A Boundary Integral Equation with the Generalized Neumann Kernel for a Certain Class of Mixed Boundary Value Problem

Abstract

We present a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving two-dimensional Laplace’s equation on multiply connected regions with mixed boundary condition. Two numerical examples are presented to verify the accuracy of the proposed method.

1. Introduction

The interplay of Riemann-Hilbert boundary value problem and integral equations with the generalized Neumann kernel on bounded multiply connected regions has been investigated in [1]. By reformulating the Dirichlet problem, the Neumann problem, and the conformal mapping as Riemann-Hilbert problems, boundary integral equations with the generalized Neumann kernel have been implemented successfully in [2] to solve the Dirichlet problem and the Neumann problem and in [3–5] to compute the conformal mapping of multiply connected regions onto the classical canonical slit domains. The mixed boundary value problem also can be reformulated as a Riemann-Hilbert problem (see [6–8]). Based on this reformulation, we present in this paper a new boundary integral equation method for two-dimensional Laplace’s equation with mixed boundary condition in bounded multiply connected regions. However, in this paper, we will consider only a certain class of mixed boundary condition where the boundary condition in each boundary component is either the Dirichlet condition or the Neumann condition but not both. This class of mixed boundary value problem has been considered in [9] for Laplace’s equation and in [10, page 78] for biharmonic equation on multiply connected regions. A similar class of mixed boundary value problem has been considered in [11, page 317] for doubly connected regions.

The presented method is an extension of the method presented in [2] for Laplace's equation with the Dirichlet boundary conditions and Laplace’s equation with the Neumann boundary conditions on multiply connected regions. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. The generalized Neumann kernel which will be considered in this paper is slightly different from the generalized Neumann kernel for integral equation associated with the Dirichlet problem and the Neumann problem which has been considered in [2]. Thus, not all of the properties of the generalized Neumann kernel which have been studied [2] are valid for the generalized Neumann kernel which will be used in this paper. For example, it is still true that 𝜆=1 is not an eigenvalue of the generalized Neumann kernel which means that the presented integral equation is uniquely solvable. However, it is no longer true that all eigenvalues of the generalized Neumann kernel are real (see Figure 3 and [2, Theorem  8]).

2. Notations and Auxiliary Material

In this section we will review the definition and some properties of the generalized Neumann kernel. For further details we refer the reader to [1–3, 5].

Let 𝐺 be a bounded multiply connected region of connectivity 𝑚+1≥1 with 0∈𝐺. The boundary Γ∶=𝜕𝐺=⋃^𝑚_𝑗=0Γ_𝑗 consists of 𝑚+1 smooth closed Jordan curves Γ₀,Γ₁,…,Γ_𝑚 where Γ₀ contains the other curves Γ₁,…,Γ_𝑚 (see Figure 1). The complement 𝐺⁻∶=ℂ⧵𝐺 of 𝐺 with respect to ℂ consists of 𝑚+1 simply connected components 𝐺₀,𝐺₁,…,𝐺_𝑚. The components 𝐺₁,…,𝐺_𝑚 are bounded and the component 𝐺₀ is unbounded where ∞∈𝐺₀. We assume that the orientation of the boundary Γ is such that 𝐺 is always on the left of Γ. Thus, the curves Γ₁,…,Γ_𝑚 always have clockwise orientations and the curve Γ₀ has a counterclockwise orientation. The curve Γ_𝑗 is parametrized by a 2𝜋-periodic twice continuously differentiable complex function 𝜂_𝑗(𝑡) with nonvanishing first derivative 𝜂̇_𝑗(𝑡)=𝑑𝜂_𝑗(𝑡)𝑑𝑡≠0,𝑡∈𝐽_𝑗∶=[0,2𝜋],𝑗=0,1,…,𝑚. (2.1) The total parameter domain 𝐽 is the disjoint union of the intervals 𝐽_𝑗. We define a parametrization of the whole boundary Γ as the complex function 𝜂 defined on 𝐽 by 𝜂(𝑡)𝜂∶=⁰(𝑡),𝑡∈𝐽₀,⋮𝜂_𝑚(𝑡),𝑡∈𝐽_𝑚. (2.2)

Figure 1 

A bounded multiply connected region 𝐺 of connectivity 𝑚+1.

