Convergence analysis of the numerical method for a singularly perturbed periodical boundary value problem
-
2180
Downloads
-
3224
Views
Authors
Musa Cakir
- Department of Mathematics, Faculty of Science, Yuzuncu Yil University, 65080, Van, Turkey.
Ilhame Amirali
- Department of Mathematics, Faculty of Art and Sciences, Duzce University, 81620, Duzce, Turkey.
Mustafa Kudu
- Department of Mathematics, Faculty of Art and Sciences, Erzincan University, 24000, Erzincan, Turkey.
Gabil M. Amiraliyev
- Department of Mathematics, Faculty of Art and Sciences, Erzincan University, 24000, Erzincan, Turkey.
Abstract
This work deals with the singularly perturbed periodical boundary value problem for a quasilinear
second-order differential equation. The numerical method is constructed on piecewise uniform Shishkin
type mesh, which gives first-order uniform convergence in the discrete maximum norm. Numerical results
supporting the theory are presented.
Share and Cite
ISRP Style
Musa Cakir, Ilhame Amirali, Mustafa Kudu, Gabil M. Amiraliyev, Convergence analysis of the numerical method for a singularly perturbed periodical boundary value problem, Journal of Mathematics and Computer Science, 16 (2016), no. 2, 248-255
AMA Style
Cakir Musa, Amirali Ilhame, Kudu Mustafa, Amiraliyev Gabil M., Convergence analysis of the numerical method for a singularly perturbed periodical boundary value problem. J Math Comput SCI-JM. (2016); 16(2):248-255
Chicago/Turabian Style
Cakir, Musa, Amirali, Ilhame, Kudu, Mustafa, Amiraliyev, Gabil M.. "Convergence analysis of the numerical method for a singularly perturbed periodical boundary value problem." Journal of Mathematics and Computer Science, 16, no. 2 (2016): 248-255
Keywords
- Singular perturbation
- periodical problem
- fitted difference method
- uniformly convergent
- boundary layer.
MSC
References
-
[1]
G. M. Amiraliyev, H. Duru, A uniformly convergent difference method for the periodical boundary value problem, Comput. Math. Appl., 46 (2003), 695-703.
-
[2]
G. M. Amiraliyev, Y. D. Mamedov, Difference schemes on the uniform mesh for singularly perturbed pseudo- parabolic equations, Turkish J. Math., 19 (1995), 207-222.
-
[3]
X. Cai, A conservative difference scheme for conservative differential equation with periodic boundary , Appl. Math. Mech. (English Ed.), 22 (2001), 1210-1215.
-
[4]
M. Çakır, G. M. Amiraliyev, A numerical method for a singularly perturbed three-point boundary value problem, J. Appl. Math., 2010 (2010 ), 17 pages.
-
[5]
E. P. Doolan, J. J. H. Miller, W. H. A. Schilders, Uniform numerical methods for problems with initial and boundary layers , Boole Press, Dublin (1980)
-
[6]
P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, G. I. Shishkin, Robust computational techniques for boundary layers, Chapman & Hall/CRC, Boca Raton (2000)
-
[7]
J. L. Gracia, E. O'Riordan, M. L. Pickett, A parameter robust second order numerical method for a singularly perturbed two-parameter problem, Appl. Numer. Math., 56 (2006), 962-980.
-
[8]
P. C. Lin, B. X. Jiang, A singular perturbation problem for periodic boundary differential equations , (Chinese) Appl. Math. Mech., 8 (1987), 929-937.
-
[9]
T. LinB, Layer-adapted meshes for reaction-convection-diffusion problems , Springer-Verlag, Berlin (2010)
-
[10]
T. LinB, H. G. Roos, Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameter, J. Math. Anal. Appl., 289 (2004), 355-366.
-
[11]
A. H. Nayfeh, Perturbation methods, Wiley-Interscience (John Wiley & Sons), New York (1973)
-
[12]
R. E. O'Malley, Singular perturbation methods for ordinary differential equations, Springer-Verlag, New York (1991)
-
[13]
A. A. Pechenkina, Solution of the periodic problem for second order ordinary differential equation with small paramete in its leading derivatives, Ural Scientific Centre, Sverdlovsk, (1980), 111-117.
-
[14]
J. I. Ramos, Exponentially-fitted methods on layer-adapted meshes, Appl. Math. Comput., 167 (2005), 1311-1330.
-
[15]
H. G. Roos, M. Stynes, L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Springer-Verlag, Berlin, Heidelberg (2008)
-
[16]
A. A. Samarskii, Theory of difference schemes , Marcel Dekker, Inc., New York (2001)
-
[17]
G. Sun, M. Stynes, A uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with multible solutions , Math. Comput., 65 (1996), 1085-1109.