Abstract
The application of the sincGalerkin method to an approximate solution of secondorder singular Dirichlettype 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 sincGalerkin 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 secondorder Dirichlettype boundary value problems.
Keywords:
sincGalerkin method; sinc basis functions; Dirichlettype 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 twopoint 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 sincGalerkin method was introduced in [10]. Sinc domain decomposition was presented in [1113] and [14]. Iterative methods for symmetric sincGalerkin 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 integrodifferential equation by means of the sinc collocation method was considered. The paper [2] illustrates the application of a sincGalerkin method to an approximate solution of linear and nonlinear secondorder ordinary differential equations, and to an approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The fully sincGalerkin method was developed for a family of complexvalued partial differential equations with timedependent boundary conditions [19]. Some novel procedures of using sinc methods to compute solutions to three types of medical problems were illustrated in [23], and sincbased algorithm was used to solve a nonlinear set of partial differential equations in [24]. A new sincGalerkin method was developed for approximating the solution of convection diffusion equations with mixed boundary conditions on halfinfinite intervals in [25]. The work which was presented in [26] deals with the sincGalerkin method for solving nonlinear fourthorder differential equations with homogeneous and nonhomogeneous boundary conditions. In [27], sinc methods were used to solve secondorder ordinary differential equations with homogeneous Dirichlettype boundary conditions.
2 Sinc functions preliminaries
Let C denote the set of all complex numbers, and for all , define the sine cardinal or sinc function by
For , the translated sinc function with evenly spaced nodes is given by
For various values of k, the sinc basis function on the whole real line is illustrated in Figure 1. For various values of h, the central function is illustrated in Figure 2.
If a function is defined over the real line, then for , the series
is called the Whittaker cardinal expansion of f whenever this series converges. The infinite strip of the complex w plane, where , is given by
In general, approximations can be constructed for infinite, semiinfinite and finite intervals. Define the function
which is a conformal mapping from , the eyeshaped domain in the zplane, onto the infinite strip , where
This is shown in Figure 3.
Figure 3. The relationship between the eyeshaped domainand the infinite strip.
For the sincGalerkin method, the basis functions are derived from the composite translated sinc functions
for . These are shown in Figure 4 for real values x. The function is an inverse mapping of . We may define the range of on the real line as
the evenly spaced nodes on the real line. The image which corresponds to these nodes is denoted by
A list of conformal mappings may be found in Table 1[6].
Figure 4. Three adjacent memberswhenandof the mapped sinc basis on the interval.
Table 1. Conformal mappings and nodes for some subintervals ofR
Definition 2.1 Let be a simply connected domain in the complex plane C, and let denote the boundary of . Let a, b be points on and ϕ be a conformal map onto such that and . If the inverse map of ϕ is denoted by φ, define
Definition 2.2 Let be the class of functions F that are analytic in and satisfy
where
and those on the boundary of satisfy
The proof of following theorems can be found in [2].
Theorem 2.1Let Γ be, , then forsufficiently small,
where
For the sincGalerkin 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
then the error bound for the quadrature rule (2.14) is
The infinite sum in (2.14) is truncated with the use of (2.16) to arrive at the inequality (2.17). Making the selections
whereis an integer part of the statement andNis the integer value which specifies the grid size, then
We used Theorems 2.1 and 2.2 to approximate the integrals that arise in the formulation of the discrete systems corresponding to a secondorder boundary value problem.
Theorem 2.3Letϕbe a conformal onetoone map of the simply connected domainonto. Then
3 The sincGalerkin method for singular Dirichlettype boundary value problems
Consider the following problem:
with Dirichlettype boundary condition
where P, Q and F are analytic on D. We consider sinc approximation by the formula
The unknown coefficients 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 coefficients by solving the linear system of equations
Let and be analytic functions on D and the inner product in (3.5) be defined as follows:
where w is the weight function. For the secondorder problems, it is convenient to take [2].
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:
where . If we choose and as given in [2] the accuracy for each equation between (3.8)(3.11) will be .
Using (3.5), (3.8)(3.11), we obtain a linear system of equations for numbers .
The linear system given in (3.5) can be expressed by means of matrices. Let , and let and be a column vector defined by
Let denote a diagonal matrix whose diagonal elements are and nondiagonal elements are zero, and also let , and denote the matrices
With these notations, the discrete system of equations in (3.5) takes the form:
Theorem 3.1Letcbe anmvector whosejth component is. Then the system (3.16) yields the following matrix system, the dimensions of which are:
Now we have a linear system ofequations of theunknown coefficients. If we solve (3.17) by usingLUorQRdecomposition methods, we can obtaincoefficients for the approximate sincGalerkin solution
4 Examples
Three examples were given in order to illustrate the performance of the sincGalerkin method to solve a singular Dirichlettype boundary value problem in this section. The discrete sinc system defined by (3.18) was used to compute the coefficients ; for each example. All of the computations were done by an algorithm which we have developed for the sincGalerkin method. The algorithm automatically compares the sincmethod with the exact solutions. It is shown in Tables 24 and Figures 57 that the sincGalerkin method is a very efficient and powerful tool to solve singular Dirichlettype boundary value problems.
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).
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).
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 sincGalerkin in comparison with the exact solutions of Eq. (4.1) for
Table 3. The numerical results for the approximate solutions obtained by sincGalerkin in comparison with the exact solutions of Eq. (4.2) for
Table 4. The numerical results for the approximate solutions obtained by sincGalerkin in comparison with the exact solutions of Eq. (4.3) for
Example 4.1 Consider the following singular Dirichlettype boundary value problem on the interval :
The exact solution of (4.1) is
We choose the weight function according to [2], , , and by taking , , for , the solutions inFigure 5 and Table 2 are achieved.
Example 4.2 Let us have the following form of a singular Dirichlettype boundary value problem on the interval :
The problem has an exact solution like
where , .By taking , , for ,we get the solutions in Figure 6 and Table 3.
Example 4.3 The following problem is given on the interval :
where the exact solution of (4.3) is .
In this case, , , and by taking , , for , we get results in Figure 7 and Table 4 .
5 Conclusion
The sincGalerkin method was employed to find the solutions of secondorder Dirichlettype 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 secondorder Dirichlettype BVPs with the sincGalerkin 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 sincGalerkin approximation
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
AS proposed main idea of the solution schema by using Sinc Method for linear BVPs. He developed computer algorithm and worked on theoretical aspect of problem. MK searched the materials about study and compared with other techniques, contributed with his experience on Nonlinear Approximation methods.
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