- Open Access
A boundary integral equation with the generalized Neumann kernel for a mixed boundary value problem in unbounded multiply connected regions
© Al-Hatemi et al.; licensee Springer 2013
Received: 29 September 2012
Accepted: 8 February 2013
Published: 14 March 2013
In this paper we propose a new method for solving the mixed boundary value problem for the Laplace equation in unbounded multiply connected regions. All simple closed curves making up the boundary are divided into two sets. The Dirichlet condition is given for one set and the Neumann condition is given for the other set. The mixed problem is reformulated in the form of a Riemann-Hilbert (RH) problem which leads to a uniquely solvable Fredholm integral equation of the second kind. Three numerical examples are presented to show the effectiveness of the proposed method.
In the present paper, we continue the research concerned with the study of mixed boundary value problems in the plane started in . We consider a mixed boundary value problem for the Laplace equation in an unbounded multiply connected regions. Recently, the interplay of the RH boundary value problem and integral equations with the generalized Neumann kernel on unbounded multiply connected regions has been investigated in . Based on the reformulations of the Dirichlet problem, the Neumann problem and conformal mappings as RH problems, boundary integral equations with the generalized Neumann kernel have been implemented successfully in  to solve the Dirichlet problem and the Neumann problem and in [4–6] to compute the conformal mappings of unbounded multiply connected regions onto the classical canonical slit domains.
The mixed boundary value problem also can be reformulated as an RH problem (see, e.g., [7–9]). Recently, Nasser et al.  have presented a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving the mixed boundary value problem in bounded multiply connected regions. The idea of this paper is to reformulate the mixed boundary value problem to the form of the RH problem in unbounded multiply connected regions. Based on this reformulation, we present a new boundary integral equation method for two-dimensional Laplace’s equation with the mixed boundary condition in unbounded multiply connected regions. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel.
This paper is organized as follows. After presenting some auxiliary materials in Section 2, we present in Section 3 the mixed boundary value problem in unbounded multiply connected regions. In Section 4, we give an explanation of an integral equation with the generalized Neumann kernel and its solvability. The reduction of the mixed boundary value problem to the form of the RH problem is given in Section 5. In Section 6, we present the solution of the mixed boundary problem via an integral equation method. In Section 7, we explain briefly the numerical implementation of the method. In Section 8, we illustrate the method by presenting two numerical examples with exact solutions and also one example without an exact solution.
2 Notations and auxiliary material
with real Hölder continuous 2π-periodic functions defined on . So, here and in what follows, we do not distinguish between functions of the form and .
3 The mixed boundary value problem
for a real function u in G. We call (6b) and (6c) Dirichlet conditions and Neumann conditions, respectively.
Problem (6a)-(6c) reduces to the Dirichlet problem for and to the Neumann problem for . Both problems have been considered in . So, we assume in this paper that and .
4 Integral equation
is a singular integral operator.
The generalized Neumann kernel for an integral equation associated with the mixed boundary value problem which will be presented in this paper is different from the generalized Neumann kernel for the integral equation considered in [1, 3]. Thus, not all of the properties of the generalized Neumann kernel which have been studied in  are valid for the generalized Neumann kernel which will be studied 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.
By using the same approach used in  for unbounded multiply connected regions, we can prove that the properties of the generalized Neumann kernel proved in , except Theorem 8, Theorem 10 and Corollary 2, are still valid for the generalized Neumann kernel formed with the function in (10) above (see ).
5 Reformulation of the mixed boundary value problem as an RH problem
Obviously, the functions are known explicitly for with . However, for with , it is required to integrate to obtain .
6 The solution of the mixed boundary value problem
where , are knowns and h, μ are unknowns. The real constants are known for and unknown for .
only the constants c, for and for are unknowns. Thus, linear equations (49a) and (49b) represent a linear system of equations in unknowns for and for .
Finally, the solution of the mixed boundary value problem can be computed from , where is given by (7).
7 Numerical implementations
Since the functions and are 2π-periodic, the integrals in the operators N and M in integral equations (45) are best discretized on an equidistant grid by the trapezoidal rule . The computational details are similar to previous works in [4, 5, 10, 14]. For analytic integrands, the rate of convergence is better than for any positive integer k (see, e.g., [, p.83]). The obtained approximate solutions of the integral equations converge to the exact solutions with a similarly rapid rate of convergence (see, e.g., [, p.322]). Since the smoothness of the integrands in (45) depends on the smoothness of the function , results of high accuracy can be obtained for very smooth boundaries.
By using the trapezoidal rule with n (an even positive integer) equidistant collocation points on each boundary component, solving integral equations (45) reduces to solving mn by mn linear systems. Since integral equations (45) are uniquely solvable, then for sufficiently large values of n, the obtained linear systems are also uniquely solvable .
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 .
8 Numerical examples
In this section, the proposed method is used to solve three mixed boundary value problems in unbounded multiply connected regions with smooth boundaries.
The values of constants , , , and for Example 2
0.1621 + 0.5940i
0.10 + 0.50i
−1.7059 + 0.3423i
−1.60 + 0.40i
0.3577 − 0.9846i
0.30 − 0.90i
1.0000 + 1.2668i
0.95 + 1.20i
−1.9306 − 1.0663i
−1.85 − 1.00i
−0.8330 − 2.1650i
−0.80 − 2.10i
The authors would like to thank the editor and referees for their helpful comments and suggestions which improved the presentation of the paper. The authors acknowledge the financial support for this research by the Malaysian Ministry of Higher Education (MOHE) through UTM GUP Vote Q.J130000.7126.01H75.
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