Abstract
In this paper the sinc-Galerkin method, as well as the sinc-collocation method, based on the double exponential transformation (DE transformation) for singularly perturbed boundary value problems of second order ordinary differential equation is considered. A large merit of the present method exists in that we can apply the standard sinc method with only a small care for perturbation parameter. Through several numerical experiments we confirmed higher efficiency of the present method than that of other methods, e.g., sinc method based on the single exponential (SE) transformation, as the number of sampling points increases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin, Ireland, 1980.
M. El-Gamel and J.R. Cannon, On the solution a of second order singularly perturbed boundary value problem by the sinc-Galerkin method. Z. angew. Math. Phys.,56 (2005), 45–58.
G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Method for Mathematical Computations. Prentice-Hall, 1977.
M.K. Kadalbajoo and Y.N. Reddy, An initial-value technique for a class of nonlinear singular perturbation problems. J. Opt. Theo. Appl.,53 (1987), 395–406.
S.-T. Liu and Y. Xu, Galerkin methods based on Hermite splines for singular perturbation problems. SIAM J. Numer. Anal.,43 (2006), 2607–2623.
J. Lund and K.L. Bowers, Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia, 1992.
M. Muhammad and M. Mori, Double exponential formulas for numerical indefinite integration. J. Comput. Appl. Math.,161 (2003), 431–448.
M. Mori and M. Sugihara, The double exponential transformation in numerical analysis. Numerical Analysis in the 20th Century, Vol. V: W. Gautschi, F. Marcellán and L. Reichal (eds.), Quadrature and Orthogonal Polynomials, J. Comput. Appl. Math.,127 (2001), 287–296.
A. Nurmuhammad, M. Muhammad, M. Mori and M. Sugihara, Double exponential transformation in the sinc-collocation method for a boundary value problem with fourth-order ordinary differential equation. J. Comput. Appl. Math.,182 (2005), 32–50.
A. Nurmuhammad, M. Muhammad and M. Mori, Sinc-Galerkin method based on the DE transformation for the boundary value problem of fourth-order ODE. J. Comput. Appl. Math.,206 (2007), 17–26.
R.E. O’Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations. Applied Mathematical Sciences,89, Springer-Verlag, 1991.
I. Radeka and D. Herceg, High-order methods for semilinear singularly perturbed boundary value problems. Novi Sad J. Math.,33 (2003), 139–161.
S.M. Roberts, A boundary-value technique for singular perturbation problems. J. Math. Anal. Appl.,87 (1982), 489–508.
H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems. Springer Series in Computational Mathematics,24, Springer 1996.
F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, Berlin, New York, 1993.
M. Sugihara, Optimality of the double exponential formula—functional analysis approach—. Numer. Math.,75 (1997), 379–395.
M. Sugihara, Double exponential transformation in the sine-collocation method for two-point boundary value problems. J. Comput. Appl. Math.,149 (2002), 239–250.
M. Sugihara, Near optimality of the sinc approximation. Math. Comput.,72 (2003), 767–786.
H. Takahasi and M. Mori, Double exponential formulas for numerical integration. Publ. RIMS Kyoto Univ.,9 (1974), 721–741.
Author information
Authors and Affiliations
Additional information
Partially supported by the Grant-in-Aid for the 21st Century COE (Center of Excellence) Research by the Ministry of Education, Culture, Sports, Science and Technology, and also by the Grant-in-Aid for Scientific Research (C) by Japan Society for the Promotion of Science.
In Pinyin notation: Aheniyazi Nuermaimaiti.
In Pinyin notation: Maimaiti Mayinuer.
About this article
Cite this article
Mori, M., Nurmuhammad, A. & Muhammad, M. DE-sinc method for second order singularly perturbed boundary value problems. Japan J. Indust. Appl. Math. 26, 41 (2009). https://doi.org/10.1007/BF03167545
Received:
Revised:
DOI: https://doi.org/10.1007/BF03167545