Steplike contrast structures for a second-order ordinary differential equation with a small parameter multiplying the highest derivative

A boundary value problem is considered for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative. The limit equation has three solutions, of which two are stable and are separated by the third unstable one. For the original problem, an asymptotic expansion of a solution is studied that undergoes a jump from one stable root of the limit equation to the other in the neighborhood of a certain point. A uniform asymptotic approximation of this solution is constructed up to an arbitrary power of the small parameter.     

Asymptotics of the solution to the Neumann problem in a thin domain with sharp edge

An asymptotic expansion of the solution to the Neumann problem for a second-order equation in a thin domain with peak-like edge is constructed and justified. Owing to the sharpness of the edge, the procedure of dimension reduction leads to a degenerate limit equation on the longitudinal cross-section of the domain and a solution has irregular behavior near the boundary. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 193–219.     

Asymptotics of a second-order differential equation with a small parameter in the case when the reduced equation has two solutions

The boundary value problem for a second-order nonlinear ordinary differential equation with a small parameter multiplying the highest derivative is examined. It is assumed that the reduced equation has two solutions with intersecting graphs. Near the intersection point, the asymptotic behavior of the solution to the original problem is fairly complex. A uniform asymptotic approximation to the solution that is accurate up to any prescribed power of the small parameter is constructed and justified.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

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.     

Initial Boundary Value Problems for Scalar and Vector Burgers Equations

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x >0, t >0 with Riemann type of initial and boundary conditions and use the Hopf–Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.     

Numerical solution of the differential equation describing the behavior of the zeroth-order boundary function

The asymptotic solution of the integro-differential plasma-sheath equation is considered. This equation is singularly perturbed because of the small coefficient multiplying the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the form of both a regular series expansion and an expansion in boundary functions. The equation for the first coefficient of the regular series has only a trivial solution. A numerical algorithm is considered for the solution of the second-order differential equation describing the behavior of the zeroth-order boundary function. The proposed algorithm efficiently solves the boundary-value problem and produces a well-behaved solution of the Cauchy problem. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 24–35, 2006.     

具有转向点曲线的椭圆型方程奇摄动边值问题

The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem,the asymptotic behavior of solution for the boundary value problem is studied.     

Asymptotic behavior of a reaction-diffusion system in bacteriology

This paper is concerned with the asymptotic behavior of the solution for a coupled system of reaction-diffusion equations which describes the bacteria growth and the diffusion of histidine and buffer concentrations. Under the basic boundary condition of Neumann type or mixed type the coupled system can have infinitely many steady-state solutions. The present paper gives some explicit information on the asymptotic limit of the time-dependent solution in relation to these steady states. This information exhibits some rather distinct properties of the solutions between the Neumann boundary problem and the Dirichlet or mixed boundary problem.     

自共轭椭圆型偏微分方程奇异摄动问题的一致收敛差分解法

本文在矩形域内考虑高阶导数项含有小参数的自共轭椭圆型第一边值问题. 本文,我们应用渐近分析方法建立了一种新的差分格式,比较了差分方程的解与微分方程的解的渐近性态,并证明了解的一致收敛性.     

具有特征边界的一类非线性椭圆型方程的奇摄动

讨论了一类具有特征边界的非线性椭圆型方程的奇摄动。利用多重尺度法和比较原理研究了边值问题的存在性及渐近性态。     

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.     

Asymptotic behavior of solutions of a free boundary problem modeling multi-layer tumor growth in presence of inhibitor

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed in little Hölder spaces. Next we investigate asymptotic behavior of the solution. By computing the spectrum of the linearized problem and using the linearized stability theorem, we give the rigorous analysis of stability and instability of all stationary flat solutions under the non-flat perturbations. The method used in proving these results is first to reduce the free boundary problem to a differential equation in a Banach space, and next use the abstract well-posedness and geometric theory for parabolic differential equations in Banach spaces to make the analysis.     

Application of the dual operator method to an equation describing the behavior of a boundary function

The dual operator is an analogue of the conjugate operator in linear theory. In this study the dual operator is applied to a second-order differential equation describing the behavior of the zero-order boundary function in the boundary function method used to derive the asymptotic solution of the singularly perturbed integro-differential plasma-sheath equation. This approach produces is a three-point difference scheme. The results of a numerical solution of the Cauchy problem are reported. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 49–60, 2007.     

一类反应扩散方程的奇摄动

讨论了一类奇摄动反应扩散方程的初边值问题. 在适当的条件下,利用不动点定理，证明了原问题解的存在唯一性及其渐近性态.     

On boundary value problem with singular inhomogeneity concentrated on the boundary

We study the asymptotic behavior of solutions to a boundary value problem for the Poisson equation with a singular right-hand side, singular potential and with alternating type of the boundary condition. Assuming that the boundary microstructure is periodic, we construct the limit problem and prove the homogenization theorem by means of the unfolding method. The proof requires that the dimension be larger than two.     

Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ?Ω of the domain as the square root of the distance to ?Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.     

Long-time behavior for a nonlinear fourth-order parabolic equation

We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.

     

半线性方程ROBIN问题的角层解(英文)

本文讨论了一类半线性方程Robin边值问题.利用微分不等式理论,研究了边值问题角层解的存在性和渐近性态.     

Existence and uniqueness of large solutions for a class of non-uniformly elliptic semilinear equations

In this paper, we study existence, uniqueness, and asymptotic behavior of large solutions of second-order degenerate elliptic semilinear problems in non-divergence form. The main particularity of the problem is the interior uniform ellipticity of the equation, which degenerates on the boundary, involving an effect on the boundary blow-up profile of the solution.