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

  • Samer AA Al-Hatemi1,

    Affiliated with

    • Ali HM Murid1, 2Email author and

      Affiliated with

      • Mohamed MS Nasser3, 4

        Affiliated with

        Boundary Value Problems20132013:54

        DOI: 10.1186/1687-2770-2013-54

        Received: 29 September 2012

        Accepted: 8 February 2013

        Published: 14 March 2013

        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 [46] 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., [79]). 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 m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq1_HTML.gif with boundary Γ = G = j = 1 m Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq2_HTML.gif consisting of m clockwise oriented smooth closed Jordan curves Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq3_HTML.gif, j = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq4_HTML.gif. The complement G : = C ¯ G ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq5_HTML.gif consists of m bounded simply connected components G j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq6_HTML.gif interior to Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq3_HTML.gif, j = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq7_HTML.gif. We assume G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq8_HTML.gif, 0 G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq9_HTML.gif (see Figure 1).
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig1_HTML.jpg
        Figure 1

        An unbounded multiply connected region G of connectivity m .

        The curves Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq3_HTML.gif are parametrized by 2π-periodic twice continuously differentiable complex-valued functions η j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq10_HTML.gif with non-vanishing first derivatives, i.e.,
        η ˙ j ( t ) = d η j ( t ) / d t 0 , t J j : = [ 0 , 2 π ] , j = 1 , 2 , , m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ1_HTML.gif
        (1)
        The total parameter domain J is the disjoint union of the intervals J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq11_HTML.gif, j = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq4_HTML.gif. We define a parametrization of the whole boundary Γ as the complex-valued function η defined on J by
        η ( t ) : = { η 1 ( t ) , t J 1 , η m ( t ) , t J m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ2_HTML.gif
        (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 ϕ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq12_HTML.gif can be interpreted via ϕ ˆ ( t ) : = ϕ ( η ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq13_HTML.gif, t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq14_HTML.gif, as a real Hölder continuous 2π-periodic function ϕ ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq15_HTML.gif of the parameter t J http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq14_HTML.gif, i.e.,
        ϕ ˆ ( t ) : = { ϕ ˆ 1 ( t ) , t J 1 , ϕ ˆ m ( t ) , t J m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ3_HTML.gif
        (3)

        with real Hölder continuous 2π-periodic functions ϕ ˆ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq16_HTML.gif defined on J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq11_HTML.gif. So, here and in what follows, we do not distinguish between functions of the form ψ ( η ( t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq17_HTML.gif and ψ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq18_HTML.gif.

        The subspace of H which consists of all piecewise constant functions defined on Γ is denoted by S, i.e., a function h S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq19_HTML.gif has the representation
        h ( t ) : = { h 1 , t J 1 , h m , t J m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ4_HTML.gif
        (4)
        where h 1 , , h m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq20_HTML.gif are real constants. For simplicity, the function h is denoted by
        h ( t ) = ( h 1 , , h m ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ5_HTML.gif
        (5)

        3 The mixed boundary value problem

        Let S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq21_HTML.gif and S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq22_HTML.gif be two subsets of the set { 1 , , m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq23_HTML.gif such that
        S d , S n , S d S n = { 1 , , m } and S d S n = . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equa_HTML.gif
        Let n be the exterior unit normal to Γ and let ϕ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq24_HTML.gif be a given function. We consider the mixed boundary value problem
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ6_HTML.gif
        (6a)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ7_HTML.gif
        (6b)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ8_HTML.gif
        (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 S n = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq25_HTML.gif and to the Neumann problem for S d = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq26_HTML.gif. Both problems have been considered in [3]. So, we assume in this paper that S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq27_HTML.gif and S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq28_HTML.gif.

