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This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

Open Access Research

The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems

Aydin Secer1* and Muhammet Kurulay2

Author affiliations

1 Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, İstanbul, 34210, Turkey

2 Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Davutpasa, İstanbul, 34210, Turkey

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Citation and License

Boundary Value Problems 2012, 2012:126  doi:10.1186/1687-2770-2012-126

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/126


Received:22 September 2012
Accepted:15 October 2012
Published:29 October 2012

© 2012 Secer and Kurulay; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
sinc-Galerkin method; sinc basis functions; Dirichlet-type boundary value problems; LU decomposition method

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M1">View MathML</a>, define the sine cardinal or sinc function by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M2">View MathML</a>

(2.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M3">View MathML</a>, the translated sinc function with evenly spaced nodes is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M4">View MathML</a>

(2.2)

For various values of k, the sinc basis function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M5">View MathML</a> on the whole real line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M6">View MathML</a> is illustrated in Figure 1. For various values of h, the central function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M7">View MathML</a> is illustrated in Figure 2.

thumbnailFigure 1. The basis functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M8">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M9">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M10">View MathML</a>.

thumbnailFigure 2. Central sinc basis function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M11">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M12">View MathML</a>.

If a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M13">View MathML</a> is defined over the real line, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M3">View MathML</a>, the series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M15">View MathML</a>

(2.3)

is called the Whittaker cardinal expansion of f whenever this series converges. The infinite strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M16">View MathML</a> of the complex w plane, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M17">View MathML</a>, is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M18">View MathML</a>

(2.4)

In general, approximations can be constructed for infinite, semi-infinite and finite intervals. Define the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M19">View MathML</a>

(2.5)

which is a conformal mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a>, the eye-shaped domain in the z-plane, onto the infinite strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M21">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M22">View MathML</a>

(2.6)

This is shown in Figure 3.

thumbnailFigure 3. The relationship between the eye-shaped domain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M23">View MathML</a>and the infinite strip<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M24">View MathML</a>.

For the sinc-Galerkin method, the basis functions are derived from the composite translated sinc functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M25">View MathML</a>

(2.7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M26">View MathML</a>. These are shown in Figure 4 for real values x. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M27">View MathML</a> is an inverse mapping of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M28">View MathML</a>. We may define the range of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M29">View MathML</a> on the real line as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M30">View MathML</a>

(2.8)

the evenly spaced nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M31">View MathML</a> on the real line. The image which corresponds to these nodes is denoted by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M32">View MathML</a>

(2.9)

A list of conformal mappings may be found in Table 1[6].

thumbnailFigure 4. Three adjacent members<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M33">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M34">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M35">View MathML</a>of the mapped sinc basis on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M36">View MathML</a>.

Table 1. Conformal mappings and nodes for some subintervals ofR

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a> be a simply connected domain in the complex plane C, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M45">View MathML</a> denote the boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a>. Let a, b be points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M45">View MathML</a> and ϕ be a conformal map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a> onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M21">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M51">View MathML</a>. If the inverse map of ϕ is denoted by φ, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M52">View MathML</a>

(2.10)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M54">View MathML</a> .

Definition 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M55">View MathML</a> be the class of functions F that are analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a> and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M57">View MathML</a>

(2.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M58">View MathML</a>

(2.12)

and those on the boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M60">View MathML</a>

(2.13)

The proof of following theorems can be found in [2].

Theorem 2.1Let Γ be<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M62">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M3">View MathML</a>sufficiently small,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M64">View MathML</a>

(2.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M65">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M66">View MathML</a>

(2.16)

then the error bound for the quadrature rule (2.14) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M67">View MathML</a>

(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)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M70">View MathML</a>is an integer part of the statement andNis the integer value which specifies the grid size, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M71">View MathML</a>

(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 domain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M20">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M21">View MathML</a>. Then

(2.21)

(2.22)

(2.23)

3 The sinc-Galerkin method for singular Dirichlet-type boundary value problems

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M77">View MathML</a>

(3.1)

with Dirichlet-type boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M78">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M81">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M81">View MathML</a> coefficients by solving the linear system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M83">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M84">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M85">View MathML</a> be analytic functions on D and the inner product in (3.5) be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M86">View MathML</a>

(3.6)

where w is the weight function. For the second-order problems, it is convenient to take [2].

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M87">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M92">View MathML</a>. If we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M93">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M94">View MathML</a> as given in [2] the accuracy for each equation between (3.8)-(3.11) will be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M95">View MathML</a>.

Using (3.5), (3.8)-(3.11), we obtain a linear system of equations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M96">View MathML</a> numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M81">View MathML</a>.

The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M98">View MathML</a> linear system given in (3.5) can be expressed by means of matrices. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M99">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M101">View MathML</a> be a column vector defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M102">View MathML</a>

(3.12)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M103">View MathML</a> denote a diagonal matrix whose diagonal elements are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M104">View MathML</a> and non-diagonal elements are zero, and also let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M106">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M107">View MathML</a> 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M112">View MathML</a>. Then the system (3.16) yields the following matrix system, the dimensions of which are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M113">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M114">View MathML</a>

(3.17)

Now we have a linear system of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M115">View MathML</a>equations of the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M116">View MathML</a>unknown coefficients. If we solve (3.17) by usingLUorQRdecomposition methods, we can obtain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M112">View MathML</a>coefficients for the approximate sinc-Galerkin solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M118">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M112">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M120">View MathML</a> 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.

thumbnailFigure 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).

thumbnailFigure 6. Approximation to the exact solution: the red colored curve displays the exact solution and the green one is the approximate solution of Eq. (4.2).

thumbnailFigure 7. Approximation to the exact solution: the red colored curve displays the exact solution and the green one is the approximate solution of Eq. (4.3).

Table 2. The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. (4.1) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M121">View MathML</a>

Table 3. The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. (4.2) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M121">View MathML</a>

Table 4. The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. (4.3) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M121">View MathML</a>

Example 4.1 Consider the following singular Dirichlet-type boundary value problem on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M124">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M125">View MathML</a>

(4.1)

The exact solution of (4.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M126">View MathML</a>

We choose the weight function according to [2], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M128">View MathML</a>, and by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M131">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M132">View MathML</a>, 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M124">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M134">View MathML</a>

(4.2)

The problem has an exact solution like

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M135">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M128">View MathML</a>.By taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M140">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M141">View MathML</a>,we get the solutions in Figure 6 and Table 3.

Example 4.3 The following problem is given on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M142">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M143">View MathML</a>

(4.3)

where the exact solution of (4.3) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M144">View MathML</a>.

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M128">View MathML</a>, and by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M149">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/126/mathml/M141">View MathML</a>, 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

Competing interests

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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