Abstract
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.
Highlights
In the present paper, we continue the research concerned with the study of mixed boundary value problems in the plane started in [ ]
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 generalized Neumann kernel used in this paper is formed with A(t) = e–iθ(t) which is different from the ones used in [, ]
Summary
We continue the research concerned with the study of mixed boundary value problems in the plane started in [ ]. 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 [ ]. Nasser et al [ ] 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 Section , we illustrate the method by presenting two numerical examples with exact solutions and one example without an exact solution
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