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Theoretical analysis of Sinc-collocation methods and Sinc-Nystr?m methods for initial value problems  [PDF]
Tomoaki Okayama
Computer Science , 2013,
Abstract: A Sinc-collocation method has been proposed by Stenger, and he also gave theoretical analysis of the method in the case of a `scalar' equation. This paper extends the theoretical results to the case of a `system' of equations. Furthermore, this paper proposes more efficient method by replacing the variable transformation employed in Stenger's method. The efficiency is confirmed by both of theoretical analysis and numerical experiments. In addition to the existing and newly-proposed Sinc-collocation methods, this paper also gives similar theoretical results for Sinc-Nystr\"{o}m methods proposed by Nurmuhammad et al. From a viewpoint of the computational cost, it turns out that the newly-proposed Sinc-collocation method is the most efficient among those methods.
Numerical Solution of Troesch’s Problem by Sinc-Collocation Method  [PDF]
Mohamed El-Gamel
Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.44098
Abstract:

A new algorithm is presented for solving Troeschs problem. The numerical scheme based on the sinc-collocation technique is deduced. The equation is reduced to systems of nonlinear algebraic equations. Some numerical experiments are made. Compared with the modified homotopy perturbation technique (MHP), the variational iteration method and the Adomian decomposition method. It is shown that the sinc-collocation method yields better results.

The Double Exponential Sinc Collocation Method for Singular Sturm-Liouville Problems  [PDF]
Philippe Gaudreau,Richard Slevinsky,Hassan Safouhi
Mathematics , 2014,
Abstract: Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with Sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential Sinc collocation method clearly illustrate the advantage of using the double exponential formula.
A variational sinc collocation method for strong-coupling problems  [PDF]
Paolo Amore
Physics , 2006, DOI: 10.1088/0305-4470/39/22/L01
Abstract: We have devised a variational sinc collocation method (VSCM) which can be used to obtain accurate numerical solutions to many strong-coupling problems. Sinc functions with an optimal grid spacing are used to solve the linear and non-linear Schr\"odinger equations and a lattice $\phi^4$ model in $(1+1)$. Our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision.
Polynomial spline collocation methods for second-order Volterra integrodifferential equations  [PDF]
Edris Rawashdeh,David Mcdowell,Leela Rakesh
International Journal of Mathematics and Mathematical Sciences , 2004, DOI: 10.1155/s016117120421033x
Abstract: We have presented a method for the construction of an approximation to the initial-value second-order Volterra integrodifferential equation (VIDE). The polynomial spline collocation methods described here give a superconvergence to the solution of the equation.
Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems  [PDF]
Philippe Gaudreau,Hassan Safouhi
Mathematics , 2015,
Abstract: Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.
Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems  [PDF]
Mohamed El-Gamel
Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.311225
Abstract: Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the sinc-collocation method for solving linear and nonlinear system of second order differential equation. The method is then tested on linear and nonlinear examples and a comparison with B-spline method is made. It is shown that the sinc-collocation method yields better results.
Numerical solution of integro differential equations based on double exponential transformation in the sinc-collocation method
M. A. Fariborzi Araghi,Gh. Kazemi Gelian
Applied Mathematics and Computational Intelligence , 2012,
Abstract: In this paper, we consider double exponential transformations to solve integro-differential equations by Sinc collocation method. Numerical examples illustrate the validity and applicability of the method. In addition, the method is easy to use and yields very accurate results.
Sinc Collocation Method for Solving the Benjamin-Ono Equation  [PDF]
Edson Pindza,Eben Maré
Journal of Computational Methods in Physics , 2014, DOI: 10.1155/2014/392962
Abstract: We propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential equations. The propagation of one soliton and the interaction of two solitons are used to validate our numerical method. The method is easy to implement and yields accurate results. 1. Introduction It is well known that nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of sciences, such as physics, biology, and chemistry. In this paper, we consider the initial value problem for the Benjamin-Ono (BO) equation of the form together with the following initial and the boundary conditions: Here is a real valued function and denotes the Hilbert transform defined and described by [1] (amongst others): where denotes the Cauchy principal value. This equation was derived by Benjamin [2] and later by Ono [3] as a model for one-dimensional waves in deep water and it has a close relation with the famous KdV equation which models long waves in shallow water [4]. We recall that the BO equation has an infinite sequence of invariants [5], the first three of which are The BO has been shown to admit rational analytical soliton solutions in and [6]. However, for a given arbitrary initial condition, finding analytical solutions of the BO equation becomes an intractable problem. Therefore the use of numerical methods plays an important role in the study of the dynamics of the BO equation. James and Weideman [7] used the Fourier method which implicitly assumes the periodicity of the boundary conditions and a method based on rational approximating function to compute numerical solutions of the BO. The rational method was shown to have spectacular accuracy for solutions that do not wander too far from the origin while, for long dated solutions, the Fourier method was shown to retain superior accuracy. Miloh et al. [8] proposed an efficient pseudospectral method for the numerical solution of the weakly nonlinear Benjamin-Ono equation for arbitrary initial conditions and suggested a practical new relationship for estimating the number of solitons in terms of arbitrary initial conditions. Thomée and Vsaudeva Murthy [9] used the Crank-Nicolson approximation in time and finite difference approximations in space to solve the BO equation. They
Solving the Coulomb Schrodinger Equation in d=2+1 via Sinc Collocation  [PDF]
Vasilios G. Koures
Physics , 1995, DOI: 10.1006/jcph.1996.0191
Abstract: We solve the non-relativistic Coulomb Shrodinger equation in d = 2+1 via sinc collocation. We get excellent convergence using a generalized sinc basis set in position space. Since convergence in position space could not be obtained with more common numerical techniques, this result helps to corroborate the conjecture that the use of a localized basis set within the context of light cone quantization can yield much better convergence. All of the computations presented here were performed on an IBM-compatible PC with an Intel 486DX2-66 microchip.
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