Open Access

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

Boundary Value Problems20122012:126

DOI: 10.1186/1687-2770-2012-126

Received: 22 September 2012

Accepted: 15 October 2012

Published: 29 October 2012

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 [1113] 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 z C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq1_HTML.gif, define the sine cardinal or sinc function by
sin c ( z ) = { sin ( π z ) π z , y 0 , 1 , y = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ1_HTML.gif
(2.1)
For h > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq2_HTML.gif, the translated sinc function with evenly spaced nodes is given by
sin c ( k , h ) ( z ) = { sin ( π z k h h ) π z k h h , z k h , 1 , z = k h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ2_HTML.gif
(2.2)
For various values of k, the sinc basis function S ( k , π / 4 ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq3_HTML.gif on the whole real line < x < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq4_HTML.gif is illustrated in Figure 1. For various values of h, the central function S ( 0 , h ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq5_HTML.gif is illustrated in Figure 2.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig1_HTML.jpg
Figure 1

The basis functions S ( k , h ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq6_HTML.gif for k = 1 , 0 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq7_HTML.gif with h = π / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq8_HTML.gif .

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig2_HTML.jpg
Figure 2

Central sinc basis function S ( 0 , h ) ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq9_HTML.gif for h = π / 2 , π / 4 , π / 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq10_HTML.gif .

If a function f ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq11_HTML.gif is defined over the real line, then for h > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq2_HTML.gif, the series
C ( f , h ) ( x ) = k = f ( k h ) sin c ( x k h h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ3_HTML.gif
(2.3)
is called the Whittaker cardinal expansion of f whenever this series converges. The infinite strip D s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq12_HTML.gif of the complex w plane, where d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq13_HTML.gif, is given by
D s { w = u + i v : | v | < d π 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ4_HTML.gif
(2.4)
In general, approximations can be constructed for infinite, semi-infinite and finite intervals. Define the function
w = ϕ ( z ) = ln ( z 1 z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ5_HTML.gif
(2.5)
which is a conformal mapping from D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif, the eye-shaped domain in the z-plane, onto the infinite strip D S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq15_HTML.gif, where
D E = z = { x + i y : | arg ( z 1 z ) | < d π 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ6_HTML.gif
(2.6)
This is shown in Figure 3.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig3_HTML.jpg
Figure 3

The relationship between the eye-shaped domain D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq16_HTML.gif and the infinite strip D S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq17_HTML.gif .

For the sinc-Galerkin method, the basis functions are derived from the composite translated sinc functions
S h ( z ) = S ( k , h ) ( z ) = sin c ( ϕ ( z ) k h h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ7_HTML.gif
(2.7)
for z D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq18_HTML.gif. These are shown in Figure 4 for real values x. The function z = ϕ 1 ( w ) = e w 1 + e w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq19_HTML.gif is an inverse mapping of w = ϕ ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq20_HTML.gif. We may define the range of ϕ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq21_HTML.gif on the real line as
Γ = { ϕ 1 ( u ) D E : < u < } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ8_HTML.gif
(2.8)
the evenly spaced nodes { k h } k = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq22_HTML.gif on the real line. The image which corresponds to these nodes is denoted by
x k = ϕ 1 ( k h ) = e k h 1 + e k h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ9_HTML.gif
(2.9)
A list of conformal mappings may be found in Table 1 [6].
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig4_HTML.jpg
Figure 4

Three adjacent members S ( k , h ) ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq23_HTML.gif when k = 1 , 0 , 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq24_HTML.gif and h = π 8 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq25_HTML.gif of the mapped sinc basis on the interval ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq26_HTML.gif .

