Convergence Research of Numerical Methods for Solving Problems With Mixed Type Operator in an Unlimited Area

Methods for solving problems of elliptic equations, based on the third Green formula, was analyzed. New methods for solving a problem with a mixed-type operator in an unbounded domain are proposed. On the basis of the proposed methods, programs for solving problems with a mixed type operator have been created. The results of computational experiments, showing the correctness of the application of methods, are presented.

Stochastic Stability Criteria for Neutral Distributed Parameter Systems with Markovian Jump

This paper deals with the problem of stochastic stability for a class of neutral distributed parameter systems with Markovian jump. In this model, we only need to know the absolute maximum of the state transition probability on the principal diagonal line; other transition rates can be completely unknown. Based on calculating the weak infinitesimal generator and combining Poincare inequality and Green formula, a stochastic stability criterion is given in terms of a set of linear matrix inequalities (LMIs) by the Schur complement lemma. Because of the existence of the neutral term, we need to construct Lyapunov functionals showing more complexity to handle the cross terms involving the Laplace operator. Finally, a numerical example is provided to support the validity of the mathematical results.

Finite-Time Robust Stability of Uncertain Genetic Regulatory Networks with Time-Varying Delays and Reaction-Diffusion Terms

This study seeks to address the finite-time robust stability of delayed genetic regulatory networks (GRNs) with uncertain parameters and reaction-diffusion terms. We employ an appropriate Lyapunov-Krasovskii functional to derive some less conservative stability criteria for GRNs under Dirichlet boundary conditions, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. The time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. In addition, we define some new variables to deal with uncertain parameters. Moreover, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed to handle integral terms. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the obtained stability criteria.

Scalar problems in junctions of rods and a plate

In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these “parasitic” eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.

On Some Problems Generated by a Sesquilinear Form

Based on the generalized Green formula for a sesquilinear nonsymmetric form for the Laplace operator, we consider spectral nonself-adjoint problems. Some of them are similar to classical problems while the other arise in problems of hydrodynamics, diffraction, and problems with surface dissipation of energy. Properties of solutions of such problems are considered. Also we study initial-boundary value problems generating considered spectral problems and prove theorems on correct solvability of such problems on any interval of time.

Asymptotic conditions at infinity for the time-periodic Stokes problem in a system of pipes

A time-periodic Stokes problem set in a domain with cylindrical outlets to infinity is studied. We derive the generalized Green formula. It enables us to impose so-called asymptotic conditions at infinity which allow to determine a unique time-periodic solution. This method was proposed and applied for a steady Stokes problem by Nazarov and co-authors.