        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
        F ( z ) = f ( z ) j = 1 m a j log ( z z j ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ9_HTML.gif
        (7)
        where f is a single-valued analytic function in G, z j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq29_HTML.gif are fixed points in G j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq6_HTML.gif, j = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq4_HTML.gif; and a 1 , , a m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq30_HTML.gif are real constants uniquely determined by ϕ (see [12]). Without lost of generality, we assume that Im f ( ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq31_HTML.gif. The constants a 1 , , a m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq30_HTML.gif are chosen to ensure that (see [[12], p.149] and [3])
        Γ j f ( η ) d η = 0 , j = 1 , 2 , , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equb_HTML.gif
        i.e., a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif are given by (see [3])
        a j = 1 2 π i Γ j F ( η ) d η , j = 1 , 2 , , m . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ10_HTML.gif
        (8)
        The constants a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif satisfy
        j = 1 m a j = j = 1 m 1 2 π i Γ j F ( η ) d η = 1 2 π i Γ F ( η ) d η = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ11_HTML.gif
        (9)

        4 Integral equation

        In this paper we assume that the function A is a continuously differentiable complex-valued function given by
        A ( t ) : = e i θ ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ12_HTML.gif
        (10)
        where θ is the real piecewise constant function
        θ ( t ) = ( θ 1 , , θ m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ13_HTML.gif
        (11)
        with either θ j = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq33_HTML.gif or θ j = π / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq34_HTML.gif, j = 1 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq35_HTML.gif. Here the function A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq36_HTML.gif is different from the ones used in [1, 3]. The generalized Neumann kernel formed with A and η ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq37_HTML.gif is defined by
        N ( s , t ) : = 1 π Im ( A ( s ) A ( t ) η ˙ ( t ) η ( t ) η ( s ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ14_HTML.gif
        (12)
        We also define a real kernel M by
        M ( s , t ) : = 1 π Re ( A ( s ) A ( t ) η ˙ ( t ) η ( t ) η ( s ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ15_HTML.gif
        (13)
        The kernel N is continuous and the kernel M has a cotangent singularity type (see [2] for more details). Hence, the operator
        N μ ( s ) : = J N ( s , t ) μ ( t ) d t , s J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ16_HTML.gif
        (14)
        is a Fredholm integral operator and the operator
        M μ ( s ) : = J M ( s , t ) μ ( t ) d t , s J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ17_HTML.gif
        (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 Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq3_HTML.gif divided by 2π) of the function A (see [2]). For the function A given by (10), the indices κ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq38_HTML.gif of A on the curves Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq3_HTML.gif and the index κ = j = 1 m κ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq39_HTML.gif of A on the whole boundary curve Γ are given by
        κ j = 0 , j = 1 , , m , κ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ18_HTML.gif
        (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 λ = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq40_HTML.gif 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 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq36_HTML.gif in (10) above (see [5]).

