Published in Lobachevskii Journal of Mathematics
AbstractThe article investigates the two point boundary-value problem for a singularly perturbed linear in homogeneous ordinary differential equation of the second order on a segment with a turning point and interior layer. The aim of the study is to construct a uniform asymptotic expansion of the solution of the two point problem with an arbitrary...
Published in Computational Mathematics and Mathematical Physics
AbstractAsymptotic analysis is used to study the existence, local uniqueness, and asymptotic Lyapunov stability of the solution to a one-dimensional nonlinear parabolic system of the activator–inhibitor type. A specific feature of the problem is the discontinuities of the first kind of the functions on the right-hand sides of the equations. The jum...
Published in Moscow University Physics Bulletin
AbstractIn this paper, we consider the existence, local uniqueness, and asymptotic stability of a solution of the boundary-layer type for a nonlinear one-dimensional initial-boundary problem with inhomogeneous Neumann conditions. The corresponding theorems are proved for different types of quasimonotone right-hand sides of the equations by the meth...
Published in Proceedings of the Steklov Institute of Mathematics
The present work is devoted to a time-optimal control problem for a singularly perturbed linear autonomous system with smooth geometric constraints on the control and an unbounded target set: cases˙𝑥subscript𝐴11𝑥subscript𝐴12𝑦subscript𝐵1𝑢formulae-sequence𝑥superscriptℝ𝑛formulae-sequence𝑦superscriptℝ𝑚𝑢superscriptℝ𝑟𝜀˙𝑦subscript𝐴21𝑥subscript𝐴22𝑦subscrip...
Published in Mathematical Notes
Abstract The Cauchy problem for a first-order differential equation with a small parameter multiplying the derivative in a Banach space is considered. The right-hand side of the equation contains the Fredholm operator perturbed by an additional operator term containing a small parameter. The asymptotic expansion of the solution in powers of the sma...
Published in Theoretical and Mathematical Physics
Abstract We study a difference–differential model of an optoelectronic oscillator that is a modification of the Ikeda equation with delay. We analyze the stability of the zero equilibrium state. We note that the number of roots of the characteristic equation of the linearized problem with the real part that is close to zero increases without bound ...
Published in Theoretical and Mathematical Physics
Abstract We study the problem of the existence and asymptotic stability of a stationary solution of an initial boundary value problem for the reaction–diffusion–advection equation assuming that the reaction and advection terms are comparable in size and have a jump along a smooth curve located inside the studied domain. The problem solution has a l...
Published in Computational Mathematics and Mathematical Physics
AbstractThe Cauchy problem for a singularly perturbed system of equations describing a transport process with diffusion in a multiphase medium is considered. A formal asymptotic expansion of its solution is constructed in the case when exchange between the phases proceeds much more rapidly than the transport and diffusion processes. The case when t...
Published in Mathematical Notes
Abstract Systems of nonlinear parabolic equations with small parameter multiplying the highest derivative and stochastic models associated with them are considered. It is shown that the vanishing viscosity method, which makes it possible to choose physical solutions to the Cauchy problem for systems of nonlinear conservation laws, has a natural jus...
Published in Theoretical and Mathematical Physics
Abstract We consider the construction of asymptotic solutions of linear equations related to equations of classical mechanics: the Hamilton–Jacobi equation and the transport equation. We show that these methods and also the theory of the mechanics of an infinitely narrow beam as a whole can be applied to some objects in bioenergy if the thin organi...