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 can be interpreted via 𝜙(𝑡)∶=𝜙(𝜂(𝑡)), 𝑡∈𝐽, as a real Hölder continuous 2𝜋-periodic functions 𝜙(𝑡) of the parameter 𝑡∈𝐽, that is, 𝜙(𝑡)𝜙∶=⁰(𝑡),𝑡∈𝐽₀,⋮𝜙_𝑚(𝑡),𝑡∈𝐽_𝑚, (2.3) with real Hölder continuous 2𝜋-periodic functions 𝜙_𝑗 defined on 𝐽_𝑗, and vice versa. So, here and in what follows, we will not distinguish between 𝜙(𝜂(𝑡)) and 𝜙(𝑡).

The subspace of 𝐻 which consists of all piecewise constant functions defined on Γ will be denoted by 𝑆, that is, a function ℎ∈𝑆 has the representation ℎ(𝑡)ℎ∶=⁰,𝑡∈𝐽₀,⋮ℎ_𝑚,𝑡∈𝐽_𝑚, (2.4) where ℎ₀,…,ℎ_𝑚 are real constants. For simplicity, the function ℎ will be denoted by ℎ(𝑡)=ℎ₀,…,ℎ_𝑚. (2.5)

In this paper we will assume that the function 𝐴 is the continuously differentiable complex function 𝐴(𝑡)∶=𝑒^{−i𝜃(𝑡)}𝜂(𝑡), (2.6) where 𝜃 is the real piecewise constant function 𝜃(𝑡)=𝜃₀,…,𝜃_𝑚, (2.7) with either 𝜃_𝑗=0 or 𝜃_𝑗=𝜋/2, 𝑗=0,1,…,𝑚. This function, 𝐴(𝑡), is a special case of the function 𝐴(𝑡) in [5, Equation (4)] in connection with numerical conformal mapping of multiply connected regions.

The generalized Neumann kernel formed with 𝐴 is defined by 𝑁()1𝑠,𝑡∶=𝜋𝐴(𝑠)Im𝐴(𝑡)𝜂̇(𝑡)𝜂(𝑡)(𝑠).−𝜂 (2.8) We define also a real kernel 𝑀 by 𝑀()1𝑠,𝑡∶=𝜋𝐴(𝑠)Re𝐴(𝑡)𝜂̇(𝑡)𝜂(𝑡)(𝑠).−𝜂 (2.9) The kernel 𝑁 is continuous and the kernel 𝑀 has a cotangent singularity type (see [1] for more details). Hence, the operator 𝐍𝜇(𝑠)∶=_𝐽𝑁(𝑠,𝑡)𝜇(𝑡)𝑑𝑡,𝑠∈𝐽, (2.10) is a Fredholm integral operator and the operator, 𝐌𝜇(𝑠)∶=_𝐽𝑀(𝑠,𝑡)𝜇(𝑡)𝑑𝑡,𝑠∈𝐽, (2.11) is a singular integral operator.

The solvability of boundary integral equations with the generalized Neumann kernel is determined by the index (winding number in other terminology) of the function 𝐴 (see [1]). For the function 𝐴 given by (2.6), the index 𝜅_𝑗 of 𝐴 on the curve Γ_𝑗 and the index 𝜅=∑^𝑚_𝑗=0𝜅_𝑗 of 𝐴 on the whole boundary Γ are given by 𝜅₀=1,𝜅_𝑗=0,𝑗=1,…,𝑚,𝜅=1. (2.12)

Although the function 𝐴(𝑡) in (2.6) is different from the function 𝐴(𝑡) in [2, Equation (11)], both functions have the same indices. So by using the same approach used in [2], we can prove that the properties of the generalized Neumann kernel proved in [2], except Theorem  8, Theorem  10, and Corollary  2, are still valid for the generalized Neumann kernel formed with the function 𝐴(𝑡) in (2.6) (see [5]). Numerical evidence show that [2, Theorem  8], which claims that the eigenvalues of the generalized Neumann kernel lies in [−1,1), is no longer true for the function 𝐴(𝑡) in (2.6) (see Figure 3).