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

        Theorem 1 For a given function γ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq41_HTML.gif, there exist unique functions h S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq19_HTML.gif and μ H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq42_HTML.gif such that
        A g = γ + h + i μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ19_HTML.gif
        (17)
        are boundary values of a unique analytic function g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif in G with g ( ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq44_HTML.gif. The function μ is the unique solution of the integral equation
        ( I N ) μ = M γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ20_HTML.gif
        (18)
        and the function h is given by
        h = [ M μ ( I N ) γ ] / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ21_HTML.gif
        (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
        F = ψ + i φ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ22_HTML.gif
        (20)
        Although, the function F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq45_HTML.gif is in general multi-valued, its derivative F http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq46_HTML.gif is a single-valued analytic function on G. The boundary values of the function F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq47_HTML.gif are given by
        η ˙ F = ψ + i φ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ23_HTML.gif
        (21)
        For the Dirichlet conditions, i.e., t J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq48_HTML.gif and j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif, the functions ψ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq50_HTML.gif are equal to the known functions ϕ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq51_HTML.gif (see (6b)). Thus, the function F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq45_HTML.gif satisfies the RH problem
        Re [ F ( η j ( t ) ) ] = ϕ j ( t ) , t J j , j S d . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ24_HTML.gif
        (22)
        The Neumann conditions can also be reduced to an RH problem by using the Cauchy-Riemann equations and integrating along the boundaries Γ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq3_HTML.gif, j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif. Let T ( ζ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq53_HTML.gif be the unit tangent vector and n ( ζ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq54_HTML.gif be the unit external normal vector to Γ at ζ Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq55_HTML.gif. Let also ν ( ζ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq56_HTML.gif be the angle between the normal vector n ( ζ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq54_HTML.gif and the positive real axis, i.e., n ( ζ ) = e i ν ( ζ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq57_HTML.gif. Then
        e i ν ( η ( t ) ) = i T ( η ( t ) ) = i η ˙ ( t ) | η ˙ ( t ) | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equc_HTML.gif
        Thus,
        u n = u n = cos ν u x + sin ν u y = Re [ e i ν ( u x i u y ) ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ25_HTML.gif
        (23)
        Since u ( z ) = Re F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq58_HTML.gif, then by the Cauchy-Riemann equations, we have
        F ( z ) = u ( z ) x i u ( z ) y . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equd_HTML.gif
        Thus, the function F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq47_HTML.gif satisfies the RH problem
        Re [ i η ˙ j ( t ) F ] = | η ˙ j ( t ) | u n , t J j , j S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ26_HTML.gif
        (24)
        If we define the real piecewise constant function
        θ ( t ) = { 0 , t J j , j S d , π / 2 , t J j , j S n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ27_HTML.gif
        (25)
        the boundary values of the function F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq45_HTML.gif satisfy on the boundary Γ the condition
        Re [ e i θ ( t ) F ( η ( t ) ) ] = ϕ ˆ ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ28_HTML.gif
        (26)
        where
        ϕ ˆ ( t ) = { ϕ j ( t ) , t J j , j S d , φ j ( t ) , t J j , j S n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ29_HTML.gif
        (27)
        is known and
        φ j ( t ) = Re [ i η ˙ j ( t ) F ( η j ( t ) ) ] = ϕ j ( t ) | η ˙ j ( t ) | , t J j , j S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ30_HTML.gif
        (28)
        The functions ϕ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq51_HTML.gif for j S d S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq59_HTML.gif are given by (6b) and (6c). The functions φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq60_HTML.gif can be then computed for t J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq48_HTML.gif and j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif by integrating the functions φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq61_HTML.gif. Then it follows from (7), (26) and (27) that the function f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq62_HTML.gif is a solution of the RH problem
        Re [ e i θ ( t ) f ( η ( t ) ) ] = ϕ ˆ ( t ) + k = 1 m a k Re [ e i θ ( t ) log ( η ( t ) z k ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ31_HTML.gif
        (29)
        or briefly,
        Re [ e i θ ( t ) f ( η ( t ) ) ] = ϕ ˆ ( t ) + k S n a k γ [ k ] ( t ) + k S d a k γ [ k ] ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ32_HTML.gif
        (30)
        where
        γ [ k ] ( t ) = Re [ e i θ ( t ) log ( η ( t ) z k ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ33_HTML.gif
        (31)
        for k = 1 , 2 , , m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq63_HTML.gif. In view of (8) and (28), the real constants a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq64_HTML.gif are known for k S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq65_HTML.gif and are given by
        a k = 1 2 π J k ϕ k ( t ) | η ˙ k ( t ) | d t , k S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ34_HTML.gif
        (32)
        However, for k S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq66_HTML.gif, the real constants a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq64_HTML.gif are unknown. Thus, the boundary condition (29) can be written as
        Re [ e i θ ( t ) f ( η ( t ) ) ] = ψ ˆ ( t ) + k S d a k γ [ k ] ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ35_HTML.gif
        (33)
        where the function ψ ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq67_HTML.gif is known and is given by
        ψ ˆ ( t ) = { ϕ j ( t ) + k S n a k γ j [ k ] ( t ) , t J j , j S d , φ j ( t ) + k S n a k γ j [ k ] ( t ) , t J j , j S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ36_HTML.gif
        (34)

        Obviously, the functions ψ ˆ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq68_HTML.gif are known explicitly for t J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq48_HTML.gif with j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif. However, for t J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq48_HTML.gif with j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif, it is required to integrate φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq61_HTML.gif to obtain φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq60_HTML.gif.

