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

Open Access Research

An efficient computer application of the sinc-Galerkin approximation for nonlinear boundary value problems

Aydin Secer1*, Muhammet Kurulay2, Mustafa Bayram1 and Mehmet Ali Akinlar3

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

3 Department of Mathematics, Bilecik University, Bilecik, 11210, Turkey

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Boundary Value Problems 2012, 2012:117  doi:10.1186/1687-2770-2012-117

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/117


Received:3 August 2012
Accepted:2 October 2012
Published:24 October 2012

© 2012 Secer et al.; 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

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.

Keywords:
Maple; sinc-Galerkin approximation; sinc basis function; nonlinear matrix system; Newton method

1 Introduction

We present here the sinc-Galerkin approximation technique using Maple to solve systems of nonlinear BVPs such as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M1">View MathML</a>

(1.1)

where NL is the nonlinear part of Eq. (1.1) which can take any form of nonlinearity, and we investigate the approximate solution on some closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M2">View MathML</a> in R.

We start by casting a given linear or nonlinear BVP into a sinc-Galerkin form accurate to the order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M3">View MathML</a>[1]. This discretization yields a set of linear or nonlinear algebraic equations that include all unknown coefficients. These equations are expressed in a nonlinear or linear matrix form depending on (1.1). If the equation is linear, the LU decomposition method can be used to find unknown coefficients. However, if it is not linear, the coefficients can be found by the Newton interpolation method for nonlinear equation systems by using Maple. The methodology is illustrated on nonlinear ordinary differential equations with Dirichlet-type boundaries. Once the solution is obtained, we compare its accuracy with the Newton method as a graphical and numerical simulation by using Maple.

We start with some literature on the sinc-Galerkin methods. The sinc methods were introduced in [2] and expanded in [1] by Frank Stenger. 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 are discussed in detail. Some kind of singular elliptic problems are solved in [9], and the symmetric sinc-Galerkin method is introduced in [10]. The sinc domain decomposition is presented in [11-14]. Also, iterative methods for symmetric sinc-Galerkin systems are given in [15-17]. Sinc methods are studied thoroughly in [18]. Applications of sinc methods can also be found in [19-21]. The article [22] summarizes the results that are obtained by sinc numerical methods of computation. In [14] a numerical solution of the Volterra integro-differential equation by means of the sinc collocation method is considered. The paper [1] illustrates the application of a sinc-Galerkin method to the approximate solution of linear and nonlinear second-order ordinary differential equations and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The fully sinc-Galerkin method is developed for a family of complex-valued partial differential equations with time-dependent boundary conditions [19]. In [23] some novel procedures of using sinc methods to compute the solutions of three types of medical problems are illustrated. In [24], the sinc-based algorithm is used to solve a nonlinear set of partial differential equations. A new sinc-Galerkin method is developed for approximating the solution of convection diffusion equations with mixed boundary conditions on half-infinite intervals in [25]. The work which is presented in [26] deals with the sinc-Galerkin method for solving nonlinear fourth-order differential equations with homogeneous and nonhomogeneous boundary conditions. In [27], the sinc methods are used to solve second-order ordinary differential equations with homogeneous Dirichlet-type boundary conditions. In the paper given in [28], the sinc-Galerkin method is applied to solving Troesch’s problem. The properties of the sinc procedure are utilized to reduce the computation of Troesch’s equation to nonlinear equations with unknown coefficients.

2 Sinc basis functions

Let C denote the set of all complex numbers, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M4">View MathML</a> define the sine cardinal or sinc function by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M5">View MathML</a>

(2.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M6">View MathML</a>, the translated sinc function with evenly spaced nodes is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M7">View MathML</a>

(2.2)

For various values of k, the sinc basis function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M8">View MathML</a> on the whole real line, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M9">View MathML</a>, is illustrated in Figure 1. For various values of h, the central function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M10">View MathML</a> is illustrated in Figure 2.

thumbnailFigure 1. The basis functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M11">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M12">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M13">View MathML</a>.

thumbnailFigure 2. Central sinc basis function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M14">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M15">View MathML</a>.

