Asymptotics of the solution to a singularly perturbed elliptic problem with a three-zone boundary layer

For a singularly perturbed elliptic boundary value problem, an asymptotic expansion of the boundary-layer solution is constructed and justified in the case when the boundary layer consists of three zones with different behavior of the solution, which is caused by the multiplicity of the root of the degenerate equation.     

Asymptotics of the solution to Robin problem

Convergence of the solution to the exterior Robin problem to the solution of the Dirichlet problem, as the impedance tends to infinity, is proved. The rate of convergence is established. A method for deriving higher order terms of the asymptotics of the solution is given.     

Asymptotics of a solution of a discontinuous singularly perturbed Cauchy problem

We construct the asymptotic expansion of a solution of the Cauchy problem for a singularly perturbed system of differential equations whose right-hand side is discontinuous on a certain surface. We consider the case where the surface of discontinuity is crossed and estimate the remainder of the constructed asymptotic expansion.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1502–1508, November, 1994.The present work was supported by the Ukrainian State Committee on Science and Technology.     

Asymptotics of a solution to an optimal boundary control problem

A problem of optimal boundary control of solutions to an elliptic-type equation with a small coefficient at the highest derivative and integral constraints on the control is considered. Asymptotic estimates for solutions to a problem that approximates the original problem are obtained.     

Asymptotics of the solution to the Neumann problem with a delta-function-like boundary function

A uniform asymptotic expansion is found for the integral ∫∫ _s ▽² udxdy, where u is the solution of the Neumann problem with a delta-function-like derivative on the boundary. A physics application of the result is discussed.     

The Cauchy-Poisson problem in an inviscid stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface response of a nonrotating inviscid stratified liquid of finite or infinite depth to the initial displacement of the free surface. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and distance from the origin of disturbance. It is shown that the asymptotic solution consists of the classical free surface gravity waves and the internal waves.     

Asymptotics of the solution of the Steklov spectral problem in a domain with a blunted peak

Mathematical Notes - We construct and justify the asymptotics of the eigenvalues and eigenfunctions of the Laplace equation with Steklov boundary conditions in a domain with an acute peak whose end...     

Asymptotics of a self-similar solution to the piston problem in the case of cylindrical symmetry

For the piston problem in gas dynamics, a high-order asymptotic expansion of the solution with respect to the viscosity and thermal conductivity is constructed and substantiated in the case of cylindrical symmetry.