On the boundary integral equation method of solving boundary value problems for the two dimensional Laplace equation

Yu. M. Sybil

doi:10.30970/ms.62.1.60-76

Yu. M. Sybil Ivan Franko National University of Lviv Lviv, Ukraine

DOI: https://doi.org/10.30970/ms.62.1.60-76

Keywords: Laplace equation, Dirichlet and Neumann boundary value problems, potentials of simple and double layer, integral equations of the first and second kind

Abstract

We consider approach based on the integral representation of solutions
in domain which consists of bounded and unbounded parts that
gives us opportunity to reduce different transmission type problems to
connected with them equivalent boundary equations of the first and the second kind.
We suppose also that solutions of some of these boundary problems are unbounded at infinity.
Interior and exterior Dirichlet and Neumann boundary value
problems for Laplace equation are restrictions of the solutions
os more general this transmission problems.
Interior Neumann and exterior Dirichlet boundary value problems we also can solve using
integral equation of the second kind that have not unique solution.
Corresponding modified equations are constructed in this case and solutions of obtained equations are unique.
We also show correctness of all obtained boundary equations of the second type
given on closed Lipschitz curve in some Hilbert spaces
without compactness of corresponding integral operators.

Author Biography

Yu. M. Sybil, Ivan Franko National University of Lviv Lviv, Ukraine

Ivan Franko National University of Lviv
Lviv, Ukraine

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