| Abstract [eng] |
The doctoral dissertation deals with the hyperbolic problem with nonlocal integral boundary conditions. The research object is the stability of finite difference approximation of the hyperbolic problem and eigenspectrum analysis. Partial differential equations of the hyperbolic type with integral conditions often occur in problems related to fluid mechanics, linear thermoelasticity, vibrations, etc. We consider hyperbolic equation on a rectangular domain, with classical initial conditions and nonlocal integral boundary conditions. We investigate the eigenstructure of the explicit finite difference scheme (FDS) for the hyperbolic problem with two integral boundary conditions, formulate and prove the sufficient stability condition of such scheme. We also investigate a class of weighted FDS with one weight parameter. We use the generalized characteristic functions to investigate eigenspectrum (complex and real) of discrete problem. We obtain the structure of eigenspectrum, formulate and prove stability conditions according to boundary parameters and weights of the scheme. We also consider a class of weighted schemes with two weights. Numerically modelling characteristic functions we obtain stability regions and restrictions on FDS weights. We obtain equivalence conditions for the Sturm–Liouville problem (which can be generalized to the evolution equations) to the algebraic eigenvalue problem. |