Theorem 2.1 (see [5]). For a given function 𝛾∈𝐻, there exist unique functions ℎ∈𝑆 and 𝜇∈𝐻 such that 𝐴𝑓=𝛾+ℎ+i𝜇 (2.13) is a boundary value of a unique analytic function 𝑓(𝑧) in 𝐺. The function 𝜇 is the unique solution of the integral equation (𝐈−𝐍)𝜇=−𝐌𝛾 (2.14) and the function ℎ is given by ()𝛾ℎ=𝐌𝜇−𝐈−𝐍2. (2.15)

3. The Mixed Boundary Value Problem

Let 𝑆_𝐝 and 𝑆_𝐧 be two subsets of the set {0,1,…,𝑚} such that 𝑆_𝐝≠∅,𝑆_𝐧≠∅,𝑆_𝐝∪𝑆_𝐧={0,1,…,𝑚},𝑆_𝐝∩𝑆_𝐧=∅. (3.1) We will consider the following class of mixed boundary value problems. Find a real function 𝑢 in 𝐺 such that Δ𝑢=0,onG,𝑢=𝜙_𝑗,onΓ_𝑗for𝑗∈𝑆_𝐝,𝜕𝑢𝜕𝐧=𝜙_𝑗,onΓ_𝑗for𝑗∈𝑆_𝐧, (3.2a)(3.2b)(3.2c)where 𝑛 is the unit exterior normal to Γ.

The problem (3.2a), (3.2b), and (3.2c) reduces to the Dirichlet problem for 𝑆_𝐧=∅ and to the Neumann problem for 𝑆_𝐝=∅. Both problems have been considered in [2]. So we have assumed in this paper that 𝑆_𝐧≠∅ and 𝑆_𝐝≠∅. The mixed boundary value problem (3.2a), (3.2b), and (3.2c) is uniquely solvable. Its unique solution 𝑢 can be regarded as a real part of an analytic function 𝐹 in 𝐺 which is not necessairly single-valued. However, the function 𝐹 can be written as 𝐹(𝑧)=𝑓(𝑧)−^𝑚_𝑗=1𝑎_𝑗log𝑧−𝑧_𝑗, (3.3) where 𝑓 is a single-valued analytic function in 𝐺, each 𝑧_𝑗 is a fixed point in 𝐺_𝑗, 𝑗=1,2,…,𝑚; and 𝑎₁,…,𝑎_𝑚 are real constants uniquely determined by 𝜙 (see [12, page 145] and [13, page 174]). Without los of generality, we assume that Im𝑓(0)=0. The constants 𝑎₁,…,𝑎_𝑚 are chosen to ensure that (see [14, page 222] and [15, page 88]) _Γ𝑘𝑓^′𝜂𝑑𝜂=0,𝑘=1,2,…,𝑚, (3.4) that is, 𝑎_𝑗 are given by (see [2]) 𝑎_𝑗=12𝜋i_Γ𝑗𝐹^′𝜂𝑑𝜂,𝑗=1,2,…,𝑚. (3.5)

We define a real constant 𝑎₀ by 𝑎₀∶=−^𝑚_𝑗=1𝑎_𝑗. (3.6) Hence, we have 𝑎₀=−^𝑚_𝑗=1𝑎_𝑗=−^𝑚_𝑗=112𝜋i_Γ𝑗𝐹^′𝜂1𝑑𝜂=−2𝜋i_Γ𝐹^′𝜂1𝑑𝜂+2𝜋i_Γ0𝐹^′𝜂𝑑𝜂. (3.7) Since (1/2𝜋i)∫_Γ𝐹^′(𝜂)𝑑𝜂=0, we obtain 𝑎₀=12𝜋i_Γ0𝐹^′𝜂𝑑𝜂. (3.8)

