On mixed boundary value problems for the Helmholtz equation

1977 ◽  
Vol 77 (1-2) ◽  
pp. 65-77 ◽  
Author(s):  
R. Kress ◽  
G. F. Roach
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Theoretic Approach ◽  
Mixed Boundary Value Problems

SynopsisExistence and uniqueness theorems are obtained for a class of mixed boundary value problems associated with the three-dimensional Helmholtz equation. In this context the boundary of the region of interest is assumed to consist of the union of a finite number of disjoint, closed, bounded Lyapunov surfaces on some of which are imposed Dirichlet conditions whilst Neumann conditions are imposed on the remainder. An integral equation method is adopted throughout. The required boundary integral equations are generated by a modified layer theoretic approach which extends the work of Brakhage and Werner [1] and Leis [2, 3].

Some interior and exterior boundary-value problems for the Helmholtz equation in a quadrant

1993 ◽  
Vol 123 (2) ◽  
pp. 275-294 ◽  
Author(s):  
E. Meister ◽  
F. Penzel ◽  
F.-O. Speck ◽  
F. S. Teixeira
Keyword(s):  
Sobolev Space ◽  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Exterior Problems ◽  
Potential Operators

SynopsisThe Dirichlet, Neumann and mixed boundary-value problems for the two-dimensional Helmholtz equation in the interior or exterior of a quadrant are considered in a Sobolev space setting. It is shown that the potential operators arising in the interior problems can be used to derive systems of boundary integral equations to the exterior problems, which can be solved explicitly.

scholarly journals Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain

2020 ◽  
Vol 27 (2) ◽  
pp. 211-231
Author(s):  
Roland Duduchava ◽  
Medea Tsaava
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Bessel Potential ◽  
Angular Domain ◽  
Mixed Boundary Value Problems ◽  
Classical Setting ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces {\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, {s>\frac{1}{p}}, {1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.

scholarly journals A unified boundary integral equation method for a class of second order elliptic boundary value problems

1984 ◽  
Vol 25 (4) ◽  
pp. 501-517 ◽  
Author(s):  
M. Rezayat ◽  
F. J. Rizzo ◽  
D. J. Shippy
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Second Order ◽  
Elliptic Type ◽  
Integral Equation Formulation

AbstractA generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.

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