A Comparative Study of Galerkin Finite Element and B-Spline Methods for Two Point Boundary Value Problems

Print
International Journal of Computer Applications
© 2013 by IJCA Journal
Volume 67 - Number 23
Year of Publication: 2013
Authors:
Dinkar Sharma
Sangeeta Arora
Sheo Kumar
10.5120/11532-5386

Dinkar Sharma, Sangeeta Arora and Sheo Kumar. Article: A Comparative Study of Galerkin Finite Element and B-Spline Methods for Two Point Boundary Value Problems. International Journal of Computer Applications 67(23):1-6, April 2013. Full text available. BibTeX

@article{key:article,
	author = {Dinkar Sharma and Sangeeta Arora and Sheo Kumar},
	title = {Article: A Comparative Study of Galerkin Finite Element and B-Spline Methods for Two Point Boundary Value Problems},
	journal = {International Journal of Computer Applications},
	year = {2013},
	volume = {67},
	number = {23},
	pages = {1-6},
	month = {April},
	note = {Full text available}
}

Abstract

In chemical engineering, deflection of beams and other area of engineering the two point boundary value problems with Neumann and mixed Robbin's boundary conditions have great importance. It is not easy task to solve numerically such type of problems. In this study a B-spline finite element has been introduced to get the solution of two point boundary value problem. Some test examples are considered for the applicability of the purposed scheme. Further the results are compared with simple Galerkin-finite element method and with the exact solution of the problems. Throughout the discussion, it is observed that the proposed technique is performing well.

References

  • Jang B (2008). Two-point boundary value problems by the extended Adomian decomposition method. Journal of Computational and Applied Mathematics 219(1), pp. 253-262.
  • Brenner H (1962). The diffusion model of longitudinal mixing in beds of finite length. Numerical values. Chemical Engineering Science 17(4), pp. 229-243.
  • Sharma D, Jiwari R and Kumar S (2011). Galerkin-finite element methods for numerical solution of advection- diffusion equation. International Journal of Pure and Applied Mathematics 70(3), pp. 389-399.
  • Noye BJ (1990). Numerical solution of partial differential equations, Lecture Notes.
  • Dogan A (2004). A Galerkin finite element approach of Burgers' equation. Applied Mathematics and Computation 157(2), pp. 331-346.
  • Sengupta TK, Talla SB and Pradhan SC (2005). Galerkin finite element methods for wave problems. Sadhana 30(5), pp. 611-623.
  • Kaneko H, Bey KS and Hou GJW (2006). Discontinuous Galerkin finite element method for parabolic problems. Applied Mathematics and Computation 182(1), pp. 388-402.
  • EI-Gebeily MA, Furati KM and O'Regan D (2009). The finite element-Galerkin method for singular self-adjoint differential equations. Journal of Computational and Applied Mathematics 223(2), pp. 735-752.
  • Jangveladze T, Kiguradze Z and Neta B (2011). Galerkin finite element method for one nonlinear integro-differential model. Applied Mathematics and Computation 217(16), pp. 6883-6892.
  • Padmaja P, Chakravarthy PP and Reddy YN (2012). A nonstandard explicit method for solving singularly perturbed two point boundary value problems via method of reduction of order. International Journal of Applied Mathematics and Mechanics 8(2), pp. 62-76.
  • Kumar R, Kumar S and Miglani A (2010). Boundary value problem in fluid saturated incompressible porous medium due to inclined load. International Journal of Applied Mathematics and Mechanics 6(13), pp. 11-27.
  • Shabani MO and Mazahery A (2011). Application of finite element method for simulation of mechanical properties in A356 alloy. International Journal of Applied Mathematics and Mechanics 7 (5), pp. 89-97.
  • Pathak AK and Doctor HD (2011). Comparison of collocation method â€" spline, finite difference and finite element method for solving two dimensional parabolic partial differential equation. International Journal of Applied Mathematics and Mechanics 7(18), pp. 56-68.
  • Mittal RC and Jiwari R (2009). Numerical study of burger-huxley equation by differential quadrature method. International Journal of Applied Mathematics and Mechanics 5(8), pp. 1-9.
  • Chun C and Sakthivel R (2010). Homotopy perturbation technique for solving two-point boundary value problems- comparison with other methods. Computer Physics Communications. 181 (6), pp. 1021-1024.
  • Keller H B (1968). Numerical methods for Two-point boundary value problems. Blaisdell Mass.
  • Sharma D, Jiwari R and Kumar S (2012). Numerical solution of two-point boundary value problems using Galerkin-Finite element method. International Journal of Non-linear Science. 13(2), pp. 204-210.
  • Fox L (1957). The numerical solution of Two-point boundary value problems in ordinary differential equations. Oxford University Press.
  • Bellman R E and Kalaba R E (1965). Quasilinearization and nonlinear boundary value problems. Rand Corp.
  • Sharma D, Jiwari R and Kumar S (2012). A comparative study of Model matrix and Finite element methods for two-point boundary value problems. International Journal of Applied Mathematics and Mechanics 8 (13), pp. 29-45.
  • Lee E S (1968). Quasilinearization and invariant imbedding. Academic Press, New York.