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Journal of Applied Mathematics
Volume 2012, Article ID 254123, 17 pages
http://dx.doi.org/10.1155/2012/254123
Research Article

A Boundary Integral Equation with the Generalized Neumann Kernel for a Certain Class of Mixed Boundary Value Problem

1Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia
2Department of Mathematics, Faculty of Science, Ibb University, P.O. Box 70270, Ibb, Yemen
3Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
4Ibnu Sina Institute for Fundamental Science Studies, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

Received 21 May 2012; Revised 29 August 2012; Accepted 12 September 2012

Academic Editor: Yuantong Gu

Copyright © 2012 Mohamed M. S. Nasser et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving two-dimensional Laplace’s equation on multiply connected regions with mixed boundary condition. Two numerical examples are presented to verify the accuracy of the proposed method.

1. Introduction

The interplay of Riemann-Hilbert boundary value problem and integral equations with the generalized Neumann kernel on bounded multiply connected regions has been investigated in [1]. By reformulating the Dirichlet problem, the Neumann problem, and the conformal mapping as Riemann-Hilbert problems, boundary integral equations with the generalized Neumann kernel have been implemented successfully in [2] to solve the Dirichlet problem and the Neumann problem and in [35] to compute the conformal mapping of multiply connected regions onto the classical canonical slit domains. The mixed boundary value problem also can be reformulated as a Riemann-Hilbert problem (see [68]). Based on this reformulation, we present in this paper a new boundary integral equation method for two-dimensional Laplace’s equation with mixed boundary condition in bounded multiply connected regions. However, in this paper, we will consider only a certain class of mixed boundary condition where the boundary condition in each boundary component is either the Dirichlet condition or the Neumann condition but not both. This class of mixed boundary value problem has been considered in [9] for Laplace’s equation and in [10, page 78] for biharmonic equation on multiply connected regions. A similar class of mixed boundary value problem has been considered in [11, page 317] for doubly connected regions.

The presented method is an extension of the method presented in [2] for Laplace's equation with the Dirichlet boundary conditions and Laplace’s equation with the Neumann boundary conditions on multiply connected regions. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. The generalized Neumann kernel which will be considered in this paper is slightly different from the generalized Neumann kernel for integral equation associated with the Dirichlet problem and the Neumann problem which has been considered in [2]. Thus, not all of the properties of the generalized Neumann kernel which have been studied [2] are valid for the generalized Neumann kernel which will be used in this paper. For example, it is still true that is not an eigenvalue of the generalized Neumann kernel which means that the presented integral equation is uniquely solvable. However, it is no longer true that all eigenvalues of the generalized Neumann kernel are real (see Figure 3 and [2, Theorem  8]).

2. Notations and Auxiliary Material

In this section we will review the definition and some properties of the generalized Neumann kernel. For further details we refer the reader to [13, 5].

Let be a bounded multiply connected region of connectivity with . The boundary consists of smooth closed Jordan curves where contains the other curves (see Figure 1). The complement of with respect to consists of simply connected components . The components are bounded and the component is unbounded where . We assume that the orientation of the boundary is such that is always on the left of . Thus, the curves always have clockwise orientations and the curve has a counterclockwise orientation. The curve is parametrized by a -periodic twice continuously differentiable complex function with nonvanishing first derivative The total parameter domain is the disjoint union of the intervals . We define a parametrization of the whole boundary as the complex function defined on by

254123.fig.001
Figure 1: A bounded multiply connected region of connectivity .

Let be the space of all real Hölder continuous functions on the boundary . In view of the smoothness of , a function can be interpreted via , , as a real Hölder continuous -periodic functions of the parameter , that is, with real Hölder continuous -periodic functions defined on , and vice versa. So, here and in what follows, we will not distinguish between and .

The subspace of which consists of all piecewise constant functions defined on will be denoted by , that is, a function has the representation where are real constants. For simplicity, the function will be denoted by

In this paper we will assume that the function is the continuously differentiable complex function where is the real piecewise constant function with either or , . This function, , is a special case of the function in [5, Equation ] in connection with numerical conformal mapping of multiply connected regions.

The generalized Neumann kernel formed with is defined by We define also a real kernel by The kernel is continuous and the kernel has a cotangent singularity type (see [1] for more details). Hence, the operator is a Fredholm integral operator and the operator, is a singular integral operator.

The solvability of boundary integral equations with the generalized Neumann kernel is determined by the index (winding number in other terminology) of the function (see [1]). For the function given by (2.6), the index of on the curve and the index of on the whole boundary are given by

Although the function in (2.6) is different from the function in [2, Equation ], both functions have the same indices. So by using the same approach used in [2], we can prove that the properties of the generalized Neumann kernel proved in [2], except Theorem  8, Theorem  10, and Corollary  2, are still valid for the generalized Neumann kernel formed with the function in (2.6) (see [5]). Numerical evidence show that [2, Theorem  8], which claims that the eigenvalues of the generalized Neumann kernel lies in , is no longer true for the function in (2.6) (see Figure 3).

