Vishik-Lyusternik Method Research Articles

АСИМПТОТИКА РЕШЕНИЯ ПЕРВОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ СИНГУЛЯРНО ВОЗМУЩЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ В ЧАСТНЫХ ПРОИЗВОДНЫХ ВТОРОГО ПОРЯДКА ПАРАБОЛИЧЕСКОГО ТИПА

The article constructs a complete uniform asymptotic expansion in a small parameter of the solution of the first boundary value problem. The first boundary value problem is posed for a singularly perturbed linear inhomogeneous second-order partial differential equation with two independent variables of parabolic type. The problem is investigated on a rectangle. The peculiarities of the problem are the presence of a small parameter in front of the heat conduction operator, the existence of corner boundary layers at the lower corners of the rectangle. It is required to construct a uniform asymptotic expansion of the solution of the first boundary value problem on a rectangle, with any degree of accuracy, as the small parameter tends to zero. The asymptotic expansion of the solution in terms of a small parameter is constructed by the Vishik–Lyusternik method. When solving the problem, we use: methods of integration of ordinary differential equations, the classical method of a small parameter, the Vishik–Lyusternik boundary function method, and the maximum principle. As usual, the problem is solved in two stages: in the first stage, a formal expansion of the solution of the first boundary value problem is constructed; and in the second stage, the remainder of the resulting expansion is estimated and this proves that the resulting expansion is indeed asymptotic over the entire rectangle. In the first stage, a formal asymptotic solution is sought in the form of a sum of six functions (solutions): an external solution defined on the entire rectangle; boundary layer solution in a small neighborhood of the lower side of the rectangle; two lateral boundary layer solutions in a small neighborhood of the lateral sides of the rectangle and two corner boundary layer solutions in the neighborhood of the lower vertices of the rectangle. All these boundary layer solutions exponentially decrease outside the boundary layers.

DEGENERATE BOUNDARY-VALUE PROBLEMS WITH A PERTURBING MATRIX FOR A DERIVATIVE

In the paper, there is considered degenerated Noether boundary value problem with a perturbing matrix for a derivative, in which the boundary condition is given by a linear vector functional. We have proposed an algorithm to consrtuct a set of linearly independent solutions of boundary value problems with a small parameter in the general case, when the number of boundary conditions given by a linear vector functional does not match with the number of unknowns in a degenerate differential system. There is used the technique of pseudoinverse Moore-Penrose matrices. Applying the Vishik-Lyusternik method, the solution of the boundary value problem is obtained as part of the Laurent series in powers of small parameter. We obtain conditions for the bifurcation of solutions of linear degenerated Noether boundary-value problems with a small parameter under the assumption that the unperturbed degenerated differential system can be reduced to central canonical form.

Асимптотическое решение задачи Коши для уравнения первого порядка с возмущенным фредгольмовым оператором

We consider the Cauchy problem for a first-order differentialequation in a Banach space. The equation contains a small parameter in the highest derivative and a Fredholm operator perturbed by an operator addition on the right-hand side. Systems with small parameter in the highest derivative describe the motion of a viscous flow, the behavior of thin and flexible plates and shells, the process of a supersonic viscous gas flow around a blunt body, etc. The presence of a boundary layer phenomenon is revealed; in this case, even a small additive has a strong influence on the behavior of the solution. Asymptotic expansion of the solution in powers of small parameter is constructed by means of the Vasil’yeva- Vishik-Lyusternik method. Asymptotic property of the expansion is proved. To construct the regular part of the expansion, the equation decomposition method is used. It is consisted in a step-by-step transition to similar problems of decreasing dimensions.

Weakly perturbed boundary-value problems for systems of integro-differential equations with impulsive action

Abstract The weakly perturbed BVP's for impulsive integro-differential systems are considered. Under the assumption that the generating problem (for ε = 0) does not have solutions on the space W12[a,b] for some inhomogeneity and using the Vishik-Lyusternik method, we establish conditions for the existence of solutions of these problems on the space D2([a,b]{τi}I) in the form of a Laurent series in powers of small parameter ε with finitely many terms with negative powers of ε, and we suggest an algorithm of construction of these solutions.

Application of perturbation theory to the solvability analysis of differential algebraic equations

Linear systems of ordinary differential equations with an identically singular or rectangular matrix multiplying the derivative of the desired vector function are considered. The structure of general solutions is discussed. A special case of perturbed systems is studied by applying the Vishik-Lyusternik method.

The Vishik–Lyusternik Method and Two Problems in Magnetohydrodynamics for Plasma in a Tokamak

The Vishik–Lyusternik method for constructing asymptotics of solutions to singularly perturbed elliptic equations is applied to two problems in magnetohydrodynamics; one of the problems is related to the inverse problem for plasma equilibrium in a tokamak and the other concerns the study of instability of the plasma discharge with the resistivity of the tokamak casing taken into account. Bibliography: 18 titles. Illustrations: 3 figures.

Steady Streaming due to a Sphere in Incompressible Fluid

In this paper steady streaming due to a sphere has been studied. Instead of using matched asymptotic method we are applying Vishik-Lyusternik method. In this paper we consider the situation when the sphere is at rest and fluid oscillating at infinity. Results have been obtained for Re=O(1) . Effect of Stokes drift is also calculated for the fluid particles.