        The functions φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq60_HTML.gif are not necessary 2π-periodic. In order to keep dealing with periodic functions numerically, we do not compute φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq60_HTML.gif directly by integrating the functions φ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq61_HTML.gif. Instead, we integrate the functions
        ψ ˆ j ( t ) = ϕ j ( t ) | η ˙ j ( t ) | + k S n a k d d t γ j [ k ] ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Eque_HTML.gif
        According to the definitions of the constants a k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq64_HTML.gif and the functions γ [ k ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq69_HTML.gif, we have
        0 2 π ψ ˆ j ( t ) d t = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equf_HTML.gif
        which implies that the functions ψ ˆ j ( t ) = φ j ( t ) + k S n a k γ j [ k ] ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq70_HTML.gif are always 2π-periodic. By using the Fourier series for t J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq48_HTML.gif with j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif, the functions ψ ˆ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq71_HTML.gif can be written as
        ψ ˆ j ( t ) = i = 1 a i [ j ] cos i t + i = 1 b i [ j ] sin i t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ37_HTML.gif
        (35)
        Then the functions ψ ˆ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq68_HTML.gif are given for t J j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq48_HTML.gif with j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif by
        ψ ˆ j ( t ) = ψ ˜ j ( t ) + c j , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ38_HTML.gif
        (36)
        where c j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq72_HTML.gif are undetermined real constants and the functions ψ ˜ j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq73_HTML.gif are given by
        ψ ˜ j ( t ) = i = 1 a i [ j ] i sin i t i = 1 b i [ j ] i cos i t , t J j , j S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ39_HTML.gif
        (37)
        Hence, the boundary condition (33) can then be written as
        Re [ e i θ ( t ) f ( η ( t ) ) ] = γ ˆ ( t ) + h ˜ ( t ) + k S d a k γ [ k ] ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ40_HTML.gif
        (38)
        where h ˜ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq74_HTML.gif is the real piecewise constant function
        h ˜ ( t ) = { 0 , t J j , j S d , c j , t J j , j S n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ41_HTML.gif
        (39)
        and the function γ ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq75_HTML.gif is given by
        γ ˆ ( t ) = { ϕ j ( t ) + k S n a k γ j [ k ] ( t ) , t J j , j S d , ψ ˜ j ( t ) , t J j , j S n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ42_HTML.gif
        (40)
        Let c : = f ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq76_HTML.gif (unknown real constant) and g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif be the analytic function defined on G by
        g ( z ) : = f ( z ) c , z G . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ43_HTML.gif
        (41)
        Then g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif is analytic on G with g ( ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq44_HTML.gif. The function g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif is a solution of the RH problem
        Re [ A ( t ) g ( η ( t ) ) ] = γ ˜ ( t ) + h ( t ) + j S d a j γ [ j ] ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ44_HTML.gif
        (42)
        where the function A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq36_HTML.gif is given by (10) and the function h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq77_HTML.gif is defined by
        h ( t ) = h ˆ ( t ) c cos θ ( t ) , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ45_HTML.gif
        (43)

        6 The solution of the mixed boundary value problem

        Let μ ( t ) : = Im [ A ( t ) g ( η ( t ) ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq78_HTML.gif. Then the boundary values of the function g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif are given on the boundary Γ by
        A ( t ) g ( η ( t ) ) = γ ˆ ( t ) + h ( t ) + j S d a j γ [ j ] ( t ) + i μ ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ46_HTML.gif
        (44)

        where γ ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq75_HTML.gif, γ [ 1 ] , , γ [ m ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq79_HTML.gif are knowns and h, μ are unknowns. The real constants a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif are known for j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif and unknown for j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif.

        For j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif, let the functions μ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq80_HTML.gif and μ [ j ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq81_HTML.gif be the unique solutions of the integral equations
        ( I N ) μ ˆ = M γ ˆ , ( I N ) μ [ j ] = M γ [ j ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ47_HTML.gif
        (45)
        respectively, h ˆ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq82_HTML.gif and h [ j ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq83_HTML.gif be given by
        h ˆ ( t ) = [ M μ ˆ ( I N ) γ ˆ ] / 2 , h [ j ] = [ M μ [ j ] ( I N ) γ [ j ] ] / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ48_HTML.gif
        (46)
        Then it follows from Theorem 1 that
        A ( t ) g ˆ ( η ( t ) ) = γ ˆ + h ˆ + i μ ˆ + j S d a j ( γ [ j ] + h [ j ] + i μ [ j ] ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ49_HTML.gif
        (47)
        are boundary values of an analytic function g ˆ ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq84_HTML.gif. By the uniqueness of the functions h and μ in (44), it follows from (44) and (47) that
        μ ( t ) = μ ˆ + j S d a j μ [ j ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ50_HTML.gif
        (48)
        and
        h ( t ) = h ˆ + j S d a j h [ j ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ51_HTML.gif
        (49a)
        Equation (49a) with the following equation (from (9)),
        j S d a j = j S n a j , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ52_HTML.gif
        (49b)
        represents a linear system of m equations. Since from (43) the function h ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq77_HTML.gif is given by
        h ( t ) = { c , t J j , j S d , c j , t J j , j S n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equg_HTML.gif

        only the constants c, a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif for j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif and c j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq72_HTML.gif for j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif are unknowns. Thus, linear equations (49a) and (49b) represent a linear system of m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq85_HTML.gif equations in m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq85_HTML.gif unknowns a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif for j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif and c j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq72_HTML.gif for j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif.