Table 1

Conformal mappings and nodes for some subintervals of R

(a,b)

ϕ(z)

z k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq27_HTML.gif

a

b

ln ( z a b z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq28_HTML.gif

a + b e k h 1 + e k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq29_HTML.gif

0

1

ln ( z 1 z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq30_HTML.gif

e k h 1 + e k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq31_HTML.gif

0

ln(z)

e k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq32_HTML.gif

0

ln(sinh(z))

ln ( e k h + e 2 k h + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq33_HTML.gif

−∞

z

kh

−∞

sinh−1(z)

kh

Definition 2.1 Let D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif be a simply connected domain in the complex plane C, and let D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq34_HTML.gif denote the boundary of D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif. Let a, b be points on D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq34_HTML.gif and ϕ be a conformal map D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif onto D S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq15_HTML.gif such that ϕ ( a ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq35_HTML.gif and ϕ ( b ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq36_HTML.gif. If the inverse map of ϕ is denoted by φ, define
Γ = { ϕ 1 ( u ) D E : < u < } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ10_HTML.gif
(2.10)

and z k = φ ( k h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq37_HTML.gif, k = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq38_HTML.gif .

Definition 2.2 Let B ( D E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq39_HTML.gif be the class of functions F that are analytic in D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif and satisfy
ψ ( L + u ) | F ( z ) | d z 0 , as  u = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ11_HTML.gif
(2.11)
where
L = { i y : | y | < d π 2 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ12_HTML.gif
(2.12)
and those on the boundary of D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif satisfy
T ( F ) = D E | F ( z ) d z | < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ13_HTML.gif
(2.13)

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

Theorem 2.1 Let Γ be ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq40_HTML.gif, F B ( D E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq41_HTML.gif, then for h > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq2_HTML.gif sufficiently small,
Γ F ( z ) d z h j = F ( z j ) ϕ ( z j ) = i 2 D F ( z ) k ( ϕ , h ) ( z ) sin ( π ϕ ( z ) / h ) d z I F , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ14_HTML.gif
(2.14)
where
| k ( ϕ , h ) | z D = | e [ i π ϕ ( z ) h sgn ( Im ϕ ( z ) ) ] | z D = e π d h . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ15_HTML.gif
(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 2.2 If there exist positive constants α, β and C such that
| F ( x ) ϕ ( x ) | C { e α | ϕ ( x ) | , x ψ ( ( , ) ) , e β | ϕ ( x ) | , x ψ ( ( 0 , ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ16_HTML.gif
(2.16)
then the error bound for the quadrature rule (2.14) is
| Γ F ( x ) d x h j = N N F ( x j ) ϕ ( x j ) | C ( e α N h α + e β N h β ) + | I F | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ17_HTML.gif
(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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ18_HTML.gif
(2.18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ19_HTML.gif
(2.19)
where https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq42_HTML.gif is an integer part of the statement and N is the integer value which specifies the grid size, then
Γ F ( x ) d x = h j = N N F ( x j ) ϕ ( x j ) + O ( e ( π α d N ) 1 / 2 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ20_HTML.gif
(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 2.3 Let ϕ be a conformal one-to-one map of the simply connected domain D E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq14_HTML.gif onto D S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq15_HTML.gif. Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ21_HTML.gif
(2.21)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ22_HTML.gif
(2.22)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ23_HTML.gif
(2.23)

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

Consider the following problem:
y + P ( x ) y + Q ( x ) y = F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ24_HTML.gif
(3.1)
with Dirichlet-type boundary condition
y ( a ) = 0 , y ( b ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ25_HTML.gif
(3.2)
where P, Q and F are analytic on D. We consider sinc approximation by the formula
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ26_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ27_HTML.gif
(3.4)
The unknown coefficients c k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq43_HTML.gif 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 c k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq43_HTML.gif coefficients by solving the linear system of equations
L y N F , S ( k , h ) ϕ ( x ) = 0 , k = N , N + 1 , , N 1 , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ28_HTML.gif
(3.5)
Let f 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq44_HTML.gif and f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq45_HTML.gif be analytic functions on D and the inner product in (3.5) be defined as follows:
f 1 , f 2 = Γ w ( x ) f 1 ( x ) f 2 ( x ) d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ29_HTML.gif
(3.6)
where w is the weight function. For the second-order problems, it is convenient to take [2].
w ( x ) = 1 ϕ ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ30_HTML.gif
(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:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ31_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ32_HTML.gif
(3.9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ33_HTML.gif
(3.10)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ34_HTML.gif
(3.11)

where w k = w ( x k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq46_HTML.gif. If we choose h = ( π d / α N ) 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq47_HTML.gif and w ( x ) = 1 / ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq48_HTML.gif as given in [2] the accuracy for each equation between (3.8)-(3.11) will be O ( N 1 / 2 e ( π d α N ) 1 / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq49_HTML.gif.

