On the stability and the attraction domain of the stationary solution of a singularly perturbed parabolic equation with a multiple root of the degenerate equation

We construct and justify the asymptotics of a boundary layer solution of a boundary value problem for a singularly perturbed second-order ordinary differential equation for the case in which the degenerate (finite) equation has an identically double root. A specific feature of the asymptotics is the presence of a three-zone boundary layer. The solution of the boundary value problem is a stationary solution of the corresponding parabolic equation. We prove the asymptotic stability of this solution and find its attraction domain.     

Asymptotics of the solution to an initial boundary value problem for a singularly perturbed parabolic equation in the case of a triple root of the degenerate equation

For a singularly perturbed parabolic equation, asymptotics of the solution to an initial boundary value problem in the case of a triple root of the degenerate equation is constructed and justified. Essential distinctions from the case of a simple root are the scale of the boundary layer variables and the three-zone structure of the boundary layer.     

On the special properties of the boundary layer in singularly perturbed problems with multiple root of the degenerate equation

Using the boundary-value problem for the singularly perturbed second-order differential equation as an example, we show that the multiplicity of the root of the degenerate equation significantly affects the asymptotics of the solution, especially in the boundary layer.     

Asymptotics of the solution of an initial-boundary value problem for a singularly perturbed parabolic equation in the case of double root of the degenerate equation

For a singularly perturbed parabolic equation, we construct and justify the asymptotics of the classical solution of an initial-boundary value problem in the case of a double root of the degenerate equation. This case substantially differs from the case of a simple root in that the scales of the boundary layer variables are different.     

Asymptotics of the solution to a stationary piecewise-smooth reaction-diffusion equation with a multiple root of the degenerate equation

A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.     

Asymptotics of solutions to the time-dependent Schrödinger equation with a small Planck constant

A regularized asymptotics of the solution to the time-dependent Schrödinger equation in which the spatial derivative is multiplied by a small Planck constant is constructed. It is shown that the asymptotics of the solution contains a rapidly oscillating boundary layer function.     

Steplike contrast structure in a singularly perturbed system of equations with different powers of small parameter

A singularly perturbed boundary value problem for a system of equations with different powers of a small parameter is considered in the one-dimensional case. The asymptotic behavior and existence of a solution with an internal transition layer are analyzed. The asymptotics are substantiated using the asymptotic method of differential inequalities.     

On one method for the analysis of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation

A sequence converging to the solution of the Cauchy problem for a singularly perturbed inhomogeneous second-order linear differential equation is constructed. This sequence is also asymptotic in the sense that the deviation (in the norm of the space of continuous functions) of its nth element from the solution of the problem is proportional to the (n + 1)th power of the perturbation parameter. A similar sequence is constructed for the case of an inhomogeneous first-order linear equation, on the example of which the application of such a sequence to the justification of the asymptotics obtained by the method of boundary functions is demonstrated.     

A steplike contrast structure in a singularly perturbed system of elliptic equations

A singularly perturbed boundary value problem for a system of elliptic equations in a two-dimensional region is considered. The asymptotics and existence of a solution with an internal transition layer are studied. The asymptotics is justified by the method of differential inequalities.     

On a singularly perturbed system of two second-order equations in the case of intersecting roots of the degenerate equation

For a singularly perturbed system of two second-order differential equations (one rapid and one slow), we prove the existence of a solution and obtain its asymptotics for the case in which the degenerate equation has two intersecting roots. In addition, we show that this solution is an asymptotically stable stationary solution of the corresponding parabolic problem and find its local attraction domain.     

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 a Class of Singularly Perturbed Weak Nonlinear Boundary Value Problem with a Multiple Root of the Degenerate Equation

A singularly perturbed boundary value problem with weak nonlinearity in the case when the degenerate equation has a multiple root is studied. The asymptotic approximation of the solution is constructed by the modified boundary layer function method. Based on the comparison principle, there exist multizonal boundary layers in the neighborhood of the endpoints. The existence of a solution is proved by using the method of asymptotic differential inequalities.     

On the application of the method of matching asymptotic expansions to a singular system of ordinary differential equations with a small parameter

We consider a bisingular initial value problem for a system of ordinary differential equations with a single small parameter, the asymptotics of whose solution can be constructed in the form of power-logarithmic series on several boundary layers and an external layer. To use the method of matching asymptotic expansions, we prove theorems that permit one to make the passage between two adjacent layers and obtain a uniform estimate of the approximation to the solution by a composite asymptotic expansion.     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

The Cauchy problem for a singularly perturbed integro-differential Fredholm equation

An initial problem is considered for an ordinary singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator. The case when the reduced equation has a smooth solution is investigated, and the solution to the reduced equation with a corner point is analyzed. The asymptotics of the solution to the Cauchy problem is constructed by the method of boundary functions. The asymptotics is validated by the asymptotic method of differential inequalities developed for a new class of problems.     

Multiplicative asymptotics of solutions of the first boundary value problem on a half-axis for a parabolic equation with a small parameter

In this work we consider the first boundary value problem for a parabolic equation of second order with a small parameter on a half-axis (i.e., we consider the one-dimensional case). We take the zero initial condition. We construct the global (that is, the caustic points are taken into account) asymptotics of a solution for the boundary value problem. The asymptotic solution of this problem has a different structure depending on the sign of the coefficient (the drift coefficient) at the derivative of first order at a boundary point. The constructed asymptotic solutions are justified.     

Integral representations and boundary value problems for a second-order elliptic system with a singular point

For a second-order elliptic system with a singular point, we obtain integral representations and inversion formulas for the case in which the singular point is an interior point of the domain. In the integral representations, we clearly extract the singular part of the solutions, which permits one to study the asymptotics of the solutions as r → 0. In addition, we give a well-posed statement of a number of boundary value problems.     

Regularization of a two-dimensional singularly perturbed parabolic problem

A regularized asymptotic expansion of the solution to a singularly perturbed two-dimensional parabolic problem in domains with boundaries containing corner points is constructed. The asymptotics of solutions to such problems contain ordinary boundary-layer functions, parabolic boundary-layer functions, and their products, which describe a corner boundary layer.     

Turning point in a system of differential equations with analytic operator

We construct uniform asymptotics for a solution of a system of singularly perturbed differential equations with turning point. We consider the case where the boundary operator analytically depends on a small parameter.     

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

This paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.