017}
}

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TY  - JOUR
T1  - Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method
AU  - Akalu Abriham Anulo
AU  - Alemayehu Shiferaw Kibret
AU  - Genanew Gofe Gonfa
AU  - Ayana Deressa Negassa
Y1  - 2017/09/18
PY  - 2017
N1  - https://doi.org/10.11648/j.mcs.20170205.12
DO  - 10.11648/j.mcs.20170205.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 66
EP  - 78
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20170205.12
AB  - 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.
VL  - 2
IS  - 5
ER  -

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Author Information

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

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Plain Text BibTeX RIS

APA Style

Akalu Abriham Anulo, Alemayehu Shiferaw Kibret, Genanew Gofe Gonfa, Ayana Deressa Negassa. (2017). Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method. Mathematics and Computer Science, 2(5), 66-78. https://doi.org/10.11648/j.mcs.20170205.12

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ACS Style

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. Math. Comput. Sci. 2017, 2(5), 66-78. doi: 10.11648/j.mcs.20170205.12

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AMA Style

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. Math Comput Sci. 2017;2(5):66-78. doi: 10.11648/j.mcs.20170205.12

Copy | Download

@article{10.11648/j.mcs.20170205.12,
  author = {Akalu Abriham Anulo and Alemayehu Shiferaw Kibret and Genanew Gofe Gonfa and Ayana Deressa Negassa},
  title = {Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method},
  journal = {Mathematics and Computer Science},
  volume = {2},
  number = {5},
  pages = {66-78},
  doi = {10.11648/j.mcs.20170205.12},
  url = {https://doi.org/10.11648/j.mcs.20170205.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20170205.12},
  abstract = {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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method
AU  - Akalu Abriham Anulo
AU  - Alemayehu Shiferaw Kibret
AU  - Genanew Gofe Gonfa
AU  - Ayana Deressa Negassa
Y1  - 2017/09/18
PY  - 2017
N1  - https://doi.org/10.11648/j.mcs.20170205.12
DO  - 10.11648/j.mcs.20170205.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 66
EP  - 78
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20170205.12
AB  - 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.
VL  - 2
IS  - 5
ER  -

Copy | Download