Using (3.5), (3.8)-(3.11), we obtain a linear system of equations for 2 N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq50_HTML.gif numbers c k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq43_HTML.gif.

The 2 N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq51_HTML.gif linear system given in (3.5) can be expressed by means of matrices. Let m = 2 N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq52_HTML.gif, and let S m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq53_HTML.gif and c m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq54_HTML.gif be a column vector defined by
S m ( x ) = ( S N S N + 1 S N ) , c m = ( c N c N + 1 c N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ35_HTML.gif
(3.12)
Let A m ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq55_HTML.gif denote a diagonal matrix whose diagonal elements are y ( x N ) , y ( x N + 1 ) , , y ( x N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq56_HTML.gif and non-diagonal elements are zero, and also let I m ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq57_HTML.gif, I m ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq58_HTML.gif and I m ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq59_HTML.gif denote the matrices
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ36_HTML.gif
(3.13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ37_HTML.gif
(3.14)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ38_HTML.gif
(3.15)
With these notations, the discrete system of equations in (3.5) takes the form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ39_HTML.gif
(3.16)
Theorem 3.1 Let c be an m-vector whose jth component is c j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq60_HTML.gif. Then the system (3.16) yields the following matrix system, the dimensions of which are ( 2 N + 1 ) × ( 2 N + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq61_HTML.gif:
Φ c = A m F w ϕ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ40_HTML.gif
(3.17)
Now we have a linear system of ( 2 N + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq62_HTML.gif equations of the ( 2 N + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq63_HTML.gif unknown coefficients. If we solve (3.17) by using LU or QR decomposition methods, we can obtain c j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq60_HTML.gif coefficients for the approximate sinc-Galerkin solution
y ( x ) y N ( x ) = k = N N c k S ( k , h ) ϕ ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ41_HTML.gif
(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 c j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq60_HTML.gif; j = N , , N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq64_HTML.gif 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.
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig5_HTML.jpg
Figure 5

Approximation to the exact solution: the red colored curve displays the exact solution and the green one is the approximate solution of Eq. ( 4.1 ).

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig6_HTML.jpg
Figure 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 ).

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Fig7_HTML.jpg
Figure 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 N = 100 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq65_HTML.gif

x

Exact solution

Sinc-Galerkin

Absolute error

0.2

0.000450466988174113

0.000450466929764516

5.8409597E − 11

0.4

0.000893654763766436

0.000893654689218907

7.4547529E − 11

0.6

0.001096474957106920

0.001096474871619300

8.5487620E − 11

0.8

0.000797109647979786

0.000797109574773798

7.3205988E − 11

Table 3

The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. ( 4.2 ) for N = 100 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq65_HTML.gif

x

Exact solution

Sinc-Galerkin

Absolute error

0.2

0.00314134396980435

0.00314134378138869

1.88415721000000E − 10

0.4

0.01128904694197050

0.01128904622846880

7.13501861405898E − 10

0.6

0.02049668664764170

0.02049668582683820

8.20803253396388E − 10

0.8

0.02205723725961330

0.02205723670616530

5.53448662985227E − 10

Table 4

The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. ( 4.3 ) for N = 100 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq65_HTML.gif

