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

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.

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 $m\ge 2$ 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).

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

Let n be the exterior unit normal to Γ and let $\varphi \in H$ 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 $S_{\mathbf{n}}=\mathrm{\varnothing }$ 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 $S_{\mathbf{d}}\ne \mathrm{\varnothing }$.

is a singular integral operator.

The generalized Neumann kernel for an integral equation associated with the mixed boundary value problem which will be presented in this paper is different from the generalized Neumann kernel for the integral equation considered in [1, 3]. Thus, not all of the properties of the generalized Neumann kernel which have been studied in [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]).

Obviously, the functions ${\hat{\psi }}_j(t)$ 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 ${\phi }_j(t)$.

where $\hat{\gamma }(t)$, ${\gamma }^{[1]},\dots ,{\gamma }^{[m]}$ are knowns and h, μ are unknowns. The real constants $a_j$ are known for $j\in S_{\mathbf{n}}$ and unknown for $j\in S_{\mathbf{d}}$.

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 $j\in S_{\mathbf{n}}$.

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

Since the functions $A_j$ and ${\eta }_j$ 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 $1/n^k$ 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].

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].

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 $0\le t\le 2\pi$.

where $u(z)$ is the exact solution of the mixed boundary value problem and $u_n(z)$ 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 $n=64$ (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.

where the values of the complex constants $z_j$ and the real constants ${\alpha }_j$, ${\beta }_j$, ${\nu }_j$ are as in Table 1.

The absolute errors $|u(z)-u_n(z)|$ at selected points in the entire domain are plotted in Figure 8. The graph of the approximate solution $u_n(z)$ is shown in Figure 9.

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 ${\varphi }_1(t)=4$, while on the ellipse-shape boundary is the Neumann condition with Neumann data $\frac{\partial u}{\partial \mathbf{n}}={\varphi }_2(t)=-4$. The graph of the approximate solution is illustrated in Figure 11.