Theorem 2.1 (see [5]). For a given function , there exist unique functions and such that is a boundary value of a unique analytic function in . The function is the unique solution of the integral equation and the function is given by

3. The Mixed Boundary Value Problem

Let and be two subsets of the set such that We will consider the following class of mixed boundary value problems. Find a real function in such that where is the unit exterior normal to .

The problem (3.2a), (3.2b), and (3.2c) reduces to the Dirichlet problem for and to the Neumann problem for . Both problems have been considered in [2]. So we have assumed in this paper that and . The mixed boundary value problem (3.2a), (3.2b), and (3.2c) is uniquely solvable. Its unique solution can be regarded as a real part of an analytic function in which is not necessairly single-valued. However, the function can be written as where is a single-valued analytic function in , each is a fixed point in , ; and are real constants uniquely determined by (see [12, page 145] and [13, page 174]). Without los of generality, we assume that . The constants are chosen to ensure that (see [14, page 222] and [15, page 88]) that is, are given by (see [2])

We define a real constant by Hence, we have Since , we obtain

The Dirichlet boundary condition is a special case of the Riemann-Hilbert boundary condition. The Neumann boundary condition also can be reduced to a Riemann-Hilbert boundary condition by using the Cauchy-Riemann equations and integrating along the boundary for . Let be the unit tangent vector and be the unit external normal vector to at . Let also be the angle between the normal vector and the positive real axis, that is, . Then, Thus Since , then by the Cauchy-Riemann equation, we have Thus (3.10) becomes Let the boundary value of the multivalued analytic function be given by Then the function is a single-valued analytic function and has the boundary value

Let be the real piecewise constant function Hence, the boundary values of the function satisfy on the boundary where and for and , the function is known and is given by The functions for are given by (3.2b) and (3.2c). The function can be then computed for and by integrating the function . Then, it follows from (3.3), (3.16), and (3.17) that the the function is a solution of the Riemann-Hilbert problem Since then, in view of (3.6), the boundary condition (3.19) can be written as where for .

In view of (3.5), (3.8), and (3.18), the constants are known for and are given by For , the real constants are unknown. Thus, the boundary condition (3.19) can be written as where the function is known and is given by It is clear that the function is known explicitly for with . However, for with it is required to integrate to obtain .

The function is not necessary -periodic. Numerically, we prefer to deal with periodic functions. So we will not compute directly by integrating the function . Instead, we will integrate the function By the definitions of the constants and the functions , we have which implies that the function is always -periodic. So, for with , we write the function using the Fourier series as Then the function is given for with by where is undetermined real constant and the function is given by Hence, the boundary condition (3.24) can be then written as where is the real piecewise constant function and the function is given by

Let (unknown real constant) and be the analytic function defined on by Thus the function is a solution of the Riemann-Hilbert problem where the function is given by (2.6) and the function is defined by Let , then the boundary value of the function is given on the boundary by where , are knowns and , are unknowns.

4. The Solution of the Mixed Boundary Value Problem

Let and let for be the unique solutions of the integral equations and , be given by Then, it follows from Theorem 2.1 that is the boundary value of an analytic function . Since the unknown functions and in (3.37) are uniquely determined by the known functions and for , it follows from (3.37) and (4.3) that In view of (3.15), (3.32), and (3.36), the function is given by Thus, we have from (4.5), (3.23), and (3.6) the linear system of equations in unknowns , for and for .

By solving the linear system (4.7a) and (4.7b), we obtain the values of the constants and the function . Then, we obtain the function from (4.4). Consequently, the boundary value of the function is given by where The function can be computed for by the Cauchy integral formula. Then the function is given by Finally, the solution of the mixed boundary value problem can be computed from where is given by (3.3).

5. Numerical Implementations

Since the functions and are -periodic, the integrands in the integral equations (4.1) are -periodic. Hence, the most efficient numerical method for solving (4.1) is generally the Nyström method with the trapezoidal rule (see e.g., [16, page 321]). We will use the trapezoidal rule with (an even positive integer) equidistant node points on each boundary component to discretize the integrals in (4.1). If the integrands in (4.1) are times continuously differentiable, then the rate of convergence of the trapezoidal rule is . For analytic integrands, the rate of convergence is better than for any positive integer (see e.g., [17, page 83]). The obtained approximate solutions of the integral equations converge to the exact solutions with a similarly rapid rate of convergence (see e.g., [16, page 322]). Since the smoothness of the integrands in (4.1) depends on the smoothness of the function , results of high accuracy can be obtained for very smooth boundaries.

