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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-Hatemi
^{1}, - Ali HM Murid
^{1, 2}Email author and - Mohamed MS Nasser
^{3, 4}

**2013**:54

https://doi.org/10.1186/1687-2770-2013-54

© Al-Hatemi et al.; licensee Springer 2013

**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 [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].

*G*of connectivity $m\ge 2$ with boundary $\mathrm{\Gamma}=\partial G={\bigcup}_{j=1}^{m}{\mathrm{\Gamma}}_{j}$ consisting of

*m*clockwise oriented smooth closed Jordan curves ${\mathrm{\Gamma}}_{j}$, $j=1,2,\dots ,m$. The complement ${G}^{-}:=\overline{\mathbb{C}}\setminus \overline{G}$ consists of

*m*bounded simply connected components ${G}_{j}$ interior to ${\mathrm{\Gamma}}_{j}$, $j=1,2,\dots ,m$. We assume $\mathrm{\infty}\in G$, $0\in G$ (see Figure 1).

*π*-periodic twice continuously differentiable complex-valued functions ${\eta}_{j}(t)$ with non-vanishing first derivatives,

*i.e.*,

*J*is the disjoint union of the intervals ${J}_{j}$, $j=1,2,\dots ,m$. We define a parametrization of the whole boundary Γ as the complex-valued function

*η*defined on

*J*by

*H*be the space of all real Hölder continuous functions on the boundary Γ. In view of the smoothness of

*η*, a function $\varphi \in H$ can be interpreted via $\stackrel{\u02c6}{\varphi}(t):=\varphi (\eta (t))$, $t\in J$, as a real Hölder continuous 2

*π*-periodic function $\stackrel{\u02c6}{\varphi}(t)$ of the parameter $t\in J$,

*i.e.*,

with real Hölder continuous 2*π*-periodic functions ${\stackrel{\u02c6}{\varphi}}_{j}$ defined on ${J}_{j}$. So, here and in what follows, we do not distinguish between functions of the form $\psi (\eta (t))$ and $\psi (t)$.

*H*which consists of all piecewise constant functions defined on Γ is denoted by

*S*,

*i.e.*, a function $h\in S$ has the representation

*h*is denoted by

## 3 The mixed boundary value problem

**n**be the exterior unit normal to Γ and let $\varphi \in H$ be a given function. We consider the mixed boundary value problem

for a real function *u* in *G*. We call (6b) and (6c) Dirichlet conditions and Neumann conditions, respectively.

Problem (6a)-(6c) reduces to the Dirichlet problem for ${S}_{\mathbf{n}}=\mathrm{\varnothing}$ and to the Neumann problem for ${S}_{\mathbf{d}}=\mathrm{\varnothing}$. Both problems have been considered in [3]. So, we assume in this paper that ${S}_{\mathbf{n}}\ne \mathrm{\varnothing}$ and ${S}_{\mathbf{d}}\ne \mathrm{\varnothing}$.

*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*is a single-valued analytic function in

*G*, ${z}_{j}$ are fixed points in ${G}_{j}$, $j=1,2,\dots ,m$; and ${a}_{1},\dots ,{a}_{m}$ are real constants uniquely determined by

*ϕ*(see [12]). Without lost of generality, we assume that $Imf(\mathrm{\infty})=0$. The constants ${a}_{1},\dots ,{a}_{m}$ are chosen to ensure that (see [[12], p.149] and [3])

*i.e.*, ${a}_{j}$ are given by (see [3])

## 4 Integral equation

*A*is a continuously differentiable complex-valued function given by

*θ*is the real piecewise constant function

*A*and $\eta (t)$ is defined by

*M*by

*N*is continuous and the kernel

*M*has a cotangent singularity type (see [2] for more details). Hence, the operator

is a singular integral operator.

*index*(the change of the argument of

*A*on the curves ${\mathrm{\Gamma}}_{j}$ divided by 2

*π*) of the function

*A*(see [2]). For the function

*A*given by (10), the indices ${\kappa}_{j}$ of

*A*on the curves ${\mathrm{\Gamma}}_{j}$ and the index $\kappa ={\sum}_{j=1}^{m}{\kappa}_{j}$ of

*A*on the whole boundary curve Γ are given by

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 $\lambda =1$ 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 $A(t)$ in (10) above (see [5]).

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

**Theorem 1**

*For a given function*$\gamma \in H$,

*there exist unique functions*$h\in S$

*and*$\mu \in H$

*such that*

*are boundary values of a unique analytic function*$g(z)$

*in*

*G*

*with*$g(\mathrm{\infty})=0$.

