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.
1 Introduction
Sinc methods were introduced by Frank Stenger in [1] and expanded upon by him in [2]. Sinc functions were first analyzed in [3] and [4]. An extensive research of sinc methods for two-point boundary value problems can be found in [5, 6]. In [7, 8], parabolic and hyperbolic problems were discussed in detail. Some kind of singular elliptic problems were solved in [9], and the symmetric sinc-Galerkin method was introduced in [10]. Sinc domain decomposition was presented in [11–13] and [14]. Iterative methods for symmetric sinc-Galerkin systems were discussed in [15, 16] and [17]. Sinc methods were discussed thoroughly in [18]. Applications of sinc methods can also be found in [19, 20] and [21]. The article [22] summarizes the results obtained to date on sinc numerical methods of computation. In [14], a numerical solution of a Volterra integro-differential equation by means of the sinc collocation method was considered. The paper [2] illustrates the application of a sinc-Galerkin method to an approximate solution of linear and nonlinear second-order ordinary differential equations, and to an approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The fully sinc-Galerkin method was developed for a family of complex-valued partial differential equations with time-dependent boundary conditions [19]. Some novel procedures of using sinc methods to compute solutions to three types of medical problems were illustrated in [23], and sinc-based algorithm was used to solve a nonlinear set of partial differential equations in [24]. A new sinc-Galerkin method was developed for approximating the solution of convection diffusion equations with mixed boundary conditions on half-infinite intervals in [25]. The work which was presented in [26] deals with the sinc-Galerkin method for solving nonlinear fourth-order differential equations with homogeneous and nonhomogeneous boundary conditions. In [27], sinc methods were used to solve second-order ordinary differential equations with homogeneous Dirichlet-type boundary conditions.
2 Sinc functions preliminaries
Let C denote the set of all complex numbers, and for all , define the sine cardinal or sinc function by
(2.1)
For , the translated sinc function with evenly spaced nodes is given by
(2.2)
For various values of k, the sinc basis function on the whole real line is illustrated in Figure 1. For various values of h, the central function is illustrated in Figure 2.
Definition 2.1 Let be a simply connected domain in the complex plane C, and let denote the boundary of . Let a, b be points on and ϕ be a conformal map onto such that and . If the inverse map of ϕ is denoted by φ, define
(2.10)
and , .
Definition 2.2 Let be the class of functions F that are analytic in and satisfy
(2.11)
where
(2.12)
and those on the boundary of satisfy
(2.13)
The proof of following theorems can be found in [2].
Theorem 2.1Let Γ be , , then forsufficiently small,
(2.14)
where
(2.15)
For the sinc-Galerkin method, the infinite quadrature rule must be truncated to a finite sum. The following theorem indicates the conditions under which an exponential convergence results.
Theorem 2.2If there exist positive constantsα, βandCsuch that
(2.16)
then the error bound for the quadrature rule (2.14) is
(2.17)
The infinite sum in (2.14) is truncated with the use of (2.16) to arrive at the inequality (2.17). Making the selections
(2.18)
(2.19)
whereis an integer part of the statement andNis the integer value which specifies the grid size, then
(2.20)
We used Theorems 2.1 and 2.2 to approximate the integrals that arise in the formulation of the discrete systems corresponding to a second-order boundary value problem.
Theorem 2.3Letϕbe a conformal one-to-one map of the simply connected domainonto . Then
(2.21)
(2.22)
(2.23)
3 The sinc-Galerkin method for singular Dirichlet-type boundary value problems
Consider the following problem:
(3.1)
with Dirichlet-type boundary condition
(3.2)
where P, Q and F are analytic on D. We consider sinc approximation by the formula
(3.3)
(3.4)
The unknown coefficients in Eq. (3.3) are determined by orthogonalizing the residual with respect to the sinc basis functions. The Galerkin method enables us to determine the coefficients by solving the linear system of equations
(3.5)
Let and be analytic functions on D and the inner product in (3.5) be defined as follows:
(3.6)
where w is the weight function. For the second-order problems, it is convenient to take [2].
(3.7)
For Eq. (3.1), we use the notations (2.21)-(2.23) together with the inner product that, given (3.5) [2], showed to get the following approximation formulas:
(3.8)
(3.9)
(3.10)
(3.11)
where . If we choose and as given in [2] the accuracy for each equation between (3.8)-(3.11) will be .