If a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M16">View MathML</a> is defined over the real line, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M6">View MathML</a> the series

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M18">View MathML</a>

(2.3)

is called the Whittaker cardinal expansion of f whenever this series converges. The infinite strip Ds of the complex w plane, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M19">View MathML</a>, is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M20">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/2012/1/117/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M21">View MathML</a>

(2.5)

which is a conformal mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a>, the eye-shaped domain in the z-plane, onto the infinite strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M23">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M24">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/2012/1/117/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M25">View MathML</a>and the infinite strip<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M26">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/2012/1/117/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M27">View MathML</a>

(2.7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M28">View MathML</a>. These are shown in Figure 4 for real values of x. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M29">View MathML</a> is an inverse mapping of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M30">View MathML</a>. We may define the range of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M31">View MathML</a> on the real line as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M32">View MathML</a>

(2.8)

For the evenly spaced nodes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M33">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/2012/1/117/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M34">View MathML</a>

(2.9)

A list of conformal mappings may be found in Table 2.1 [6].

thumbnailFigure 4. Three adjacent members<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M35">View MathML</a>when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M36">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M37">View MathML</a>of the mapped sinc basis on the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M38">View MathML</a>.

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a> be a simply connected domain in the complex plane C, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M40">View MathML</a> denote the boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a>. Let a, b be points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M40">View MathML</a> and ϕ be a conformal map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a> onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M46">View MathML</a>. If the inverse map of ϕ is denoted by φ, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M47">View MathML</a>

(2.10)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M49">View MathML</a> .

We can use Table 1 to choose convenient conformal map according to boundary conditions.

Table 1. Conformal mappings and nodes for several subintervals ofR

Definition 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M57">View MathML</a> be the class of functions F that are analytic in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a> and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M59">View MathML</a>

(2.11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M60">View MathML</a>

(2.12)

and on the boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a> it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M62">View MathML</a>

(2.13)

The proof of following theorems can be found in [1].

Theorem 2.1Let Γ be<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M64">View MathML</a>, then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M6">View MathML</a>sufficiently small,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M66">View MathML</a>

(2.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M67">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/2012/1/117/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M68">View MathML</a>

(2.16)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M69">View MathML</a>

(2.17)

The infinite sum in (2.14) is truncated with the use of (2.16) to arrive at (2.17).

Making the selections

(2.18)

(2.19)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M72">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/2012/1/117/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M73">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 the 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/2012/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M22">View MathML</a>onto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M23">View MathML</a>. Then

(2.21)

(2.22)

(2.23)

3 Convergence analysis

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M79">View MathML</a>

(3.1)

with Dirichlet-type boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M80">View MathML</a>

(3.2)

where P, Q, R, 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/2012/1/117/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M83">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/2012/1/117/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M83">View MathML</a> coefficients by solving the nonlinear system of equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M85">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M87">View MathML</a> be analytic functions on D. The inner product in (3.5) is defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M88">View MathML</a>

(3.6)

where w is the weight function. For the second-order problems, it is convenient to take [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M89">View MathML</a>

(3.7)

For Eq. (3.1), we use the notations (2.21)-(2.23) together with the inner product given in (3.5) [1] to get the following approximation formulas:

(3.8)

(3.9)

(3.10)

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M94">View MathML</a>etc. The choices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M96">View MathML</a> yield <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M97">View MathML</a>[1] accuracy for each of the approximations in (3.8)-(3.11).

Using (3.5), (3.8)-(3.11), we obtain a nonlinear system of equations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M98">View MathML</a> numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M83">View MathML</a>.

The nonlinear system with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M100">View MathML</a> unknowns given in (3.5) can be expressed by means of matrices. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M101">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M102">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M104">View MathML</a> be column vectors defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M105">View MathML</a>

(3.12)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M106">View MathML</a> denote a diagonal matrix whose diagonal elements are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M107">View MathML</a> and non-diagonal elements are zero, and also let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M110">View MathML</a> denote the matrices

(3.13)

(3.14)

(3.15)

With these notations, the discrete system in (3.5) takes the form:

(3.16)

Theorem 3.1Letc, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M115">View MathML</a>be an m-vector whosejth component is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M117">View MathML</a>then the system (3.16) yields the following matrix system whose dimensions are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M118">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M119">View MathML</a>

(3.17)

Now we have a nonlinear system with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M120">View MathML</a>equations in the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M120">View MathML</a>unknown coefficients. If we solve (3.17) with the Newton method (for nonlinear equation systems) by using Maple, we can obtain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116">View MathML</a>coefficients for the approximate sinc-Galerkin solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M123">View MathML</a>

