×
Mori, Masatake; Nurmuhammad, Ahniyaz; Muhammad, Mayinur

DE-sinc method for second order singularly perturbed boundary value problems. (English) Zbl 1169.65074

Japan J. Ind. Appl. Math. 26, No. 1, 41-63 (2009).
Summary: 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 transformation, as the number of sampling points increases.

Cited in 3 Documents

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

Keywords:

double exponential transformation; DE formula; DE-sinc method; singular perturbation; sinc-Galerkin method; sinc-collocation method; numerical experiments
PDF BibTeX XML Cite
\textit{M. Mori} et al., Japan J. Ind. Appl. Math. 26, No. 1, 41--63 (2009; Zbl 1169.65074)
Full Text: DOI Euclid

References:

[1] 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. · Zbl 0459.65058
[2] 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. · Zbl 1058.65082
[3] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Method for Mathematical Computations. Prentice-Hall, 1977. · Zbl 0361.65002
[4] 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. · Zbl 0594.34017
[5] S.-T. Liu and Y. Xu, Galerkin methods based on Hermite splines for singular perturbation problems. SIAM J. Numer. Anal.,43 (2006), 2607–2623. · Zbl 1107.65064
[6] J. Lund and K.L. Bowers, Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia, 1992. · Zbl 0753.65081
[7] M. Muhammad and M. Mori, Double exponential formulas for numerical indefinite integration. J. Comput. Appl. Math.,161 (2003), 431–448. · Zbl 1038.65018
[8] 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. · Zbl 0971.65015
[9] 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. · Zbl 1073.65064
[10] 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. · Zbl 1117.65113
[11] R.E. O’Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations. Applied Mathematical Sciences,89, Springer-Verlag, 1991.
[12] I. Radeka and D. Herceg, High-order methods for semilinear singularly perturbed boundary value problems. Novi Sad J. Math.,33 (2003), 139–161. · Zbl 1087.65076
[13] S.M. Roberts, A boundary-value technique for singular perturbation problems. J. Math. Anal. Appl.,87 (1982), 489–508. · Zbl 0481.65048
[14] 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. · Zbl 0844.65075
[15] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, Berlin, New York, 1993. · Zbl 0803.65141
[16] M. Sugihara, Optimality of the double exponential formula–functional analysis approach–. Numer. Math.,75 (1997), 379–395. · Zbl 0868.41019
[17] M. Sugihara, Double exponential transformation in the sine-collocation method for two-point boundary value problems. J. Comput. Appl. Math.,149 (2002), 239–250. · Zbl 1014.65063
[18] M. Sugihara, Near optimality of the sinc approximation. Math. Comput.,72 (2003), 767–786. · Zbl 1013.41009
[19] H. Takahasi and M. Mori, Double exponential formulas for numerical integration. Publ. RIMS Kyoto Univ.,9 (1974), 721–741. · Zbl 0293.65011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.