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A boundary integral equation with the generalized Neumann kernel for a mixed boundary value problem in unbounded multiply connected regions

Samer AA Al-Hatemi1, Ali HM Murid12* and Mohamed MS Nasser34

Author Affiliations

1 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, 81310 UTM, Malaysia

2 UTM Centre for Industrial and Applied Mathematics, Universiti Teknologi Malaysia, Johor Bahru, Johor, 81310 UTM, Malaysia

3 Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

4 Department of Mathematics, Faculty of Science, Ibb University, P.O. Box 70270, Ibb, Yemen

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Boundary Value Problems 2013, 2013:54  doi:10.1186/1687-2770-2013-54


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/54


Received:29 September 2012
Accepted:8 February 2013
Published:14 March 2013

© 2013 Al-Hatemi et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

Keywords:
mixed boundary value problem; RH problem; Fredholm integral equation; generalized Neumann kernel

1 Introduction

In the present paper, we continue the research concerned with the study of mixed boundary value problems in the plane started in [1]. 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 [2]. 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 [3] 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.[1] 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

In this section, we review some properties of the generalized Neumann kernel from [2,3,5,10].

We consider an unbounded multiply connected region G of connectivity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M1">View MathML</a> with boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M2">View MathML</a> consisting of m clockwise oriented smooth closed Jordan curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M4">View MathML</a>. The complement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M5">View MathML</a> consists of m bounded simply connected components <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M6">View MathML</a> interior to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M8">View MathML</a>. We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M10">View MathML</a> (see Figure 1).

thumbnailFigure 1. An unbounded multiply connected regionGof connectivitym.

The curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3">View MathML</a> are parametrized by 2π-periodic twice continuously differentiable complex-valued functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M12">View MathML</a> with non-vanishing first derivatives, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M13">View MathML</a>

(1)

The total parameter domain J is the disjoint union of the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M4">View MathML</a>. We define a parametrization of the whole boundary Γ as the complex-valued function η defined on J by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M16">View MathML</a>

(2)

Let H be the space of all real Hölder continuous functions on the boundary Γ. In view of the smoothness of η, a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M17">View MathML</a> can be interpreted via <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M19">View MathML</a>, as a real Hölder continuous 2π-periodic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M20">View MathML</a> of the parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M19">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M22">View MathML</a>

(3)

with real Hölder continuous 2π-periodic functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M23">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M14">View MathML</a>. So, here and in what follows, we do not distinguish between functions of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M26">View MathML</a>.

The subspace of H which consists of all piecewise constant functions defined on Γ is denoted by S, i.e., a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M27">View MathML</a> has the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M28">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M29">View MathML</a> are real constants. For simplicity, the function h is denoted by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M30">View MathML</a>

(5)

3 The mixed boundary value problem

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M32">View MathML</a> be two subsets of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M33">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M34">View MathML</a>

Let n be the exterior unit normal to Γ and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M35">View MathML</a> be a given function. We consider the mixed boundary value problem

(6a)

(6b)

(6c)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M39">View MathML</a> and to the Neumann problem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M40">View MathML</a>. Both problems have been considered in [3]. So, we assume in this paper that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M42">View MathML</a>.

The mixed boundary value problem (6a)-(6c) is uniquely solvable [11]. Its unique solution u can be regarded as a real part of an analytic function F in G which is not necessary single-valued. The function F can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M43">View MathML</a>

(7)

where f is a single-valued analytic function in G, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M44">View MathML</a> are fixed points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M4">View MathML</a>; and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M47">View MathML</a> are real constants uniquely determined by ϕ (see [12]). Without lost of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M48">View MathML</a>. The constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M47">View MathML</a> are chosen to ensure that (see [[12], p.149] and [3])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M50">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a> are given by (see [3])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M52">View MathML</a>

(8)

The constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a> satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M54">View MathML</a>

(9)

4 Integral equation

In this paper we assume that the function A is a continuously differentiable complex-valued function given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M55">View MathML</a>

(10)

where θ is the real piecewise constant function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M56">View MathML</a>

(11)

with either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M57">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M59">View MathML</a>. Here the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60">View MathML</a> is different from the ones used in [1,3]. The generalized Neumann kernel formed with A and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M61">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M62">View MathML</a>

(12)

We also define a real kernel M by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M63">View MathML</a>

(13)

The kernel N is continuous and the kernel M has a cotangent singularity type (see [2] for more details). Hence, the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M64">View MathML</a>

(14)

is a Fredholm integral operator and the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M65">View MathML</a>

(15)

is a singular integral operator.