The Dirichlet boundary condition is a special case of the Riemann-Hilbert boundary condition. The Neumann boundary condition also can be reduced to a Riemann-Hilbert boundary condition by using the Cauchy-Riemann equations and integrating along the boundary Γ_𝑗 for 𝑗∈𝑆_𝐧. Let 𝐓(𝜁) be the unit tangent vector and 𝐧(𝜁) be the unit external normal vector to Γ at 𝜁∈Γ. Let also 𝜈(𝜁) be the angle between the normal vector 𝐧(𝜁) and the positive real axis, that is, 𝐧(𝜁)=𝑒^i𝜈(𝜁). Then, 𝑒^{i𝜈(𝜂(𝑡))}𝜂(𝑡)𝜂̇(𝑡)=−i𝐓=−i𝜂̇(𝑡.) (3.9) Thus 𝜕𝑢𝜕𝐧=∇𝑢⋅𝐧=cos𝜈𝜕𝑢𝜕𝑥+sin𝜈𝜕𝑢𝑒𝜕𝑦=Re^i𝜈𝜕𝑢𝜕𝑥−i𝜕𝑢.𝜕𝑦 (3.10) Since 𝑢(𝑧)=Re𝐹(𝑧), then by the Cauchy-Riemann equation, we have 𝐹^′(𝑧)=𝜕𝑢(𝑧)(𝑧)𝜕𝑥−i𝜕𝑢.𝜕𝑦 (3.11) Thus (3.10) becomes Re−i𝜂̇𝐹^′=𝜂̇𝜕𝑢.𝜕𝐧 (3.12) Let the boundary value of the multivalued analytic function 𝐹 be given by 𝐹=𝜓+i𝜑. (3.13) Then the function 𝐹^′(𝑧) is a single-valued analytic function and has the boundary value 𝜂̇𝐹^′=𝜓^′+i𝜑^′. (3.14)

Let 𝜃(𝑡) be the real piecewise constant function 𝜃(𝑡)=0,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐝,𝜋2,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧. (3.15) Hence, the boundary values of the function 𝐹(𝑧) satisfy on the boundary ΓRe𝑒^{−i𝜃(𝑡)}𝐹𝜂(𝑡)=𝜙(𝑡), (3.16) where 𝜙(𝑡)=𝜙_𝑗(𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐝,𝜑_𝑗(𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧, (3.17) and for 𝑡∈𝐽_𝑗 and 𝑗∈𝑆_𝐧, the function 𝜑^′_𝑗(𝑡) is known and is given by 𝜑^′_𝑗(𝑡)=Re−i𝜂̇_𝑗(𝑡)𝐹^′𝜂_𝑗(𝑡)=𝜙_𝑗(𝑡)𝜂̇_𝑗(𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧. (3.18) The functions 𝜙_𝑗(𝑡) for 𝑗∈𝑆_𝐝∪𝑆_𝐧 are given by (3.2b) and (3.2c). The function 𝜑_𝑗(𝑡) can be then computed for 𝑡∈𝐽_𝑗 and 𝑗∈𝑆_𝐧 by integrating the function 𝜑^′_𝑗(𝑡). Then, it follows from (3.3), (3.16), and (3.17) that the the function 𝑓(𝑧) is a solution of the Riemann-Hilbert problem Re𝑒^{−i𝜃(𝑡)}𝑓𝜂(𝑡)=𝜙(𝑡)+^𝑚_𝑘=1𝑎_𝑘Re𝑒^{−i𝜃(𝑡)}log𝜂(𝑡)−𝑧_𝑘. (3.19) Since log𝜂(𝑡)−𝑧_𝑘(𝑡)𝑧=log𝜂+log1−^𝑘𝜂(𝑡), (3.20) then, in view of (3.6), the boundary condition (3.19) can be written as Re𝑒^{−i𝜃(𝑡)}𝑓𝜂(𝑡)=𝜙(𝑡)+^𝑚_𝑘=0𝑎_𝑘𝛾^[𝑘](𝑡), (3.21) where 𝛾^[0](𝑡)=−Re𝑒^{−i𝜃(𝑡)}log𝜂(𝑡),𝛾^[𝑘](𝑡)=Re𝑒^{−i𝜃(𝑡)}𝑧log1−^𝑘𝜂(𝑡), (3.22) for 𝑘=1,2,…,𝑚.

In view of (3.5), (3.8), and (3.18), the constants 𝑎_𝑘 are known for 𝑘∈𝑆_𝐧 and are given by 𝑎_𝑘=12𝜋_𝐽𝑘𝜙_𝑘(𝑡)𝜂̇_𝑘(𝑡)𝑑𝑡,𝑘∈𝑆_𝐧. (3.23) For 𝑘∈𝑆_𝐝, the real constants 𝑎_𝑘 are unknown. Thus, the boundary condition (3.19) can be written as Re𝑒^{−i𝜃(𝑡)}𝑓𝜂(𝑡)=𝜓(𝑡)+_{𝑘∈𝑆𝐝}𝑎_𝑘𝛾^[𝑘](𝑡),𝑡∈𝐽, (3.24) where the function 𝜓(𝑡) is known and is given by 𝜓(𝑡)=𝜙_𝑗(𝑡)+∑_{𝑘∈𝑆𝐧}𝑎_𝑘𝛾_𝑗^[𝑘](𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐝,𝜑_𝑗(𝑡)+∑_{𝑘∈𝑆𝐧}𝑎_𝑘𝛾_𝑗^[𝑘](𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧. (3.25) It is clear that the function 𝜓_𝑗(𝑡) is known explicitly for 𝑡∈𝐽_𝑗 with 𝑗∈𝑆_𝐝. However, for 𝑡∈𝐽_𝑗 with 𝑗∈𝑆_𝐧 it is required to integrate 𝜑^′_𝑗(𝑡) to obtain 𝜑_𝑗(𝑡).