        By obtaining the values of the constants a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif, we obtain the functions μ from (48) and h from (49a). Consequently, the boundary values of the function g are given by
        A ( t ) g ( η ( t ) ) = γ ( t ) + h ( t ) + i μ ( t ) , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ53_HTML.gif
        (50)
        where
        γ ( t ) = γ ˆ ( t ) + j S d a j γ [ j ] ( t ) , t J . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ54_HTML.gif
        (51)
        The function g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif can be computed for z G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq86_HTML.gif by the Cauchy integral formula. Then the function f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq62_HTML.gif is computed from
        f ( z ) = c + g ( z ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ55_HTML.gif
        (52)

        Finally, the solution of the mixed boundary value problem can be computed from u ( z ) = Re F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq58_HTML.gif, where F ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq45_HTML.gif is given by (7).

        7 Numerical implementations

        Since the functions A j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq87_HTML.gif and η j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq88_HTML.gif 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 1 / n k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq89_HTML.gif 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 η ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq37_HTML.gif, 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 μ ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq80_HTML.gif and μ [ j ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq81_HTML.gif for j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif, which give approximations to h ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq90_HTML.gif and h [ j ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq83_HTML.gif for j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif from (46). By solving (49a) and (49b), we get approximations to the constants c, a j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq32_HTML.gif for j S d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq49_HTML.gif and c j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq72_HTML.gif for j S n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq52_HTML.gif. These give approximations to the boundary values of the function g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif from (50). Then the values of g ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq43_HTML.gif for z G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq86_HTML.gif 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 f ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq62_HTML.gif are computed from (52). Finally, in view of (7), the approximate solution of the mixed boundary value problem can be computed from
        u ( z ) = Re F ( z ) = Re f ( z ) j = 1 m a j log | z z j | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ56_HTML.gif
        (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)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equh_HTML.gif
        where 0 t 2 π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq91_HTML.gif.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig2_HTML.jpg
        Figure 2

        The region for Example 1.

        We assume that the conditions on the boundaries Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq92_HTML.gif, Γ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq93_HTML.gif are the Neumann conditions and the conditions on the boundaries Γ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq94_HTML.gif, Γ 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq95_HTML.gif are the Dirichlet conditions. The functions ϕ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq96_HTML.gif in (6b)-(6c) are obtained based on choosing an exact solution of the form
        u ( z ) = Re 1 z i 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equi_HTML.gif
        We use the error norm
        0 2 π | u ( 1 + i + 0.5 e i t ) u n ( 1 + i + 0.5 e i t ) | 2 d t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ57_HTML.gif
        (54)
        where u ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq97_HTML.gif is the exact solution of the mixed boundary value problem and u n ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq98_HTML.gif 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 n = 64 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq99_HTML.gif (256 calculation points on the whole boundary), we obtain error norm less that 10−15. The absolute errors | u ( z ) u n ( z ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq100_HTML.gif at selected points in the entire domain are plotted in Figure 4. The graph of the approximate solution u n ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq98_HTML.gif is illustrated in Figure 5.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig3_HTML.jpg
        Figure 3

        The error norm ( 54 ) for Example 1.

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig4_HTML.jpg
        Figure 4

        The absolute error for Example 1.

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig5_HTML.jpg
        Figure 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 Γ = G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq101_HTML.gif is parametrized by
        η j ( t ) = z j + e i ν j ( α j cos t + i β j sin t ) , j = 1 , , 6 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ58_HTML.gif
        (55)
        where the values of the complex constants z j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq29_HTML.gif and the real constants α j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq102_HTML.gif, β j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq103_HTML.gif, ν j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq104_HTML.gif are as in Table 1.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig6_HTML.jpg
        Figure 6

        The region for Example 2.

