Boundary Integral Equations of no Stationary Boundary Value Problems for the Klein-Gordon Equation

The non-stationary boundary value problems for Klein-Gordon equation with Dirichlet or Neumann conditions on the boundary of the domain of definition are considered; a uniqueness of boundary value problems is proved. Based on the generalized functions method, boundary integral equations method is developed to solve the posed problems in strengths of shock waves. Dynamic analogs of Green’s formulas for solutions in the space of generalized functions are obtained and their regular integral representations are constructed in 2D and 3D over space cases. The singular boundary integral equations are obtained which resolve these tasks.

Solution of inverse non-stationary boundary value problems of diffraction of plane pressure wave on convex surfaces based on analytical solution

Research in the field of unsteady interaction of shock waves propagating in continuous media with various deformable barriers are of considerable scientific interest, since so far there are only a few scientific works dealing with solving problems of this class only for the simplest special cases. In this work, on the basis of analytical solution, we study the inverse non-stationary boundary-value problem of diffraction of plain pressure wave on convex surface in form of parabolic cylinder immersed in liquid and exposed to plane acoustic pressure wave. The purpose of the work is to construct approximate models for the interaction of an acoustic wave in an ideal fluid with an undeformable obstacle, which may allow obtaining fundamental solutions in a closed form, formulating initial-boundary value problems of the motion of elastic shells taking into account the influence of external environment in form of integral relationships based on the constructed fundamental solutions, and developing methods for their solutions. The inverse boundary problem for determining the pressure jump (amplitude pressure) was also solved. In the inverse problem, the amplitude pressure is determined from the measured pressure in reflected and incident waves on the surface of the body using the least squares method. The experimental technique described in this work can be used to study diffraction by complex obstacles. Such measurements can be beneficial, for example, for monitoring the results of numerical simulations.

Investigation of convective structures and phase transition induced by non-stationary boundary conditions in a horizontal layer of water

The evolution of convective structures and the phase transition induced by non-stationary boundary conditions in a horizontal water layer bounded by flat heat-exchange surfaces were studied by shear interferometry and numerical simulation methods. Numerical modeling of the temperature field as a field of isotherms in the mode of monotonous cooling of horizontal walls was performed. The problem of fragmentary reconstruction of hilbertograms and shear interferograms images from a numerical model of the isotherm field was solved. The hydrodynamics of convective currents, the coevolution of temperature fields, interference and Hilbert structures have been modeled and studied taking into account the inversion of water density in the vicinity of the isotherm (+4°C), under conditions of phase transition and growth of the ice layer on the lower heat transfer plane. The simulation was performed using a proprietary software package. The relevance of this kind of research is due to the special importance of convection in geodynamics, physics of the atmosphere and the ocean, in hydrodynamic and thermophysical processes associated with the formation and growth of crystals.