The function 𝜑_𝑗(𝑡) is not necessary 2𝜋-periodic. Numerically, we prefer to deal with periodic functions. So we will not compute 𝜑_𝑗(𝑡) directly by integrating the function 𝜑^′_𝑗(𝑡). Instead, we will integrate the function 𝜓^′_𝑗(𝑡)=𝜙_𝑗(𝑡)𝜂̇_𝑗(𝑡)+_{𝑘∈𝑆𝐧}𝑎_𝑘𝑑𝛾𝑑𝑡^𝑗[𝑘](𝑡). (3.26) By the definitions of the constants 𝑎_𝑘 and the functions 𝛾^[𝑘], we have ₀^2𝜋𝜓^′_𝑗(𝑡)𝑑𝑡=0, (3.27) which implies that the function 𝜓_𝑗(𝑡)=𝜑_𝑗(𝑡)+∑_{𝑘∈𝑆𝐧}𝑎_𝑘𝛾_𝑗^[𝑘](𝑡) is always 2𝜋-periodic. So, for 𝑡∈𝐽_𝑗 with 𝑗∈𝑆_𝐧, we write the function 𝜓^′_𝑗(𝑡) using the Fourier series as 𝜓^′_𝑗(𝑡)=^∞_𝑖=1𝑎_𝑖^[𝑗]cos𝑖𝑡+^∞_𝑖=1𝑏_𝑖^[𝑗]sin𝑖𝑡. (3.28) Then the function 𝜓_𝑗(𝑡) is given for 𝑡∈𝐽_𝑗 with 𝑗∈𝑆_𝐧 by 𝜓_𝑗(𝑡)=𝜓_𝑗(𝑡)+𝑐_𝑗, (3.29) where 𝑐_𝑗 is undetermined real constant and the function 𝜓_𝑗(𝑡) is given by 𝜓_𝑗(𝑡)=^∞_𝑖=1𝑎_𝑖^[𝑗]𝑖sin𝑖𝑡−^∞_𝑖=1𝑏_𝑖^[𝑗]𝑖cos𝑖𝑡,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧. (3.30) Hence, the boundary condition (3.24) can be then written as Re𝑒^{−i𝜃(𝑡)}𝑓𝜂(𝑡)=𝛾(𝑡)+ℎ(𝑡)+_{𝑘∈𝑆𝐝}𝑎_𝑘𝛾^[𝑘](𝑡),𝑡∈𝐽, (3.31) where ℎ(𝑡) is the real piecewise constant function ℎ(𝑡)=0,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐝,𝑐_𝑗,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧, (3.32) and the function 𝛾(𝑡) is given by 𝛾(𝑡)=𝜙_𝑗(𝑡)+∑_{𝑘∈𝑆𝐧}𝑎_𝑘𝛾_𝑗^[𝑘](𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐝,𝜓_𝑗(𝑡),𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧. (3.33)

Let 𝑐∶=𝑓(0) (unknown real constant) and 𝑔(𝑧) be the analytic function defined on 𝐺 by 𝑔(𝑧)𝑓(𝑧)∶=−𝑐𝑧,𝑧∈𝐺. (3.34) Thus the function 𝑔(𝑧) is a solution of the Riemann-Hilbert problem Re𝐴(𝑡)𝑔𝜂(𝑡)=𝛾(𝑡)+ℎ(𝑡)+_{𝑗∈𝑆𝐝}𝑎_𝑗𝛾^[𝑗](𝑡),𝑡∈𝐽, (3.35) where the function 𝐴(𝑡) is given by (2.6) and the function ℎ(𝑡) is defined by ℎ(𝑡)=ℎ(𝑡)−𝑐cos𝜃(𝑡),𝑡∈𝐽. (3.36) Let 𝜇(𝑡)∶=Im[𝐴(𝑡)𝑔(𝜂(𝑡))], then the boundary value of the function 𝑔(𝑧) is given on the boundary Γ by 𝐴(𝑡)𝑔𝜂(𝑡)=𝛾(𝑡)+ℎ(𝑡)+_{𝑗∈𝑆𝐝}𝑎_𝑗𝛾^[𝑗](𝑡)+i𝜇(𝑡),𝑡∈𝐽, (3.37) where 𝛾, 𝛾^[𝑗] are knowns and ℎ, 𝜇 are unknowns.

