The jump problem for the laplace equation

AbstractThe boundary value problem for the Laplace equation outside several cuts in a plane is studied. The jump of the solution of the Laplace equation and the jump of its normal derivative are specified of the cuts. The problem is studied under different conditions at infinity, which lead to different uniqueness and existence theorems. The solution of this problem is constructed in the explicit form by means of single-layer and angular potentials. The singularities at the ends of the cuts are investigated

The impedance problem for the propagative helmholtz equation in interior multiply connected domain

AbstractThe impedance problem for the propagative Helmholtz equation in the interior multiply connected domain is studied in two and three dimensions by a special modification of a boundary integral equation method. Additional boundaries are introduced inside interior parts of the boundary of the domain. The solution of the problem is obtained in the form of a single layer potential on the whole boundary. The density in the potential satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers

Mixed boundary value problems of diffraction by a half-plane with an obstacle perpendicular to the boundary

The paper is devoted to the analysis of wave diffraction problems modelled by classes of mixed boundary conditions and the Helmholtz equation, within a half-plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener-Hopf plus/minus Hankel operators and Wiener-Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well-posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener-Hopf equations

Method of interior boundaries in a mixed problem of acoustic scattering

The mixed problem for the Helmholtz equation in the exterior of several bodies (obstacles) is studied in 2 and 3 dimensions. The Dirichlet boundary condition is given on some obstacles and the impedance boundary condition is specified on the rest. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called ‘Method of interior boundaries’, because additional boundaries are introduced inside scattering bodies, where impedance boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact our method holds for any positive wave numbers. The Neumann, Dirichlet, impedance problems and mixed Dirichlet–Neumann problem are particular cases of our problem

The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges endpoints of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution

The Neumann problem for the 2-D Helmholtz equation in a domain, bounded by closed and open curves

The Neumann problem for the dissipative Helmholtz equation in a connected plane region bounded by closed and open curves is studied. The existence of classical solution is proved by potential theory. The problem is reduced to the Fredholm equation of the second kind, which is uniquely solvable. Our approach holds for both internal and external domains

Wave diffraction by wedges having arbitrary aperture angle

The problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background. In fact, we may recognize significant research on this topic for more than one century. Despite this fact, up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate Sobolev space setting. In the present paper, we are considering the corresponding boundary value problems for the Helmholtz equation, with complex wave number, admitting combinations of Dirichlet and Neumann boundary conditions. The main ideas are based on a convenient combination of potential representation formulas associated with (weighted) Mellin pseudo-differential operators in appropriate Sobolev spaces, and a detailed Fredholm analysis. Thus, we prove that the problems have unique solutions (with continuous dependence on the data), which are represented by the single and double layer potentials, where the densities are solutions of derived pseudo-differential equations on the half-line