Shuvalova Y. Solvability and Numerical Implementation of Boundary Integral Equations in Vibration Problems for Thin Elastic Plates

The object of study is the process of oscillations of thin elastic plates. Its purpose is to prove the results about unique solvability of boundary equation systems appearing at solution of dynamic problems of elastic plates in the Kirchhoff model by the potential theory methods. Numerical experiment on the basis of the theoretical results was also the objective of this study. The thesis used variational methods, methods of potential theory, functional analysis, theory of differential equations in partial derivatives, methods of the theory of functions of complex variable, method of discrete singularities. The first time the continuous mathematical model of elastic plates, based on the use of boundary integral equations, was built. Model allows finding the solution at any time in any part of the plate. The theorems about solvability of boundary equation systems for dynamic problem for elastic plate with different boundary conditions are proved in one-parameter scale of functional spaces of Sobolev type. The theorems are containing estimates of smoothness of the solution for a time variable. The method was improved for numerical calculation for the case of boundary integral equations that include space and time variables. Research results are a further step in the development of potential method for non-stationary boundary value problems and can be used for numerical solution of various systems of non-stationary boundary equations, in the educational process for training specialists in the field of numerical methods and methods of mathematical physics.