4. The Solution of the Mixed Boundary Value Problem

Let 𝜇 and let 𝜇^[𝑗] for 𝑗∈𝑆_𝐝 be the unique solutions of the integral equations (𝐈−𝐍)𝜇=−𝐌𝛾,(𝐈−𝐍)𝜇^[𝑗]=−𝐌𝛾^[𝑗], (4.1) and ℎ, ℎ^[𝑗] be given by 𝐌()𝛾ℎ=𝜇−𝐈−𝐍2,ℎ^[𝑗]=𝐌𝜇^[𝑗]−(𝐈−𝐍)𝛾^[𝑗]2. (4.2) Then, it follows from Theorem 2.1 that 𝐴(𝑡)𝑔𝜂(𝑡)=𝛾+ℎ+i𝜇+_{𝑗∈𝑆𝐝}𝑎_𝑗𝛾^[𝑗]+ℎ^[𝑗]+i𝜇^[𝑗] (4.3) is the boundary value of an analytic function 𝑔(𝑧). Since the unknown functions 𝜇 and ℎ in (3.37) are uniquely determined by the known functions 𝛾 and 𝛾^[𝑗] for 𝑗∈𝑆_𝐝, it follows from (3.37) and (4.3) that 𝜇(𝑡)=𝜇(𝑡)+_{𝑗∈𝑆𝐝}𝑎_𝑗𝜇^[𝑗](𝑡),ℎ(𝑡)=ℎ(𝑡)+_{𝑗∈𝑆𝐝}𝑎_𝑗ℎ^[𝑗](𝑡). (4.4)(4.5) In view of (3.15), (3.32), and (3.36), the function ℎ(𝑡) is given by ℎ(𝑡)=−𝑐,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐝,𝑐_𝑗,𝑡∈𝐽_𝑗,𝑗∈𝑆_𝐧. (4.6) Thus, we have from (4.5), (3.23), and (3.6) the linear system ℎ(𝑡)−_{𝑗∈𝑆𝐝}𝑎_𝑗ℎ^[𝑗](𝑡)=ℎ(𝑡),_{𝑗∈𝑆𝐝}𝑎_𝑗=−_{𝑗∈𝑆𝐧}𝑎_𝑗, (4.7a)(4.7b) of 𝑚+2 equations in 𝑚+2 unknowns 𝑐, 𝑎_𝑗 for 𝑗∈𝑆_𝐝 and 𝑐_𝑗 for 𝑗∈𝑆_𝐧.

By solving the linear system (4.7a) and (4.7b), we obtain the values of the constants 𝑎_𝑗 and the function ℎ(𝑡). Then, we obtain the function 𝜇 from (4.4). Consequently, the boundary value of the function 𝑔 is given by 𝐴(𝑡)𝑔𝜂(𝑡)=𝛾(𝑡)+ℎ(𝑡)+i𝜇(𝑡),𝑡∈𝐽, (4.8) where 𝛾(𝑡)=𝛾(𝑡)+_{𝑗∈𝑆𝐝}𝑎_𝑗𝛾^[𝑗](𝑡),𝑡∈𝐽. (4.9) The function 𝑔(𝑧) can be computed for 𝑧∈𝐺 by the Cauchy integral formula. Then the function 𝑓(𝑧) is given by 𝑓(𝑧)=𝑐+𝑧𝑔(𝑧). (4.10) Finally, the solution of the mixed boundary value problem can be computed from 𝑢(𝑧)=Re𝐹(𝑧) where 𝐹(𝑧) is given by (3.3).