The solvability of boundary integral equations with the generalized Neumann kernel is determined by the index (the change of the argument of A on the curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3">View MathML</a> divided by 2π) of the function A (see [2]). For the function A given by (10), the indices <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M67">View MathML</a> of A on the curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3">View MathML</a> and the index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M69">View MathML</a> of A on the whole boundary curve Γ are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M70">View MathML</a>

(16)

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 [3] are valid for the generalized Neumann kernel which will be studied in this paper. For example, it is still true that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M71">View MathML</a> 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 [3] for unbounded multiply connected regions, we can prove that the properties of the generalized Neumann kernel proved in [3], except Theorem 8, Theorem 10 and Corollary 2, are still valid for the generalized Neumann kernel formed with the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60">View MathML</a> in (10) above (see [5]).

Thus, we have from [5] the following theorem (see also [2,10]).

Theorem 1For a given function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M73">View MathML</a>, there exist unique functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M75">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M76">View MathML</a>

(17)

are boundary values of a unique analytic function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a>inGwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M78">View MathML</a>. The functionμis the unique solution of the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M79">View MathML</a>

(18)

and the functionhis given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M80">View MathML</a>

(19)

5 Reformulation of the mixed boundary value problem as an RH problem

The mixed boundary value problem can be reduced to an RH problem as follows. Let the boundary values of the multi-valued analytic function F be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M81">View MathML</a>

(20)

Although, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82">View MathML</a> is in general multi-valued, its derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M83">View MathML</a> is a single-valued analytic function on G. The boundary values of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M84">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M85">View MathML</a>

(21)

For the Dirichlet conditions, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M88">View MathML</a> are equal to the known functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M89">View MathML</a> (see (6b)). Thus, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82">View MathML</a> satisfies the RH problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M91">View MathML</a>

(22)

The Neumann conditions can also be reduced to an RH problem by using the Cauchy-Riemann equations and integrating along the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M94">View MathML</a> be the unit tangent vector and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M95">View MathML</a> be the unit external normal vector to Γ at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M96">View MathML</a>. Let also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M97">View MathML</a> be the angle between the normal vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M95">View MathML</a> and the positive real axis, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M99">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M100">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M101">View MathML</a>

(23)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M102">View MathML</a>, then by the Cauchy-Riemann equations, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M103">View MathML</a>

Thus, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M84">View MathML</a> satisfies the RH problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M105">View MathML</a>

(24)

If we define the real piecewise constant function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M106">View MathML</a>

(25)

the boundary values of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82">View MathML</a> satisfy on the boundary Γ the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M108">View MathML</a>

(26)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M109">View MathML</a>

(27)

is known and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M110">View MathML</a>

(28)

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M89">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M112">View MathML</a> are given by (6b) and (6c). The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113">View MathML</a> can be then computed for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a> by integrating the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M116">View MathML</a>. Then it follows from (7), (26) and (27) that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M117">View MathML</a> is a solution of the RH problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M118">View MathML</a>

(29)

or briefly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M119">View MathML</a>

(30)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M120">View MathML</a>

(31)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M121">View MathML</a>. In view of (8) and (28), the real constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M122">View MathML</a> are known for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M123">View MathML</a> and are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M124">View MathML</a>

(32)

However, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M125">View MathML</a>, the real constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M122">View MathML</a> are unknown. Thus, the boundary condition (29) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M127">View MathML</a>

(33)

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M128">View MathML</a> is known and is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M129">View MathML</a>

(34)

Obviously, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M130">View MathML</a> are known explicitly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a>. However, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a>, it is required to integrate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M116">View MathML</a> to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113">View MathML</a>.