Using (3.5), (3.8)-(3.11), we obtain a linear system of equations for numbers .
The linear system given in (3.5) can be expressed by means of matrices. Let , and let and be a column vector defined by
(3.12)
Let denote a diagonal matrix whose diagonal elements are and non-diagonal elements are zero, and also let , and denote the matrices
(3.13)
(3.14)
(3.15)
With these notations, the discrete system of equations in (3.5) takes the form:
(3.16)
Theorem 3.1Letcbe anm-vector whosejth component is . Then the system (3.16) yields the following matrix system, the dimensions of which are :
(3.17)
Now we have a linear system ofequations of theunknown coefficients. If we solve (3.17) by usingLUorQRdecomposition methods, we can obtaincoefficients for the approximate sinc-Galerkin solution
(3.18)
4 Examples
Three examples were given in order to illustrate the performance of the sinc-Galerkin method to solve a singular Dirichlet-type boundary value problem in this section. The discrete sinc system defined by (3.18) was used to compute the coefficients ; for each example. All of the computations were done by an algorithm which we have developed for the sinc-Galerkin method. The algorithm automatically compares the sinc-method with the exact solutions. It is shown in Tables 2-4 and Figures 5-7 that the sinc-Galerkin method is a very efficient and powerful tool to solve singular Dirichlet-type boundary value problems.
Figure 5
Approximation to the exact solution: the red colored curve displays the exact solution and the green one is the approximate solution of Eq. (4.1).
Example 4.1 Consider the following singular Dirichlet-type boundary value problem on the interval :
(4.1)
The exact solution of (4.1) is
We choose the weight function according to [2], , , and by taking , , for , the solutions inFigure 5 and Table 2 are achieved.
Example 4.2 Let us have the following form of a singular Dirichlet-type boundary value problem on the interval :
(4.2)
The problem has an exact solution like
where , .By taking , , for ,we get the solutions in Figure 6 and Table 3.
Example 4.3 The following problem is given on the interval :
(4.3)
where the exact solution of (4.3) is .
In this case, , , and by taking , , for , we get results in Figure 7 and Table 4 .
5 Conclusion
The sinc-Galerkin method was employed to find the solutions of second-order Dirichlet-type boundary value problems on some closed real interval. The main purpose was to find the solution of boundary value problems which arise from the singular problems. The examples show that the accuracy improves with increasing number of sinc grid points N. We have also developed a very efficient and rapid algorithm to solve second-order Dirichlet-type BVPs with the sinc-Galerkin method on the Maple computer algebra system. All of the above computations and graphical representations were prepared by using Maple.
We give the Maple code in the Appendix section.
Appendix: Maple code which we developed for the sinc-Galerkin approximation
References
Stenger F: Approximations via Whittaker’s cardinal function. J. Approx. Theory 1976, 17: 222-240. 10.1016/0021-9045(76)90086-1
Lund J, Bowers KL, McArthur KM: Symmetrization of the sinc-Galerkin method with block techniques for elliptic equations. IMA J. Numer. Anal. 1989, 9: 29-46. 10.1093/imanum/9.1.29
Ng M, Bai Z: A hybrid preconditioner of banded matrix approximation and alternating-direction implicit iteration for symmetric sinc-Galerkin linear systems. Linear Algebra Appl. 2003, 366: 317-335.
Narasimhan S, Majdalani J, Stenger F: A first step in applying the sinc collocation method to the nonlinear Navier Stokes equations. Numer. Heat Transf., Part B 2002, 41: 447-462. 10.1080/104077902753725902
El-Gamel M, Behiry SH, Hashish H: Numerical method for the solution of special nonlinear fourth-order boundary value problems. Appl. Math. Comput. 2003, 145: 717-734. 10.1016/S0096-3003(03)00269-8
Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, İstanbul, 34210, Turkey
Aydin Secer
Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Davutpasa, İstanbul, 34210, Turkey
The authors declare that they have no competing interests.
Authors’ contributions
AS proposed main idea of the solution schema by using Sinc Method for linear BVPs. He developed computer algorithm and worked on theoretical aspect of problem. MK searched the materials about study and compared with other techniques, contributed with his experience on Nonlinear Approximation methods.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Secer, A., Kurulay, M. The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems.
Bound Value Probl2012, 126 (2012). https://doi.org/10.1186/1687-2770-2012-126