Sinc Methods and Chebyshev Cardinal Functions for Solving Singular Boundary Value Problems

Yıl 2013, Cilt: 1 Sayı: 1, 1 - 6, 26.04.2013

Hossein Pourbashash , H. Kheiri A. Jodeyri A. Jodeyri Akbarfam S. S. Irandoust-pakchin

Öz

In this paper we consider boundary value problems with singularity in equation or solution. To solve these problems, we apply single exponential and double exponential transformations of sinc-Galerkin and Chebyshev cardinal functions. Numerical examples highlight efficiency of Chebyshev cardinal functions and sinc-Galerkin method in problems with singularity in equations. It is illustrated that in problems with singular solutions, Chebyshev cardinal functions is not applicable. However, sinc-Galerkin method overcomes to this difficultly.

Anahtar Kelimeler

Sinc method; Singular points; Double and single exponential transformation; Chebyshev cardinal functions

Original Research Paper

Öz

Kaynakça

• F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993.
• F. Stenger, “Summary of sinc numerical methods,” J. Comput. Appl. Math., 121, 379–420, 2000.
• A. Mohsen, M. El-Gamel, “On the Galerkin and collocation methods for two-point boundary value problems using sinc bases,” Computers and Mathematics with Applications 56, 930-941, 2008.
• M. El-Gamel, “A comparison between the sinc-Galerkin and the modified decomposition methods for solving two-point boundary-value problems,” J. of Computational Physics 223, 369-383, 2007.
• X. Wu, W. Kong, C. Li, “Sinc collocation method with boundary treatment for toe-point boundary value problems,” J. of Comput. and Appl. Math., 196, 229-240, 2006.
• K. Michael, “Fast iterative methods for symmetric sinc-Galerkin system,” IMA J. Numer. Anal., 19, 357-373, 1999.
• C. Ralph and K. Bowers, “The sinc-Galerkin method for fourth-order differential equations,” SIAM J. Numer. Anal., 28, 760-788, 1991.
• J. Lund, K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, PA, 1992.
• H. Takahasi, M. Mori, “Double exponential formulas for numerical integration,” Publ. Res. Inst. Math. Sci., 9, 721–741, 1974.
• M. Sugihara, “Optimality of the double exponential formula—functional analysis approach,” Numer. Math., 75, 379–395, 1997.
• M. Mori, M. Sugihara, “The Double exponential transformation in numerical analysis,” in: W. Gautschi, F. Marcelliàn, L.Reichel (Eds.), Numerical Analysis in the 20th Century, vol.V, Quadrature and Orthogonal Polynomials; J. Comput. Appl.Math., 127, 287–296, 2001.
• M. Muhammad, M. Mori, “Double exponential formulas for numerical indefinite integration,” J. Comput. Appl. Math., 161, 431–448, 2003.
• M. Sugihara, “Double exponential transformation in the sinc-collocation method for two-point boundary value problems,” J. Comput. Appl. Math., 149, 239–250, 2002.
• A. Nurmuhammad, M. Muhammad, M. Mori, M. Sugihara, “DE-sinc Method for Second Order Singularly Perturbed Boundary Value Problems,” Japan J. Indust. Appl. Math., 26, 41-63, 2009.
• M. Sugihara, “Near optimality of the sinc approximation,” Math. Comput., 72, 767-786, 2003.
• M. El-Gamel, A. Zayed, “Sinc–Galerkin method for solving nonlinear boundary-value problems,” Comput. Math. Appl., 48, 1285–1298, 2004.
• F. Keinert, “Uniform approximation to | | by sinc functions,” J. Approx. Theory 66, 44–52, 1991.
• F. Stenger, Sincpack-Summary of Basic Sinc Methods, Department of Computer Science, University of Utah, Salt Lake City, UT, 1995.
• M.S. Sababheh, A.M.N. Al-khaled, “Some convergence results on sinc interpolation,” J. Inequal. Pure Appl. Math., 4, 32–48, 2003.
• E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci., 20, 130–141, 1963.
• D. G. Dastidar, S. K. Majumdar, “The solution of painleve equations in chebishev series,” F. C. Auluk, F.N.A, 4(2), 155-160, 1972.
• M. Okayama, T. Matsuo, M. Sugihara, “Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind,” J. Comput. Appl. Math., 234, 1211-1227, 2010.
• K. Tanaka, M. Sugihara, K. Murota, “Function classes for successful DE-sinc approximations,” Math. Comput. 78, 1553-1571, 2009.
• M. Dehghan, M. Lakestani, “The use of Chebyshev cardinal functions for solution of the secondorder one-dimensional telegraph equation,” Numerical Methods for Partial Differential Equations, 25 (4),931–938, 2009.
• M. Lakestani, M. Dehghan, “Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions,” Computer Physics Communications 181, 957-966, 2010.
• M. Lakestani, M. Dehghan, “The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement,” Journal of Computational and Applied Mathematics, 235(3), 669–678, 2010.
• M. Lakestani, M. Dehghan, “Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions,” International Journal of Computer Mathematics 87 (6), 1389–1394, 2010.
• M. Lakestani, B. N. Saray, “Numerical solution of telegraph equation using interpolating scaling functions, Computers and Mathematics with Applications,” 60, 1964-1972, 2010.
• J. P. Boyd, Chebyshev and Fourier Spectral Methods, DOVER Publications, Inc., 2000.