The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113">View MathML</a> are not necessary 2π-periodic. In order to keep dealing with periodic functions numerically, we do not compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M113">View MathML</a> directly by integrating the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M116">View MathML</a>. Instead, we integrate the functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M140">View MathML</a>

According to the definitions of the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M122">View MathML</a> and the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M142">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M143">View MathML</a>

which implies that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M144">View MathML</a> are always 2π-periodic. By using the Fourier series for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a>, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M147">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M148">View MathML</a>

(35)

Then the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M130">View MathML</a> are given for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M86">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M152">View MathML</a>

(36)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153">View MathML</a> are undetermined real constants and the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M154">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M155">View MathML</a>

(37)

Hence, the boundary condition (33) can then be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M156">View MathML</a>

(38)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M157">View MathML</a> is the real piecewise constant function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M158">View MathML</a>

(39)

and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M159">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M160">View MathML</a>

(40)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M161">View MathML</a> (unknown real constant) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> be the analytic function defined on G by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M163">View MathML</a>

(41)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> is analytic on G with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M78">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> is a solution of the RH problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M167">View MathML</a>

(42)

where the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60">View MathML</a> is given by (10) and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M169">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M170">View MathML</a>

(43)

6 The solution of the mixed boundary value problem

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M171">View MathML</a>. Then the boundary values of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> are given on the boundary Γ by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M173">View MathML</a>

(44)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M159">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M175">View MathML</a> are knowns and h, μ are unknowns. The real constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a> are known for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a> and unknown for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a>.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a>, let the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M181">View MathML</a> be the unique solutions of the integral equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M182">View MathML</a>

(45)

respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M184">View MathML</a> be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M185">View MathML</a>

(46)

Then it follows from Theorem 1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M186">View MathML</a>

(47)

are boundary values of an analytic function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M187">View MathML</a>. By the uniqueness of the functions h and μ in (44), it follows from (44) and (47) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M188">View MathML</a>

(48)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M189">View MathML</a>

(49a)

Equation (49a) with the following equation (from (9)),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M190">View MathML</a>

(49b)

represents a linear system of m equations. Since from (43) the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M169">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M192">View MathML</a>

only the constants c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a> are unknowns. Thus, linear equations (49a) and (49b) represent a linear system of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M197">View MathML</a> equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M197">View MathML</a> unknowns <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a>.

By obtaining the values of the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a>, we obtain the functions μ from (48) and h from (49a). Consequently, the boundary values of the function g are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M204">View MathML</a>

(50)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M205">View MathML</a>

(51)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> can be computed for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M207">View MathML</a> by the Cauchy integral formula. Then the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M117">View MathML</a> is computed from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M209">View MathML</a>

(52)

Finally, the solution of the mixed boundary value problem can be computed from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M102">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M82">View MathML</a> is given by (7).

7 Numerical implementations

Since the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M213">View MathML</a> 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 [13]. The computational details are similar to previous works in [4,5,10,14]. For analytic integrands, the rate of convergence is better than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M214">View MathML</a> for any positive integer k (see, e.g., [[15], 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., [[13], p.322]). Since the smoothness of the integrands in (45) depends on the smoothness of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M61">View MathML</a>, 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 [13].

In this paper, the linear systems are solved using the Gauss elimination method. By solving the linear systems, we obtain approximations to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M181">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a>, which give approximations to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a> from (46). By solving (49a) and (49b), we get approximations to the constants c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M153">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M93">View MathML</a>. These give approximations to the boundary values of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> from (50). Then the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M77">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M207">View MathML</a> are calculated by the Cauchy integral formula. For points z which are not close to the boundary Γ, the integrals in the Cauchy integral formula are approximated by the trapezoidal rule. However, for points z near the boundary Γ, the integrand is nearly singular. For the latter case, the integral in the Cauchy integral formula can be calculated accurately using the method suggested in [[16], Eq. (23)]. Then approximate values of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M117">View MathML</a> are computed from (52). Finally, in view of (7), the approximate solution of the mixed boundary value problem can be computed from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M230">View MathML</a>

(53)

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 [14].

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.

Example 1 In this example, we consider an unbounded multiply connected region of connectivity 4 bounded by the four circles (see Figure 2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M232">View MathML</a>.

thumbnailFigure 2. The region for Example 1.

