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Discrete mathematical model of hypersingular integral equation on a system of intervals
(НТУ "ХПІ", 2018) Polyanskaya, Tatyana Semenovna; Naboka, Olena Oleksiyivna
We consider a hypersingular integral equation on a system of intervals, which is reduced to a system of hypersingular integral equations on the standard interval (−1,1). The discretization of this system is carried out on the basis of the method of discrete singularities. The unique solvability of the discrete problem is proved and an estimate of the rate of convergence of the solution of this problem to the exact solution of the system of hypersingular integral equations is given.
Unifirm attractor for wave equation with non-linear damping depending explicitly on time
(НТУ "ХПІ", 2018) Naboka, Olena Oleksiyivna
The paper deals with long-time behavior of the solutions to the initial-boundary value problem for a non-autonomous non-linear wave equation. The peculiarity of the equation is the non-linear damping term depending explicitly on time. The problem is studied in the framework of the theory of processes and their attractors. The family of processes generated by the initial-boundary value problem is introduced. It is proved that this family is uniformly (with respect to the time-dependent damping coefficient) dissipative and asymptotically compact, thus possesses a uniqueuniform attractor. The attractor is a compact set in the common phase space of the processes.