5. Numerical Implementations

Since the functions 𝐴_𝑗 and 𝜂_𝑗 are 2𝜋-periodic, the integrands in the integral equations (4.1) are 2𝜋-periodic. Hence, the most efficient numerical method for solving (4.1) is generally the Nyström method with the trapezoidal rule (see e.g., [16, page 321]). We will use the trapezoidal rule with 𝑛 (an even positive integer) equidistant node points on each boundary component to discretize the integrals in (4.1). If the integrands in (4.1) are 𝑘 times continuously differentiable, then the rate of convergence of the trapezoidal rule is 𝑂(1/𝑛^𝑘). For analytic integrands, the rate of convergence is better than 1/𝑛^𝑘 for any positive integer 𝑘 (see e.g., [17, page 83]). The obtained approximate solutions of the integral equations converge to the exact solutions with a similarly rapid rate of convergence (see e.g., [16, page 322]). Since the smoothness of the integrands in (4.1) depends on the smoothness of the function 𝜂(𝑡), results of high accuracy can be obtained for very smooth boundaries.

By using the Nyström method with the trapezoidal rule, the integral equations (4.1) reduce to (𝑚+1)𝑛 by (𝑚+1)𝑛 linear systems. Since the integral equations (4.1) are uniquely solvable, then for sufficiently large values of 𝑛 the obtained linear systems are also uniquely solvable [16]. The linear systems are solved using the Gauss elimination method. The computational details are similar to previous works in [2–5].

By solving the linear systems, we obtain approximations to 𝜇 and 𝜇^[𝑗] for 𝑗∈𝑆_𝐝 which give approximations to ℎ and ℎ^[𝑗] for 𝑗∈𝑆_𝐝 through (4.2). By solving (4.7a) and (4.7b) we get approximations to the constants 𝑐, 𝑎_𝑗 for 𝑗∈𝑆_𝐝 and 𝑐_𝑗 for 𝑗∈𝑆_𝐧. These approximations allow us to obtain approximations to the boundary value of the function 𝑔(𝑧) from (4.8). The values of 𝑔(𝑧) for 𝑧∈𝐺 will be calculated by the Cauchy integral formula. The approximate values of the function 𝑓(𝑧) are then computed from (4.10). Finally, in view of (3.3), the solution of the mixed boundary value problem can be computed from 𝑢(𝑧)=Re𝐹(𝑧)=Re𝑓(𝑧)−^𝑚_𝑗=1𝑎_𝑗log𝑧−𝑧_𝑗. (5.1)

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 [18].

6. Numerical Examples

In this section, the proposed method will be used to solve two mixed boundary value problems in bounded multiply connected regions with smooth boundaries.

Example 6.1. In this example we consider a bounded multiply connected region of connectivity 4 bounded by the four circles (see Figure 2(a)) Γ₀∶𝜂₀(𝑡)=1+2𝑒^i𝑡,Γ₁∶𝜂₁(𝑡)=1+0.25𝑒^−i𝑡,Γ₂∶𝜂₂(𝑡)=1+i+0.25𝑒^−i𝑡,Γ₃∶𝜂₂(𝑡)=1−i+0.25𝑒^−i𝑡, (6.1) where 0≤𝑡≤2𝜋.

(a)

(b)

Figure 2 

The regions of Example 6.1 (a) and Example 2 (b).

(a)

(b)

Figure 3 

The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing the integral equations (4.1) with 𝑛=128 for Example 6.1 (a) and with 𝑛=256 for Example 6.2 (b).

We assume that the condition on the boundaries Γ₀,Γ₁ is the Neumann condition and the condition on the boundaries Γ₂,Γ₃ is the Dirichlet condition. The functions 𝜙_𝑗 in (3.2a), (3.2b), and (3.2c) are obtained based on choosing an exact solution of the form 𝑢(𝑧)=Re(𝑧−1)³. (6.2) This example has been considered in [9, page 894] using the regularized meshless method. The domain here is shifted by one unit to the right to ensure that 0∈𝐺. To compare our method with the method presented in [9], we use the same error norm used in [9], namely, ₀^2𝜋𝑢1+0.5𝑒^i𝑠−𝑢_𝑛1+0.5𝑒^i𝑠2𝑑𝑠, (6.3) where 𝑢(𝑧) is the exact solution of the mixed boundary value problem and 𝑢_𝑛(𝑧) is the approximate solution obtained with 𝑛 node points on each boundary component. The error norm versus the total number of calculation points 4𝑛 by using the presented method is shown in Figure 4(a) where the integral in (6.3) is discretized by the trapezoidal rule. By using only 𝑛=16 (64 calculation points on the whole boundary), we obtain error norm less than 10⁻⁹. In [9], the error norm is only less than 10⁻³ when 200 boundary points are used. The absolute errors |𝑢(𝑧)−𝑢_𝑛(𝑧)| for selected points in the entire domain are plotted in Figure 5(a) (compare the results with [9, Figure  10]).

