The sinc-Galerkin method and its applications on singular Dirichlet-type boundary value problems
- Aydin Secer^{1}Email author and
- Muhammet Kurulay^{2}
https://doi.org/10.1186/1687-2770-2012-126
© Secer and Kurulay; licensee Springer. 2012
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
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
Conformal mappings and nodes for some subintervals of R
(a,b) | ϕ(z) | ${\mathit{z}}_{\mathit{k}}$ | |
---|---|---|---|
a | b | $ln(\frac{z-a}{b-z})$ | $\frac{a+b{e}^{kh}}{1+{e}^{kh}}$ |
0 | 1 | $ln(\frac{z}{1-z})$ | $\frac{{e}^{kh}}{1+{e}^{kh}}$ |
0 | ∞ | ln(z) | ${e}^{kh}$ |
0 | ∞ | ln(sinh(z)) | $ln({e}^{kh}+\sqrt{{e}^{2kh}+1})$ |
−∞ | ∞ | z | kh |
−∞ | ∞ | sinh^{−1}(z) | kh |
and ${z}_{k}=\phi (kh)$, $k=\mp 1,\mp 2,\dots $ .
The proof of following theorems can be found in [2].
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.
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.
3 The sinc-Galerkin method for singular Dirichlet-type boundary value problems
where ${w}_{k}=w({x}_{k})$. If we choose $h={(\pi d/\alpha N)}^{1/2}$ and $w(x)=1/{\varphi}^{\prime}(x)$ as given in [2] the accuracy for each equation between (3.8)-(3.11) will be $O({N}^{1/2}{e}^{-{(\pi d\alpha N)}^{1/2}})$.
Using (3.5), (3.8)-(3.11), we obtain a linear system of equations for $2N+1$ numbers ${c}_{k}$.
4 Examples
The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. ( 4.1 ) for $\mathit{N}\mathbf{=}\mathbf{100}$
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 |
The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. ( 4.2 ) for $\mathit{N}\mathbf{=}\mathbf{100}$
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 |
The numerical results for the approximate solutions obtained by sinc-Galerkin in comparison with the exact solutions of Eq. ( 4.3 ) for $\mathit{N}\mathbf{=}\mathbf{100}$
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 |
We choose the weight function according to [2], $\varphi (x)=ln(\frac{1}{1-x})$, $w(x)=\frac{1}{{\varphi}^{\prime}(x)}$, and by taking $d=\pi /2$, $h=\frac{2}{\sqrt{N}}$, ${x}_{k}=\frac{{e}^{kh}}{1+{e}^{kh}}$ for $N=8,16,32,100$, the solutions inFigure 5 and Table 2 are achieved.
where $\varphi (x)=ln(\frac{1}{1-x})$, $w(x)=\frac{1}{{\varphi}^{\prime}(x)}$.By taking $d=\pi /2$, $h=\frac{2}{\sqrt{N}}$, ${x}_{k}=\frac{{e}^{kh}}{1+{e}^{kh}}$ for $N=8,16,32,100$,we get the solutions in Figure 6 and Table 3.
where the exact solution of (4.3) is $y(x)=\frac{{x}^{2}{\mathrm{e}}^{4}-{x}^{2}{\mathrm{e}}^{-1}+15{\mathrm{e}}^{x}-6x{\mathrm{e}}^{4}+6x{\mathrm{e}}^{-1}-7{\mathrm{e}}^{4}-8{\mathrm{e}}^{-1}}{{\mathrm{2}(\mathrm{e}}^{4}-{\mathrm{e}}^{-1})}$.
In this case, $\varphi (x)=ln(\frac{x+1}{4-x})$, $w(x)=\frac{1}{{\varphi}^{\prime}(x)}$, and by taking $d=\pi /2$, $h=\frac{2}{\sqrt{N}}$, ${x}_{k}=\frac{-1+4{e}^{kh}}{1+{e}^{kh}}$ for $N=8,16,32,100$, 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
Declarations
Authors’ Affiliations
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