Improvement of a Sinc-collocation method for Fredholm integral equations of the second kind

A Sinc-collocation scheme for Fredholm integral equations of the second kind was proposed by Rashidinia–Zarebnia in 2005. In this paper, two improved versions of the Sinc-collocation scheme are presented. The first version is obtained by improving the scheme so that it becomes more practical, and natural from a theoretical view point. Then it is rigorously proved that the convergence rate of the modified scheme is exponential, as suggested in the literature. In the second version, the variable transformation employed in the original scheme, the “tanh transformation”, is replaced with the “double exponential transformation”. It is proved that the replacement improves the convergence rate drastically. Numerical examples which support the theoretical results are also given.

[1]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .

[2]  P. Kythe,et al.  Computational Methods for Linear Integral Equations , 2002 .

[3]  Masaaki Sugihara,et al.  Recent developments of the Sinc numerical methods , 2004 .

[4]  Jalil Rashidinia,et al.  Solution of a Volterra integral equation by the Sinc-collocation method , 2007 .

[5]  Masaaki Sugihara,et al.  Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration , 2013, Numerische Mathematik.

[6]  M. Golberg,et al.  Discrete projection methods for integral equations , 1996 .

[7]  M. Mori,et al.  The double-exponential transformation in numerical analysis , 2001 .

[8]  L. Delves,et al.  Computational methods for integral equations: Frontmatter , 1985 .

[9]  Masaaki Sugihara,et al.  Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation , 2005 .

[10]  F. Stenger.  Numerical Methods Based on Sinc and Analytic Functions , 1993 .

[11]  W. Rudin.  Real and complex analysis , 1968 .

[12]  Jalil Rashidinia,et al.  Numerical solution of linear integral equations by using Sinc-collocation method , 2005, Appl. Math. Comput..

[13]  F. Stenger.  Summary of Sinc numerical methods , 2000 .

[14]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[15]  R. Kress.  Linear Integral Equations , 1989 .

[16]  Kazuo Murota,et al.  Function classes for double exponential integration formulas , 2009, Numerische Mathematik.

[17]  W. Hackbusch.  Integral Equations: Theory and Numerical Treatment , 1995 .

[18]  Kazuo Murota,et al.  Function classes for successful DE-Sinc approximations , 2009, Math. Comput..

[19]  Masaaki Sugihara,et al.  Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind , 2010, J. Comput. Appl. Math..

[20]  Jalil Rashidinia,et al.  Convergence of approximate solution of system of Fredholm integral equations , 2007 .