(a)

(b)

Figure 4 

The error norm (6.3) for Example 1 (a) and the error norm (6.6) for Example 6.2 (b) versus the total number of node points.

(a)  

(b)

Figure 5 

The absolute error |𝑢(𝑧)−𝑢𝑛(𝑧)| for the entire domain obtained with 𝑛=64 for Example 6.1 (a) and with 𝑛=256 for Example 6.2 (b).

Example 6.2. In this example we consider a bounded multiply connected region of connectivity 7 (see Figure 2(b)). The boundary Γ=𝜕𝐺 is parametrized by 𝜂_𝑗(𝑡)=𝑧_𝑗+𝑒^i𝜎𝑗𝛼_𝑗cos𝑡+i𝛽_𝑗sin𝑡,𝑗=0,1,…,6, (6.4) where the values of the complex constants 𝑧_𝑗 and the real constants 𝛼_𝑗, 𝛽_𝑗, 𝜎_𝑗 are as in Table 1.

Table 1 

The values of constants 𝛼_𝑗, 𝛽_𝑗, 𝑧_𝑗, 𝜎_𝑗, and 𝜁_𝑗 in (6.4).

The region in this example has been considered in [2, 19, 20] for the Dirichlet problem and the Neumann problem. In this example, we will consider a mixed boundary value problem where we assume that the condition on the boundaries Γ₀, Γ₁, Γ₂, and Γ₃ is the Dirichlet condition and the condition on the boundaries Γ₄, Γ₅, and Γ₆ is the Neumann condition. The functions 𝜙_𝑗 in (3.2a), (3.2b), and (3.2c) are obtained based on choosing an exact solution of the form 𝑢(𝑧)1=1+2Re𝑧−𝜁₀+⁶_𝑗=1𝑗−7/2log𝑧−𝜁_𝑗², (6.5) where the values of the complex constants 𝜁_𝑗 are as in Table 1. For the error, we use the error norm (see Figure 4(b)): ₀^2𝜋𝑢−0.35−0.35i+2.7𝑒^i𝑠−𝑢_𝑛−0.35−0.35i+2.7𝑒^i𝑠2𝑑𝑠. (6.6) The absolute errors |𝑢(𝑧)−𝑢_𝑛(𝑧)| for selected points in the entire domain are plotted in Figure 5(b).

7. Conclusions

A new uniquely solvable boundary integral equation with the generalized Neumann kernel has been presented in this paper for solving a certain class of mixed boundary value problem on multiply connected regions. Two numerical examples are presented to illustrate the accuracy of the presented method.

The presented method can be applied to mixed boundary value problem with both the Dirichlet condition and the Neumann condition on the same boundary component Γ_𝑘. For this case, the function 𝐴(𝑡) is discontinuous on 𝐽_𝑘, where 𝐴(𝑡)=𝜂_𝑘(𝑡) on the part of Γ_𝑘 corresponding to the Dirichlet condition and 𝐴(𝑡)=−i𝜂_𝑘(𝑡) on the part of Γ_𝑘 corresponding to the Neumann condition. Hence, the Riemann-Hilbert problem (3.35) will be a problem with discontinuous coefficient 𝐴(𝑡). The solvability of Riemann-Hilbert problems with discontinuous coefficients is different from the ones with continuous coefficients (see e.g., [6, page 449] and [13, page 271]). Furthermore, the discontinuity of the function 𝐴(𝑡) implies that we cannot apply the theory of the generalized Neumann kernel developed in [21] for simply connected regions and in [1] for multiply connected regions. Hence, further investigations are required. This will be considered in future work.

Acknowledgments

The authors would like to thank the anonymous referees for suggesting several improvements. This work was supported in part by the Deanship of Scientific Research at King Khalid University through Project no. KKU-SCI-11-022. This support is gratefully acknowledged.