Steady Streaming Due to Transverse Vibration and Torsional Oscillation of a Cylinder

In this paper steady streaming due to small-amplitude transverse vibrations and torsional oscillation of a cylinder in an infinite viscous incompressible fluid is studied. The Vishik–Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier–Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of order of 1. The effect of the Stokes drift of fluid particles is also considered.

Steady Flow Produced by the Vibration of a Sphere in an Infinite Viscous Incompressible Fluid

In this paper steady streaming due to a vibrating sphere in an incompressible fluid is considered. The Vishik–Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier–Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of O (1). The effect of the Stokes drift of fluid particles is also considered.

Bifurcation of solutions of singular Fredholm boundary value problems

We obtain conditions for the bifurcation of solutions of linear singular Fredholm boundary value problems with a small parameter under the assumption that the unperturbed singular differential system can be reduced to central canonical form. Using the Vishik-Lyusternik method and the technique of Moore-Penrose pseudoinverse matrices, we suggest an algorithm for finding a family of linearly independent solutions of such boundary value problems for the general case in which the number of boundary conditions specified by a linear vector functional does not coincide with the number of unknowns in the singular differential system.

Steady Streaming Due to the Vibrating Wall in an Infinite Viscous Incompressible Fluid

In this paper steady streaming due to the vibrating wall in an infinite incompressible fluid is studied. The Vishik–Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier–Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of order of 1. The effect of the Stokes drift of fluid particles is also considered.

Effect of Rotation on Steady Viscous Flow between Two Cylinders

In this paper steady viscous flow between two infinitely long circular cylinders produced by small-amplitude transverse vibrations of the inner cylinder about the axis of the outer cylinder and torsional oscillation of the outer cylinder about its axis is studied. The Vishik–Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier–Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of order of unity. The effect of the Stokes drift of fluid particles is also studied. It is shown that it is nonzero not only within the boundary layers but also in higher order terms of the expansion of the averaged outer flow.

Existence Conditions for Bounded Solutions of Weakly Perturbed Linear Impulsive Systems

The weakly perturbed linear nonhomogeneous impulsive systems in the form , and i ∈ ℤ are considered. Under the assumption that the generating system (for ε = 0) does not have solutions bounded on the entire real axis for some nonhomogeneities and using the Vishik‐Lyusternik method, we establish conditions for the existence of solutions of these systems bounded on the entire real axis in the form of a Laurent series in powers of small parameter ε with finitely many terms with negative powers of ε, and we suggest an algorithm of construction of these solutions.

Asymptotic expansion of solutions for a certain class of singularly-perturbed Douglis-Nirenberg elliptic systems

The author obtains an asymptotic expansion of solutions of Problem (1.1), (1.3) using the Vishik-Lyusternik method.

Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half‐space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik–Lyusternik approach, combined with two‐timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non‐oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier–Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are considered. These examples are still brief and illustrative, while the core of the paper is devoted to the adaptation of the Vishik–Lyusternik method to the development of the general theory of vibrational boundary layers.

The Vishik–Lyusternik method in the mixed problem for parabolic operators unresolved with respect to the highest time derivative

We consider the mixed problem for parabolic operators unresolved with respect to the highest time derivative with boundary conditions of general type and zero initial conditions. We present an analog of the Shapiro–Lopatinskii condition that allows one to obtain two-sided a priori estimates in specially constructed function spaces. In the case considered in this paper the characteristic equation in the half-space has two groups of roots with different asymptotics. Because of this, the crucial role in the study of the problem is played by the Vishik–Lyusternik method in the form presented by Volevich (2006).

Viscous boundary layers in flows through a domain with permeable boundary

The effect of small viscosity on nearly inviscid flows of an incompressible fluid through a given domain with permeable boundary is studied. The Vishik–Lyusternik method is applied to construct a boundary layer asymptotic at the outlet in the limit of vanishing viscosity. Mathematical problems with both consistent and inconsistent initial and boundary conditions at the outlet are considered. It is shown that in the former case, the viscosity leads to a boundary layer only at the outlet. In the latter case, in the leading term of the expansion there is a boundary layer at the outlet and there is no boundary layer at the inlet, but in higher order terms another boundary layer appears at the inlet. To verify the validity of the expansion, a number of simple examples are presented. The examples demonstrate that asymptotic solutions are in quite good agreement with exact or numerical solutions.

ASYMPTOTIC SOLUTION OF A VARIATIONAL INEQUALITY MODELING FRICTION

The problem of minimizing the nondifferentiable functional is considered. An asymptotic solution of the corresponding variational inequality is constructed and justified under the assumption that or is a small parameter. Also, formal asymptotic representations are obtained for singular surfaces which characterize a change in the type of boundary conditions. For a modification of the Vishik-Lyusternik method is used, and exponential boundary layers arise. If , then the boundary layer has only power growth; the principal term of the asymptotic expansion of the solution of the problem in a multidimensional region and the complete asymptotic expansion for the case are obtained.

Vishik-Lyusternik method for elliptic boundary-value problems in domains with conical points. III. Problem with degeneracy at a conical point

Vishik-Lyusternik method for elliptic boundary-value problems in domains with conical points. III. Problem with degeneracy at a conical point

The Vishik-Lyusternik method for elliptic boundary-value problems in regions with conical points

The Vishik-Lyusternik method for elliptic boundary-value problems in regions with conical points