(3.18)

4 Examples

In this section, three examples are given to illustrate the performance of the sinc-Galerkin method by solving nonlinear Dirichlet-type boundary value problems. Each of these problems have been chosen to simulate how the solutions change in different zero boundary intervals. In the following examples, the discrete sinc system defined by (3.18) is used to compute the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M125">View MathML</a>. The computations are done by the algorithm which we developed for sinc-Galerkin method by using Maple. The algorithm automatically compares the sinc-method to the Newton method. The following examples show that the sinc-Galerkin method is a very efficient and powerful tool for nonlinear Dirichlet-type boundary value problems.

Example 4.1 Consider the following nonlinear Dirichlet-type boundary value problem on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M126">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M127">View MathML</a>

(4.1)

We choose the weight function according to [1], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M129">View MathML</a> and by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M132">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M133">View MathML</a>, the solutions presented in Figure 5 and Table 2.

thumbnailFigure 5. The red-colored curve displays the Newton solution and the green one is an approximate solution of Eq. (4.1).

Table 2. The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the Newton solutions of Eq. (4.1) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M134">View MathML</a>

Example 4.2 Let us have the following form of nonlinear Dirichlet-type boundary value problem on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M126">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M136">View MathML</a>

(4.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M129">View MathML</a> and by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M141">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M142">View MathML</a> we get the solutions presented in Figure 6 and Table 3.

thumbnailFigure 6. The red-colored curve displays the Newton solution and the green one is an approximate solution of Eq. (4.2).

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/2012/1/117/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M143">View MathML</a>

Example 4.3 In this case, we take the problem to be given on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M144">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M145">View MathML</a>

(4.3)

where we chose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M146">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M147">View MathML</a> and by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M131">View MathML</a> , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M150">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M151">View MathML</a> we get the results presented in Figure 7 and Table 4.

thumbnailFigure 7. The red-colored curve displays the Newton solution and the green one is an approximate solution of Eq. (4.3).

Table 4. The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the Newton solutions of Eq. (4.3) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M134">View MathML</a>

5 Discussion

A new efficient computer application of sinc-Galerkin method has been presented for nonlinear BVPs. The main advantage of our technique compared to other methods (e.g., Newton’s method) is that the solution is independent of the singularity conditions and valid for Dirichlet-type boundary conditions. The order of accuracy used in this paper is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M3">View MathML</a>. We have used different N node points for all figures presented in this paper. Even though the numerical solution looks complex for even <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M154">View MathML</a> node points, Maple handles it very well. In the Appendix, a useful Maple program is given to explain the technique and to show how the same solution can be used for different boundary conditions. By using the same program, substituting N and other parameters (like equations, boundaries), different solutions and graphics can be produced. The total time taken on a 3.5 GHz Pentium I7 processor with 8 Core and 8 GB RAM for producing figures and numerical results is less than 20 seconds.

6 Conclusion

In this study, the sinc-Galerkin method has been employed to find the solutions of second-order nonlinear Dirichlet-type boundary value problems on some closed real interval and the method has been compared to the Newton method. Our main purpose is to find the solution of boundary value problems which arise from the singular problems for which the Newton method does not converge at singular points. The powerful side of our method is that it can easily compute solutions even if the equation has singularities. The Newton method can fail when computing some complicated forms of governing equations; on the other hand, our method can easily handle this situation. The examples show that the accuracy improves by increasing the number of sinc grid points N. The method presented here is simple and gives a numerical solution, which is valid for various boundary conditions. We have developed a very efficient algorithm to solve second-order nonlinear Dirichlet-type boundary value problems with sinc-Galerkin method in Maple Computer Algebra System. Several nonlinear BVPs have been solved by using our technique in less than 20 seconds. All computations and graphical representations have been prepared automatically by our algorithm.

Appendix: A computer application of numeric solutions for nonlinear boundary value problems (NBVPs)

We demonstrate below how to solve and simulate for a nonlinear BVP. For example, the following Maple code computes and simulates Example 4.3.

Set all parameters as default values

> restart:

For drawing approximation graphics, we must type the following line

> with(plots):

A user has to specify with (linalg) for linear algebra operations in Maple

> with(linalg):

A user can define the grid point size N for sinc-Galerkin approximation

> N:=48:

The boundary conditions are given as Eq. (4.3).