        Table 1

        The values of constants α j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq102_HTML.gif , β j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq103_HTML.gif , z j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq29_HTML.gif , ν j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq104_HTML.gif and ζ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq105_HTML.gif for Example 2

        j

        α j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq106_HTML.gif

        β j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq107_HTML.gif

        z j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq108_HTML.gif

        ν j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq109_HTML.gif

        ζ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq110_HTML.gif

        1

        0.3626

        -0.1881

        0.1621 + 0.5940i

        3.3108

        0.10 + 0.50i

        2

        0.5061

        -0.6053

        −1.7059 + 0.3423i

        0.5778

        −1.60 + 0.40i

        3

        0.6051

        -0.7078

        0.3577 − 0.9846i

        4.1087

        0.30 − 0.90i

        4

        0.7928

        -0.3182

        1.0000 + 1.2668i

        2.6138

        0.95 + 1.20i

        5

        0.3923

        -0.4491

        −1.9306 − 1.0663i

        4.4057

        −1.85 − 1.00i

        6

        0.2976

        -0.6132

        −0.8330 − 2.1650i

        5.7197

        −0.80 − 2.10i

        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 Γ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq92_HTML.gif, Γ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq93_HTML.gif, Γ 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq94_HTML.gif are the Dirichlet conditions and the conditions on the boundaries Γ 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq95_HTML.gif, Γ 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq111_HTML.gif, Γ 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq112_HTML.gif are the Neumann conditions. The functions ϕ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq96_HTML.gif in (6b)-(6c) are obtained based on choosing an exact solution of the form
        u ( z ) = 1 + j = 1 6 ( j 7 / 2 ) log ( | z ζ j | 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equj_HTML.gif
        where the values of the complex constants ζ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq105_HTML.gif are as in Table 1. For the error, we use the error norm (see Figure 7)
        0 2 π | u ( 2.0 0.25 i + 1.5 e i t ) u n ( 2.0 0.25 i + 1.5 e i t ) | 2 d t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equ59_HTML.gif
        (56)
        The absolute errors | u ( z ) u n ( z ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq100_HTML.gif at selected points in the entire domain are plotted in Figure 8. The graph of the approximate solution u n ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq98_HTML.gif is shown in Figure 9.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig7_HTML.jpg
        Figure 7

        The error norm ( 56 ) for Example 2.

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig8_HTML.jpg
        Figure 8

        The absolute error for Example 2.

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig9_HTML.jpg
        Figure 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):
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Equk_HTML.gif
        We assume the Dirichlet condition on the star-shape boundary with Dirichlet data ϕ 1 ( t ) = 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq113_HTML.gif, while on the ellipse-shape boundary is the Neumann condition with Neumann data u n = ϕ 2 ( t ) = 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq114_HTML.gif. The graph of the approximate solution is illustrated in Figure 11.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig10_HTML.jpg
        Figure 10

        The region for Example 3.

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig11_HTML.jpg
        Figure 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 ( t ) = e i θ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq115_HTML.gif 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 [ 1 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq116_HTML.gif, is no longer true for the function A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq36_HTML.gif 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 Γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq117_HTML.gif. For this case, the function A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq36_HTML.gif is discontinuous on J k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq118_HTML.gif, where A ( t ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq119_HTML.gif on the part of Γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq117_HTML.gif corresponding to the Dirichlet condition and A ( t ) = i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq120_HTML.gif on the part of Γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq117_HTML.gif corresponding to the Neumann condition. Hence, the RH problem (42) will be a problem with discontinuous coefficient  A ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq36_HTML.gif. Thus, further investigations are required. This will be considered in future work.
        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig12_HTML.jpg
        Figure 12

        The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing integral equations ( 45 ) with n = 256 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq121_HTML.gif for the region in Example 1.

        http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_Fig13_HTML.jpg
        Figure 13

        The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing integral equations ( 45 ) with n = 256 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-54/MediaObjects/13661_2012_Article_296_IEq121_HTML.gif for the region in Example 2.

        Declarations

        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.

        Authors’ Affiliations

        (1)
        Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia
        (2)
        UTM Centre for Industrial and Applied Mathematics, Universiti Teknologi Malaysia
        (3)
        Department of Mathematics, Faculty of Science, King Khalid University
        (4)
        Department of Mathematics, Faculty of Science, Ibb University

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