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
sinc-Galerkin method sinc basis functions Dirichlet-type boundary value problems LU decomposition method1 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
References
- Stenger F: Approximations via Whittaker’s cardinal function. J. Approx. Theory 1976, 17: 222-240. 10.1016/0021-9045(76)90086-1MathSciNetView ArticleGoogle Scholar
- Stenger F: A sinc-Galerkin method of solution of boundary value problems. Math. Comput. 1979, 33: 85-109.MathSciNetGoogle Scholar
- Whittaker ET: On the functions which are represented by the expansions of the interpolation theory. Proc. R. Soc. Edinb. 1915, 35: 181-194.View ArticleGoogle Scholar
- Whittaker JM Cambridge Tracts in Mathematics and Mathematical Physics 33. In Interpolation Function Theory. Cambridge University Press, London; 1935.Google Scholar
- Lund J: Symmetrization of the sinc-Galerkin method for boundary value problems. Math. Comput. 1986, 47: 571-588. 10.1090/S0025-5718-1986-0856703-9MathSciNetView ArticleGoogle Scholar
- Lund J, Bowers KL: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia; 1992.View ArticleGoogle Scholar
- Lewis DL, Lund J, Bowers KL: The space-time sinc-Galerkin method for parabolic problems. Int. J. Numer. Methods Eng. 1987, 24: 1629-1644. 10.1002/nme.1620240903MathSciNetView ArticleGoogle Scholar
- McArthur KM, Bowers KL, Lund J: Numerical implementation of the sinc-Galerkin method for second-order hyperbolic equations. Numer. Methods Partial Differ. Equ. 1987, 3: 169-185. 10.1002/num.1690030303MathSciNetView ArticleGoogle Scholar
- Bowers KL, Lund J: Numerical solution of singular Poisson problems via the sinc-Galerkin method. SIAM J. Numer. Anal. 1987, 24(1):36-51. 10.1137/0724004MathSciNetView ArticleGoogle Scholar
- Lund J, Bowers KL, McArthur KM: Symmetrization of the sinc-Galerkin method with block techniques for elliptic equations. IMA J. Numer. Anal. 1989, 9: 29-46. 10.1093/imanum/9.1.29MathSciNetView ArticleGoogle Scholar
- Lybeck, NJ: Sinc domain decomposition methods for elliptic problems. PhD thesis, Montana State University, Bozeman, Montana (1994)Google Scholar
- Lybeck NJ, Bowers KL: Domain decomposition in conjunction with sinc methods for Poisson’s equation. Numer. Methods Partial Differ. Equ. 1996, 12: 461-487. 10.1002/(SICI)1098-2426(199607)12:4<461::AID-NUM4>3.0.CO;2-KMathSciNetView ArticleGoogle Scholar
- Morlet AC, Lybeck NJ, Bowers KL: The Schwarz alternating sinc domain decomposition method. Appl. Numer. Math. 1997, 25: 461-483. 10.1016/S0168-9274(97)00068-8MathSciNetView ArticleGoogle Scholar
- Morlet AC, Lybeck NJ, Bowers KL: Convergence of the sinc overlapping domain decomposition method. Appl. Math. Comput. 1999, 98: 209-227. 10.1016/S0096-3003(97)10168-0MathSciNetView ArticleGoogle Scholar
- Alonso N, Bowers KL: An alternating-direction sinc-Galerkin method for elliptic problems. J. Complex. 2009, 25: 237-252. 10.1016/j.jco.2009.02.006MathSciNetView ArticleGoogle Scholar
- Ng M: Fast iterative methods for symmetric sinc-Galerkin systems. IMA J. Numer. Anal. 1999, 19: 357-373. 10.1093/imanum/19.3.357MathSciNetView ArticleGoogle Scholar
- Ng M, Bai Z: A hybrid preconditioner of banded matrix approximation and alternating-direction implicit iteration for symmetric sinc-Galerkin linear systems. Linear Algebra Appl. 2003, 366: 317-335.MathSciNetView ArticleGoogle Scholar
- Stenger F: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York; 1993.View ArticleGoogle Scholar
- Koonprasert, S: The sinc-Galerkin method for problems in oceanography. PhD thesis, Montana State University, Bozeman, Montana (2003)Google Scholar
- McArthur KM, Bowers KL, Lund J: The sinc method in multiple space dimensions: model problems. Numer. Math. 1990, 56: 789-816.MathSciNetView ArticleGoogle Scholar
- Stenger F: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 1981, 23: 165-224. 10.1137/1023037MathSciNetView ArticleGoogle Scholar
- Stenger F: Summary of sinc numerical methods. J. Comput. Appl. Math. 2000, 121: 379-420. 10.1016/S0377-0427(00)00348-4MathSciNetView ArticleGoogle Scholar
- Stenger F, O’Reilly MJ: Computing solutions to medical problems via sinc convolution. IEEE Trans. Autom. Control 1998, 43: 843. 10.1109/9.679023View ArticleGoogle Scholar
- Narasimhan S, Majdalani J, Stenger F: A first step in applying the sinc collocation method to the nonlinear Navier Stokes equations. Numer. Heat Transf., Part B 2002, 41: 447-462. 10.1080/104077902753725902View ArticleGoogle Scholar
- Mueller JL, Shores TS: A new sinc-Galerkin method for convection-diffusion equations with mixed boundary conditions. Comput. Math. Appl. 2004, 47: 803-822. 10.1016/S0898-1221(04)90066-1MathSciNetView ArticleGoogle Scholar
- El-Gamel M, Behiry SH, Hashish H: Numerical method for the solution of special nonlinear fourth-order boundary value problems. Appl. Math. Comput. 2003, 145: 717-734. 10.1016/S0096-3003(03)00269-8MathSciNetView ArticleGoogle Scholar
- Lybeck NJ, Bowers KL: Sinc methods for domain decomposition. Appl. Math. Comput. 1996, 75: 4-13.MathSciNetView ArticleGoogle Scholar
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