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 RiemannHilbert (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 kernel1 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 twodimensional 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
Figure 1. An unbounded multiply connected regionGof connectivitym.
The curves
The total parameter domain J is the disjoint union of the intervals
Let H be the space of all real Hölder continuous functions on the boundary Γ. In view of
the smoothness of η, a function
with real Hölder continuous 2πperiodic functions
The subspace of H which consists of all piecewise constant functions defined on Γ is denoted by S, i.e., a function
where
3 The mixed boundary value problem
Let
Let n be the exterior unit normal to Γ and let
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
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 singlevalued. The function F can be written as
where f is a singlevalued analytic function in G,
i.e.,
The constants
4 Integral equation
In this paper we assume that the function A is a continuously differentiable complexvalued function given by
where θ is the real piecewise constant function
with either
We also define a real kernel M by
The kernel N is continuous and the kernel M has a cotangent singularity type (see [2] for more details). Hence, the operator
is a Fredholm integral operator and the operator
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
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
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
Thus, we have from [5] the following theorem (see also [2,10]).
Theorem 1For a given function
are boundary values of a unique analytic function
and the functionhis given by
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 multivalued analytic function F be given by
Although, the function
For the Dirichlet conditions, i.e.,
The Neumann conditions can also be reduced to an RH problem by using the CauchyRiemann
equations and integrating along the boundaries
Thus,
Since
Thus, the function
If we define the real piecewise constant function
the boundary values of the function
where
is known and
The functions
or briefly,
where
for
However, for
where the function
Obviously, the functions
The functions
According to the definitions of the constants
which implies that the functions
Then the functions
where
Hence, the boundary condition (33) can then be written as
where
and the function
Let
Then
where the function
6 The solution of the mixed boundary value problem
Let
where
For
respectively,
Then it follows from Theorem 1 that
are boundary values of an analytic function
and
Equation (49a) with the following equation (from (9)),
represents a linear system of m equations. Since from (43) the function
only the constants c,
By obtaining the values of the constants
where
The function
Finally, the solution of the mixed boundary value problem can be computed from
7 Numerical implementations
Since the functions
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
In this paper, we have considered regions with smooth boundaries. For some ideas on how to solve numerically boundary integral equations with the generalized Neumann kernel on regions with piecewise smooth boundaries, see [14].
8 Numerical examples
In this section, the proposed method is used to solve three mixed boundary value problems in unbounded multiply connected regions with smooth boundaries.
Example 1 In this example, we consider an unbounded multiply connected region of connectivity 4 bounded by the four circles (see Figure 2)
where
Figure 2. The region for Example 1.
We assume that the conditions on the boundaries
We use the error norm
where
Figure 3. The error norm (54) for Example 1.
Figure 4. The absolute error for Example 1.
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
where the values of the complex constants
Figure 6. The region for Example 2.
Table 1. The values of constants
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
where the values of the complex constants
The absolute errors
Figure 7. The error norm (56) for Example 2.
Figure 8. The absolute error for Example 2.
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):
We assume the Dirichlet condition on the starshape boundary with Dirichlet data
Figure 10. The region for Example 3.
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
Figure 12. The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing
integral equations (45) with
Figure 13. The eigenvalues of the coefficient matrix of the linear systems obtained by discretizing
integral equations (45) with
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editor and referees for their helpful comments and suggestions which improved the presentation of the paper. The authors acknowledge the financial support for this research by the Malaysian Ministry of Higher Education (MOHE) through UTM GUP Vote Q.J130000.7126.01H75.
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