The Helmholtz equation with oblique derivative and the Dirichlet condition on sides of slits

Abstract:

The boundary value problem for the 2-d Helmholtz equation outside slits is considered. The Dirichlet boundary condition is specified on one side of each slit and the skew derivative boundary condition is given on the other side. The tangent derivative is multiplied by the purely imaginary constant in the skew derivative boundary condition. The uniqueness of the solution is proved. The solvability of the problem is proved in the case when the absolute value of the imaginary constant mentioned above is less then one. The integral representation for a solution of the problem in the form of potentials is obtained in this case. The densities in potentials are found by solving of a uniquely solvable system of Fredholm integral equations of the second kind and index zero. The problem considered generalizes the mixed Dirichlet-Neumann problem.

Publication language:russian, pages:26

Research direction:

Mathematical problems and theory of numerical methods