By using the Nyström method with the trapezoidal rule, the integral equations (4.1) reduce to by linear systems. Since the integral equations (4.1) are uniquely solvable, then for sufficiently large values of the obtained linear systems are also uniquely solvable [16]. The linear systems are solved using the Gauss elimination method. The computational details are similar to previous works in [25].

By solving the linear systems, we obtain approximations to and for which give approximations to and for through (4.2). By solving (4.7a) and (4.7b) we get approximations to the constants , for and for . These approximations allow us to obtain approximations to the boundary value of the function from (4.8). The values of for will be calculated by the Cauchy integral formula. The approximate values of the function are then computed from (4.10). Finally, in view of (3.3), the solution of the mixed boundary value problem can be computed from

In this paper, we have considered regions with smooth boundaries. For some ideas on how to solve numerically boundary integral equations with the generalized Neumann kernel on regions with piecewise smooth boundaries, see [18].

6. Numerical Examples

In this section, the proposed method will be used to solve two mixed boundary value problems in bounded multiply connected regions with smooth boundaries.

Example 6.1. In this example we consider a bounded multiply connected region of connectivity bounded by the four circles (see Figure 2(a)) where .

fig2
Figure 2: The regions of Example 6.1 (a) and Example 2 (b).
fig3
Figure 3: The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing the integral equations (4.1) with for Example 6.1 (a) and with for Example 6.2 (b).

We assume that the condition on the boundaries is the Neumann condition and the condition on the boundaries is the Dirichlet condition. The functions in (3.2a), (3.2b), and (3.2c) are obtained based on choosing an exact solution of the form This example has been considered in [9, page 894] using the regularized meshless method. The domain here is shifted by one unit to the right to ensure that . To compare our method with the method presented in [9], we use the same error norm used in [9], namely, where is the exact solution of the mixed boundary value problem and is the approximate solution obtained with node points on each boundary component. The error norm versus the total number of calculation points by using the presented method is shown in Figure 4(a) where the integral in (6.3) is discretized by the trapezoidal rule. By using only ( calculation points on the whole boundary), we obtain error norm less than . In [9], the error norm is only less than when 200 boundary points are used. The absolute errors for selected points in the entire domain are plotted in Figure 5(a) (compare the results with [9, Figure  10]).

fig4
Figure 4: The error norm (6.3) for Example 1 (a) and the error norm (6.6) for Example 6.2 (b) versus the total number of node points.
fig5
Figure 5: The absolute error for the entire domain obtained with for Example 6.1 (a) and with for Example 6.2 (b).

Example 6.2. In this example we consider a bounded multiply connected region of connectivity (see Figure 2(b)). The boundary is parametrized by where the values of the complex constants and the real constants , , are as in Table 1.

tab1
Table 1: The values of constants , , , , and in (6.4).

The region in this example has been considered in [2, 19, 20] for the Dirichlet problem and the Neumann problem. In this example, we will consider a mixed boundary value problem where we assume that the condition on the boundaries , , , and is the Dirichlet condition and the condition on the boundaries , , and is the Neumann condition. The functions in (3.2a), (3.2b), and (3.2c) are obtained based on choosing an exact solution of the form where the values of the complex constants are as in Table 1. For the error, we use the error norm (see Figure 4(b)): The absolute errors for selected points in the entire domain are plotted in Figure 5(b).

7. Conclusions

A new uniquely solvable boundary integral equation with the generalized Neumann kernel has been presented in this paper for solving a certain class of mixed boundary value problem on multiply connected regions. Two numerical examples are presented to illustrate the accuracy of the presented method.

The presented method can be applied to mixed boundary value problem with both the Dirichlet condition and the Neumann condition on the same boundary component . For this case, the function is discontinuous on , where on the part of corresponding to the Dirichlet condition and on the part of corresponding to the Neumann condition. Hence, the Riemann-Hilbert problem (3.35) will be a problem with discontinuous coefficient . The solvability of Riemann-Hilbert problems with discontinuous coefficients is different from the ones with continuous coefficients (see e.g., [6, page 449] and [13, page 271]). Furthermore, the discontinuity of the function implies that we cannot apply the theory of the generalized Neumann kernel developed in [21] for simply connected regions and in [1] for multiply connected regions. Hence, further investigations are required. This will be considered in future work.

Acknowledgments

The authors would like to thank the anonymous referees for suggesting several improvements. This work was supported in part by the Deanship of Scientific Research at King Khalid University through Project no. KKU-SCI-11-022. This support is gratefully acknowledged.

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