scholarly journals On the unique solvability of a class of modified boundary integral equations

1984 ◽  
Vol 27 (3) ◽  
pp. 303-311 ◽  
Author(s):  
R. E. Kleinman ◽  
G. F. Roach
Keyword(s):  
Helmholtz Equation ◽  
Integral Equations ◽  
Unique Solvability ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Transmission Problem

In a recent paper the authors considered the transmission problem for the Helmholtz equation by using a reformulation of the problem in terms of a pair of coupled boundary integral equations with modified Green's functions as kernels. In this note we settle the question of the unique solvability of these modified boundary integral equations.

scholarly journals On the solution of the Helmholtz equation on regions with corners

2016 ◽  
Vol 113 (33) ◽  
pp. 9171-9176 ◽  
Author(s):  
Kirill Serkh ◽  
Vladimir Rokhlin
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Potential Theory ◽  
Bessel Functions ◽  
Boundary Integral Equations ◽  
Numerical Algorithms ◽  
Boundary Integral ◽  
Numerical Examples ◽  
Polygonal Domains

In this paper we solve several boundary value problems for the Helmholtz equation on polygonal domains. We observe that when the problems are formulated as the boundary integral equations of potential theory, the solutions are representable by series of appropriately chosen Bessel functions. In addition to being analytically perspicuous, the resulting expressions lend themselves to the construction of accurate and efficient numerical algorithms. The results are illustrated by a number of numerical examples.

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