> a:=4:

> b:=5:

> Boundaries:=y(a)=0,y(b)=0;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M155">View MathML</a>

P, Q and R are the variable coefficients of Eq. (1.1). In Maple for Eq. (4.3) they are defined as follows:

> P(x):=1;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M156">View MathML</a>

> Q(x):=1;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M157">View MathML</a>

> R(x):=1;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M158">View MathML</a>

F is right side of Eq. (4.3)

> F(x):=cos(Pi*x^2)*x;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M159">View MathML</a>

We can write a nonlinear part of Eq. (1.1) as follows. User can define any form of nonlinearity in this section.

> NLPart:=-exp(-sin(y(x)))*y(x)^2/(1+y(x));

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M160">View MathML</a>

The main form of Eq. (1.1)

> Equation:=P(x)*diff(y(x),x$2)+Q(x)*diff(y(x),x$1)+R(x)*NLPart=F(x);

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M161">View MathML</a>

If the user needs, the main equation can be written in the latex format

In order to compare our method with the Newton interpolation (for nonlinear ODE) method, we first solve Eq. (4.3) numerically as follows:

Prepare the plot of the Newton solution

> PlotNewtonSolution:=odeplot(NewtonSolution,a....b):

To define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M166">View MathML</a> matrices given in Eqs. (3.13)-(3.15), we use piecewise functions in Maple in the following way:

> delta[0]:=unapply(piecewise(j=k,1,j<>k,0),j,k):

> delta[1]:=unapply(piecewise(j=k,0,j<>k,((-1)^(k-j))/(k-j)),j,k):

> delta[2]:=unapply(piecewise(j=k,(-Pi^2)/3,j<>k,-2*(-1)^(k-j)/(k-j)^2),j,k):

The parameters for sinc-approximation given [1]

> d:=Pi/2:

> h:=2/sqrt(N):

The evenly spaced nodes given (2.9) and Table 1 are defined as follows:

> xk:=unapply((a+b*exp(k*h))/(1+exp(k*h)),k);

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M167">View MathML</a>

The conformal map in Table 1 for sinc-Galerkin method and its derivatives is computed as follows:

> phi:=unapply(log((x-a)/(b-x)),x);

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M168">View MathML</a>

> Dphi:=unapply(simplify(diff(phi(x),x)),x):

> D2phi:=unapply(simplify(diff(phi(x),x$2)),x):

The weight function and its derivatives are computed for using an inner product to discretization Eq. (4.3)

> w:=unapply(1/Dphi(x),x):

> Dw:=unapply(simplify(diff(w(x),x$1)),x):

> D2w:=unapply(simplify(diff(w(x),x$2)),x):

By using sinc-discretization in (3.16), the matrix system with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M169">View MathML</a> dimensions defined in (3.17) is obtained by the following iteration:

If we want to obtain solutions of linear BVPs, we can use the following lines. They can reduce time complexity. Here, the linear solution is given as a comment (“#”).

> #for Linear system

> #vars:=seq(c[i],i=-N..N):

> #A,b:=LinearAlgebra[GenerateMatrix](evalf(MatrixSystem),[vars]):

> #c:=linsolve(A,b);

In this paper, we want to solve nonlinear problems. Then we use fsolve function given by Maple to find unknown <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116">View MathML</a> coefficients (3.17)-(3.18) from nonlinear matrix systems. This function can solve any nonlinear systems by using the Newton method (for nonlinear equation systems).

> c:=fsolve(evalf(MatrixSystem)):

Finally, we have unknown <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/117/mathml/M116">View MathML</a> coefficients for the approximate sinc-Galerkin solution (3.18)

We define plot of Eq. (4.3) obtained by the sinc-Galerkin solution

> Sinc-GalerkinPlot:=plot({ApproximateSol(x)},x=a..b,color=green,thickness=1):

Simulation: Figure 5, Figure 6, and Figure 7 are obtained as

Enter the number of digits here

> Digits := 15:

Tables 2, 3, and 4 are obtained by the following code:

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. He developed computer algorithm and worked on theoretical aspect of problem. MK searched the materials about study and compared with other techniques. MAA contributed us with his experience on Nonlinear Approximation methods. MB contributed us with his experience on Nonlinear Approximation methods, suggested us some valuable techniques.

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