We assume that the conditions on the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M234">View MathML</a> are the Neumann conditions and the conditions on the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M235">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M236">View MathML</a> are the Dirichlet conditions. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M237">View MathML</a> in (6b)-(6c) are obtained based on choosing an exact solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M238">View MathML</a>

We use the error norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M239">View MathML</a>

(54)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M240">View MathML</a> is the exact solution of the mixed boundary value problem and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M241">View MathML</a> is the approximate solution obtained with n collocation points. The error norm vs. the total number of calculation points n by using the presented method is shown in Figure 3, where the integral in (54) is discretized by the trapezoidal rule. By using only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M242">View MathML</a> (256 calculation points on the whole boundary), we obtain error norm less that 10−15. The absolute errors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M243">View MathML</a> at selected points in the entire domain are plotted in Figure 4. The graph of the approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M241">View MathML</a> is illustrated in Figure 5.

thumbnailFigure 3. The error norm (54) for Example 1.

thumbnailFigure 4. The absolute error for Example 1.

thumbnailFigure 5. The graph of the approximate solution for Example 1.

Example 2 In this example, we consider an unbounded multiply connected region of connectivity 6 (see Figure 6). The boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M245">View MathML</a> is parametrized by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M246">View MathML</a>

(55)

where the values of the complex constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M44">View MathML</a> and the real constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M250">View MathML</a> are as in Table 1.

thumbnailFigure 6. The region for Example 2.

Table 1. The values of constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M248">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M249">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M44">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M250">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M255">View MathML</a>for Example 2

The region in this example has been considered in [3,17,18] for the Dirichlet problem and the Neumann problem. In this example, we consider a mixed boundary value problem where we assume that the conditions on the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M235">View MathML</a> are the Dirichlet conditions and the conditions on the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M266">View MathML</a> are the Neumann conditions. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M237">View MathML</a> in (6b)-(6c) are obtained based on choosing an exact solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M268">View MathML</a>

where the values of the complex constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M255">View MathML</a> are as in Table 1. For the error, we use the error norm (see Figure 7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M270">View MathML</a>

(56)

The absolute errors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M243">View MathML</a> at selected points in the entire domain are plotted in Figure 8. The graph of the approximate solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M241">View MathML</a> is shown in Figure 9.

thumbnailFigure 7. The error norm (56) for Example 2.

thumbnailFigure 8. The absolute error for Example 2.

thumbnailFigure 9. The graph of the approximate solution for Example 2.

Example 3 This example aims to give an impression how the method works for a problem with an unknown exact solution. We assume that the boundaries of an unbounded doubly connected region are represented as follows (see Figure 10):

We assume the Dirichlet condition on the star-shape boundary with Dirichlet data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M274">View MathML</a>, while on the ellipse-shape boundary is the Neumann condition with Neumann data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M275">View MathML</a>. The graph of the approximate solution is illustrated in Figure 11.

thumbnailFigure 10. The region for Example 3.

thumbnailFigure 11. The graph of the approximate solution for Example 3.

9 Conclusion

We have constructed a new boundary integral equation with the generalized Neumann kernel for solving a mixed boundary value problem in unbounded multiply connected regions. The generalized Neumann kernel used in this paper is formed with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M276">View MathML</a> which is different from the ones used in [1,3]. Numerical evidences show that Theorem 8 in [3], which claims that the eigenvalues of the generalized Neumann kernel lie in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M277">View MathML</a>, is no longer true for the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60">View MathML</a> of this paper (see Figures 12 and 13). Three numerical examples are presented to illustrate the accuracy of the presented method. The numerical examples illustrate that the proposed method yields approximations of high accuracy. This paper only applies to the explicitly mixed Dirichlet and Neumann boundary conditions in unbounded multiply connected regions, but the method can be extended to a boundary with both mixed boundary conditions in a boundary component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M279">View MathML</a>. For this case, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60">View MathML</a> is discontinuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M281">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M282">View MathML</a> on the part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M279">View MathML</a> corresponding to the Dirichlet condition and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M284">View MathML</a> on the part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M279">View MathML</a> corresponding to the Neumann condition. Hence, the RH problem (42) will be a problem with discontinuous coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M60">View MathML</a>. Thus, further investigations are required. This will be considered in future work.

thumbnailFigure 12. The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing integral equations (45) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M287">View MathML</a>for the region in Example 1.

thumbnailFigure 13. The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing integral equations (45) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/54/mathml/M287">View MathML</a>for the region in Example 2.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read and approved the final manuscript.

Acknowledgements

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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