A “sinc-Galerkin” method of solution of boundary value problems (original) (raw)
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Boundary Value Problems, 2012
A powerful technique based on the sinc-Galerkin method is presented for obtaining numerical solutions of second-order nonlinear Dirichlet-type boundary value problems (BVPs). The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. Without any numerical integration, the differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products; therefore, the evaluation is based on solving a matrix system. The solution is obtained by constructing the nonlinear (or linear) matrix system using Maple and the accuracy is compared with the Newton method. The main aspect of the technique presented here is that the obtained solution is valid for various boundary conditions in both linear and nonlinear equations and it is not affected by any singularities that can occur in variable coefficients or a nonlinear part of the equation. This is a powerful side of the method when being compared to other models.
The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems
Boundary Value Problems, 2012
The application of the sinc-Galerkin method to an approximate solution of second-order singular Dirichlet-type boundary value problems were discussed in this study. The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. The differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products without any numerical integration which is needed to solve matrix system. This study shows that the sinc-Galerkin method is a very effective and powerful tool in solving such problems numerically. At the end of the paper, the method was tested on several examples with second-order Dirichlet-type boundary value problems.
Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems
Journal of Mathematics, 2013
This paper presents a modified Galerkin method based on sinc basis functions to numerically solve nonlinear boundary value problems. The modifications allow for the accurate approximation of the solution with accurate derivatives at the endpoints. The algorithm is applied to well-known problems: Bratu and Thomas-Fermi problems. Numerical results demonstrate the clear advantage of the suggested modifications in obtaining accurate numerical solutions as well as accurate derivatives at the endpoints.
Abstract and Applied Analysis, 2013
An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their appropriate derivatives. In this work, PDEs have been converted to algebraic equation systems with new accurate explicit approximations of inner products without the need to calculate any numeric integrals. The solution of this system of algebraic equations has been reduced to the solution of a matrix equation system via Maple. The accuracy of the solutions has been compared with the exact solutions of the test problem. Computational results indicate that the technique presented in this study is valid for linear partial differential equations with various types of boundary conditions.
Sinc and the numerical solution of fifth-order boundary value problems
Applied Mathematics and Computation, 2007
Sinc methods are a family of self-contained methods of approximation, which have several advantages over classical methods of approximation in the case of the presence of end-point singularities. In this paper we present a fast and accurate numerical scheme for the fifth-order boundary value problems with two-point boundary conditions. The method is then tested on linear and nonlinear examples and a comparison with sixth-degree B-spline functions is made. It is shown that the Sinc-Galerkin method yields better results.
Journal of Computational Physics, 2007
One of the new techniques used in solving boundary-value problems involving ordinary differential equations is the Sinc-Galerkin method. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. A less known technique that has been around for almost two decades is the decomposition method. In this paper we solve boundary-value problems of higher order using these two methods and then compare the results. It is shown that the Sinc-Galerkin method in many instances gives better results.
In this paper, the initial and boundary value problems are solved by Galerkin weighted residual method. In the case of initial value problem, accuracy of Galerkin method is shown over exact solution. Accuracy is also continued to improve over the solutions by some standard numerical methods. It is shown that there is an astonishing accuracy of the Galerkin’s approximation method with even two terms in the case of initial value problem. Again In the cases of boundary value problem, some aspects of boundary problem are shown in solving them by Galerkin weighted residual approximation method. In this situation, the result of our calculation shows that basis functions are very dense in a space containing the actual solution. Galerkin finite element method is also introduced in solving boundary value problem. Resulting accuracy is also tested. Galerkin finite element method is found to be so effective that in this method an extraordinary accuracy is achieved with modest effort.
Sinc-Galerkin method for solving hyperbolic partial differential equations
An International Journal of Optimization and Control: Theories & Applications (IJOCTA)
In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.
International Journal of Engineering Research and Technology (IJERT), 2014
https://www.ijert.org/a-numerical-technique-of-initial-and-boundary-value-problems-by-galerkins-weighted-method-and-comparison-of-the-other-approximate-numerical-methods https://www.ijert.org/research/a-numerical-technique-of-initial-and-boundary-value-problems-by-galerkins-weighted-method-and-comparison-of-the-other-approximate-numerical-methods-IJERTV3IS20565.pdf In this paper, the initial and boundary value problems are solved by Galerkin weighted residual method. In the case of initial value problem, accuracy of Galerkin method is shown over exact solution. Accuracy is also continued to improve over the solutions by some standard numerical methods. It is shown that there is an astonishing accuracy of the Galerkin's approximation method with even two terms in the case of initial value problem. Again In the cases of boundary value problem, some aspects of boundary problem are shown in solving them by Galerkin weighted residual approximation method. In this situation, the result of our calculation shows that basis functions are very dense in a space containing the actual solution. Galerkin finite element method is also introduced in solving boundary value problem. Resulting accuracy is also tested. Galerkin finite element method is found to be so effective that in this method an extraordinary accuracy is achieved with modest effort. Keywords-Galerkin weighted residual, Galerkin finite element method, initial value problem, boundary value problem.