*The function*

*μ*

*is the unique solution of the integral equation*

*and the function*

*h*

*is given by*

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

*F*be given by

*G*. The boundary values of the function ${F}^{\prime}(z)$ are given by

*i.e.*, $t\in {J}_{j}$ and $j\in {S}_{\mathbf{d}}$, the functions ${\psi}_{j}$ are equal to the known functions ${\varphi}_{j}(t)$ (see (6b)). Thus, the function $F(z)$ satisfies the RH problem

*i.e.*, $\mathbf{n}(\zeta )={e}^{\mathrm{i}\nu (\zeta )}$. Then

Obviously, the functions ${\stackrel{\u02c6}{\psi}}_{j}(t)$ are known explicitly for $t\in {J}_{j}$ with $j\in {S}_{\mathbf{d}}$. However, for $t\in {J}_{j}$ with $j\in {S}_{\mathbf{n}}$, it is required to integrate ${\phi}_{j}^{\prime}(t)$ to obtain ${\phi}_{j}(t)$.

*π*-periodic. In order to keep dealing with periodic functions numerically, we do not compute ${\phi}_{j}(t)$ directly by integrating the functions ${\phi}_{j}^{\prime}(t)$. Instead, we integrate the functions

*π*-periodic. By using the Fourier series for $t\in {J}_{j}$ with $j\in {S}_{\mathbf{n}}$, the functions ${\stackrel{\u02c6}{\psi}}_{j}^{\prime}(t)$ can be written as

*G*by

*G*with $g(\mathrm{\infty})=0$. The function $g(z)$ is a solution of the RH problem

## 6 The solution of the mixed boundary value problem

where $\stackrel{\u02c6}{\gamma}(t)$, ${\gamma}^{[1]},\dots ,{\gamma}^{[m]}$ are knowns and *h*, *μ* are unknowns. The real constants ${a}_{j}$ are known for $j\in {S}_{\mathbf{n}}$ and unknown for $j\in {S}_{\mathbf{d}}$.

*h*and

*μ*in (44), it follows from (44) and (47) that

*m*equations. Since from (43) the function $h(t)$ is given by

only the constants *c*, ${a}_{j}$ for $j\in {S}_{\mathbf{d}}$ and ${c}_{j}$ for $j\in {S}_{\mathbf{n}}$ are unknowns. Thus, linear equations (49a) and (49b) represent a linear system of $m+1$ equations in $m+1$ unknowns ${a}_{j}$ for $j\in {S}_{\mathbf{d}}$ and ${c}_{j}$ for $j\in {S}_{\mathbf{n}}$.

*μ*from (48) and

*h*from (49a). Consequently, the boundary values of the function

*g*are given by

Finally, the solution of the mixed boundary value problem can be computed from $u(z)=ReF(z)$, where $F(z)$ is given by (7).

## 7 Numerical implementations

Since the functions ${A}_{j}$ and ${\eta}_{j}$ are 2*π*-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 $1/{n}^{k}$ 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 $\eta (t)$, 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].

*c*, ${a}_{j}$ for $j\in {S}_{\mathbf{d}}$ and ${c}_{j}$ for $j\in {S}_{\mathbf{n}}$. These give approximations to the boundary values of the function $g(z)$ from (50). Then the values of $g(z)$ for $z\in G$ 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 $f(z)$ are computed from (52). Finally, in view of (7), the approximate solution of the mixed boundary value problem can be computed from

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

*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 $n=64$ (256 calculation points on the whole boundary), we obtain error norm less that 10

^{−15}. The absolute errors $|u(z)-{u}_{n}(z)|$ at selected points in the entire domain are plotted in Figure 4. The graph of the approximate solution ${u}_{n}(z)$ is illustrated in Figure 5.

**Example 2**In this example, we consider an unbounded multiply connected region of connectivity 6 (see Figure 6). The boundary $\mathrm{\Gamma}=\partial G$ is parametrized by

**The values of constants**
${\mathit{\alpha}}_{\mathit{j}}$
**,**
${\mathit{\beta}}_{\mathit{j}}$
**,**
${\mathit{z}}_{\mathit{j}}$
**,**
${\mathit{\nu}}_{\mathit{j}}$
**and**
${\mathit{\zeta}}_{\mathit{j}}$
**for Example 2**

j | ${\mathit{\alpha}}_{\mathit{j}}$ | ${\mathit{\beta}}_{\mathit{j}}$ | ${\mathit{z}}_{\mathit{j}}$ | ${\mathit{\nu}}_{\mathit{j}}$ | ${\mathit{\zeta}}_{\mathit{j}}$ |
---|---|---|---|---|---|

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 |

**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):

## 9 Conclusion

## 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

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