x

Exact solution

Sinc-Galerkin

Absolute error

−0.8

−0.768735600700030

−0.768735573640717

2.7059313E − 8

−0.6

−1.494977232326020

−1.494977256431750

2.4105730E − 8

−0.4

−2.178172723246240

−2.178172789883010

6.6636770E − 8

−0.2

−2.817647649506660

−2.817647724013040

7.4506380E − 8

0.0

−3.412578267829700

−3.412578329155590

6.1325890E − 8

0.2

−3.961958455904090

−3.961958531301040

7.5396950E − 8

0.4

−4.464559333163800

−4.464559424139430

9.0975630E − 8

0.6

−4.918879941496040

−4.918880051407700

1.0991166E − 7

0.8

−5.323087006521950

−5.323087129044260

1.2252231E − 7

1.0

−5.674941361858750

−5.674941494327020

1.3246827E − 7

1.2

−5.971708083510550

−5.971708201060930

1.1755038E − 7

1.4

−6.210046727765300

−6.210046817516560

8.9751260E − 8

1.6

−6.385877267459800

−6.385877325019590

5.7559790E − 8

1.8

−6.494216346163350

−6.494216361246050

1.5082700E − 8

2.0

−6.528977278586750

−6.528977261410670

1.7176080E − 8

Example 4.1 Consider the following singular Dirichlet-type boundary value problem on the interval [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq66_HTML.gif:
d 2 d x 2 y ( x ) + y ( x ) x ( x 1 ) = 72 1 , 045 x 2 + 12 1 , 045 x 3 + 1 209 x 4 + 1 / 19 x 5 , y ( 0 ) = 0 , y ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ42_HTML.gif
(4.1)
The exact solution of (4.1) is
y ( x ) = 1 , 834 , 592 887 , 331 , 445 x + 917 , 296 887 , 331 , 445 x 2 + 458 , 648 887 , 331 , 445 x 3 188 , 072 34 , 571 , 355 x 4 + 1 , 4131 29 , 252 , 685 x 5 + 32 278 , 597 x 6 + 1 817 x 7 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equa_HTML.gif

We choose the weight function according to [2], ϕ ( x ) = ln ( 1 1 x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq67_HTML.gif, w ( x ) = 1 ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq68_HTML.gif, and by taking d = π / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq69_HTML.gif, h = 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq70_HTML.gif, x k = e k h 1 + e k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq71_HTML.gif for N = 8 , 16 , 32 , 100 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq72_HTML.gif, 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 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq66_HTML.gif:
d 2 d x 2 y ( x ) 1 x d d x y ( x ) + y ( x ) x ( x + 1 ) = x 3 , y ( 0 ) = 0 , y ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ43_HTML.gif
(4.2)
The problem has an exact solution like
y ( x ) = 1 144 ( 14 ln ( x + 1 ) x + 14 ln ( x + 1 ) 14 x + 6 x 2 12 x 2 ln ( 2 ) 2 x 3 + 4 x 3 ln ( 2 ) + x 4 2 x 4 ln ( 2 ) + 9 x 5 18 x 5 ln ( 2 ) ) / ( 1 + 2 ln ( 2 ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equb_HTML.gif

where ϕ ( x ) = ln ( 1 1 x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq73_HTML.gif, w ( x ) = 1 ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq68_HTML.gif.By taking d = π / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq69_HTML.gif, h = 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq70_HTML.gif, x k = e k h 1 + e k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq74_HTML.gif for N = 8 , 16 , 32 , 100 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq75_HTML.gif,we get the solutions in Figure 6 and Table 3.

Example 4.3 The following problem is given on the interval [ 1 , 4 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq76_HTML.gif:
d 2 d x 2 y ( x ) d d x y ( x ) = ( x 4 ) , y ( 1 ) = 0 , y ( 4 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equ44_HTML.gif
(4.3)

where the exact solution of (4.3) is y ( x ) = x 2 e 4 x 2 e 1 + 15 e x 6 x e 4 + 6 x e 1 7 e 4 8 e 1 2 ( e 4 e 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq77_HTML.gif.

In this case, ϕ ( x ) = ln ( x + 1 4 x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq78_HTML.gif, w ( x ) = 1 ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq68_HTML.gif, and by taking d = π / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq69_HTML.gif, h = 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq70_HTML.gif, x k = 1 + 4 e k h 1 + e k h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq79_HTML.gif for N = 8 , 16 , 32 , 100 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_IEq75_HTML.gif, 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

https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-126/MediaObjects/13661_2012_Article_230_Equc_HTML.gif

Declarations

Authors’ Affiliations

(1)
Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University
(2)
Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University

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Copyright

© Secer and Kurulay; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. 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.