Sinc-Galerkin and Sinc-Collocation methods in the solution of nonlinear two-point boundary value problems
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Sinc-Galerkin and Sinc-Collocation methods in the solution of nonlinear two-point boundary value problems Jalil Rashidinia · Mohammad Nabati
Received: 11 September 2012 / Accepted: 11 October 2012 / Published online: 16 April 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving singular and nonsingular nonlinear second-order two-point boundary value problems (BVPs) with nonhomogeneous boundary conditions is given. We developed the Sinc-Galerkin and Sinc-Collocation methods to approximate the nonlinear two-point BVPs. These method are tested on the test examples and compared with several existing methods. The demonstrated results show that both of the presented methods more or less have the same accuracy, but in comparison with the other existing methods are considerable accurate. Keywords
Sinc-Galerkin · Sinc-Collocation · Singular nonlinear BVP
Mathematics Subject Classification (2000)
65L10 · 65L60 · 34B16
1 Introduction Application of Sinc function to numerical solution of boundary value problems (BVPs) has been studied by Stenger (1979) more than 30 years ago. In addition, the Sinc approach is more efficient for solution of singular and nonsingular problems. In comparison with the finite difference, finite element and boundary element methods, the Sinc approach has been more suitable for handling singularities in boundary layers and semi-infinite domain (Morlet 1995), however the approximation by Sinc function depends only on parameters of the problem, regardless of whether it is singular or nonsingular. These approximations yield
Communicated by Domingo Tarzia. J. Rashidinia (B) · M. Nabati School of Mathematics, Iran University of Science and Technology, Hengam, Narmak, 168613114 Tehran, Iran e-mail: [email protected] M. Nabati e-mail: [email protected]
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both an affective and rapidly convergent scheme and circumvent the instability problems that one typically encounters in some difference methods (Sababheh et al. 2003). It is well known that the approximations by Sinc functions are typified by errors of the form O(exp(− hc )) where c is positive constant and h is a step size. The above reasons and the good accuracy for problems with singularities cause many authors to use these approximations for solving problems. Bialecki (1991) used Sinc-Collocation approach to solve a two-point linear BVPs. Gamel et al. used Sinc-Galerkin approach for solving linear and nonlinear ordinary and partial differential equations (Mohsen and EL-Gamel 2008; El-Gamel and zayed 2004; El-Gamel et al. 2004). Dehghan (2007) used Sinc-Collocation method for solving nonlinear system of second-order BVPs. The books by Stenger (1993, 2010) and Lund and Bowers (1992) provide excellent overviews of existing methods based on Sinc functions for solving ODEs, PDEs and integral equations. In our previous work, we applied the Sinc-Collocation method for solution of linear and nonline
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