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Open Access Research

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

Samer AA Al-Hatemi1, Ali HM Murid12* and Mohamed MS Nasser34

Author Affiliations

1 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru, Johor, 81310 UTM, Malaysia

2 UTM Centre for Industrial and Applied Mathematics, Universiti Teknologi Malaysia, Johor Bahru, Johor, 81310 UTM, Malaysia

3 Department of Mathematics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

4 Department of Mathematics, Faculty of Science, Ibb University, P.O. Box 70270, Ibb, Yemen

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Boundary Value Problems 2013, 2013:54  doi:10.1186/1687-2770-2013-54

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