Front motion in the parabolic reaction-diffusion problem

A singularly perturbed initial-boundary value problem is considered for a parabolic equation known in applications as the reaction-diffusion equation. An asymptotic expansion of solutions with a moving front is constructed, and an existence theorem for such solutions is proved. The asymptotic expansion is substantiated using the asymptotic method of differential inequalities, which is extended to the class of problems under study. The method is based on well-known comparison theorems and is a development of the idea of using formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

Stationary internal layers in a reaction-advection-diffusion integro-differential system

A class of singularly perturbed nonlinear integro-differential problems with solutions involving internal transition layers (contrast structures) is considered. An asymptotic expansion of these solutions with respect to a small parameter is constructed, and the stability of stationary solutions to the associated integro-parabolic problems is investigated. The asymptotics are substantiated using the asymptotic method of differential inequalities, which is extended to the new class of problems. The method is based on well-known theorems about differential inequalities and on the idea of using formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.     

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 a system of singularly perturbed equations in the case of a multiple root of the degenerate equation

We construct and justify the asymptotics of the solution of a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations that contain distinct powers of a small parameter multiplying second-order derivatives for the case of a multiple root of the degenerate equation. The root multiplicity results in changes in the structure of the asymptotics of the boundary layer solution as compared with the case of a simple root, in particular, in changes in the scale of the boundary layer variables.     

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.     

Equivalence of Two Sets of Transition Points Corresponding to Solutions with Interior Transition Layers

We establish the equivalence of two sets of transition points corresponding to solutions of singularly perturbed boundary-value problems with interior boundary layers. The first set appears in the formalism for constructing the asymptotics of the solution of a boundary-value problem and the second, in the direct scheme formalism for constructing the asymptotics of the solution of a variational problem.     

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.     

Self-adjoint Extensions for the Neumann Laplacian and Applications

A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so-called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self-adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains, The error estimates for proposed approximations of shape functionals are provided.     

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.     

A singular approach to a class of impulsive differential equation

In this paper, a singular approach to study the solutions of an impulsive differential equation from a qualitative and quantitative point of view is proposed. In the approach, a suitable singular perturbation term is introduced and a singularly perturbed system with infinite initial values is defined, in which, the reduced problem of the singularly perturbed system is exactly the impulsive differential equation under consideration. Then the boundary layer function method is applied to construct the uniformly valid asymptotic solutions to the singularly perturbed system. Based on the continuous asymptotic solution, the discontinuous solutions of the impulsive differential equation are described and approximated. An example, namely, a classical Lotka-Volterra prey-predator model with one pulse is carried out to illustrate the main results.     

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.     

Sensitivity analysis of hyperbolic optimal control problems

The aim of this paper is to perform sensitivity analysis of optimal control problems defined for the wave equation. The small parameter describes the size of an imperfection in the form of a small hole or cavity in the geometrical domain of integration. The initial state equation in the singularly perturbed domain is replaced by the equation in a smooth domain. The imperfection is replaced by its approximation defined by a suitable Steklov??s type differential operator. For approximate optimal control problems the well-posedness is shown. One term asymptotics of optimal control are derived and justified for the approximate model. The key role in the arguments is played by the so called ??hidden regularity?? of boundary traces generated by hyperbolic solutions.     

Inner turning point in the theory of singular Perturbations

We construct the uniform asymptotics of a solution of a singularly perturbed differential equation of Liouville type with an interior turning point.     

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.     

Moving fronts in integro-parabolic reaction-advection-diffusion equations

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.     

Singularly perturbed problems with double singularity

A class of singularly perturbed initial and boundary value problems for systems of linear differential equations with singularities of various types is studied. The asymptotics of the solutions of these problems is constructed; in contrast to known results, it involves boundary layers of new types that are dependent not only on the spectrum of the limit operator. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 494–501, October, 1997. Translated by N. K. Kulman     

System of differential equations with a strong turning point

The uniform asymptotics of a solution of a system of singularly perturbed differential equations with strong turning point is constructed. We study the case where the boundary operator is analytic with respect to a small parameter.