On the solution of the equations of perturbed motion of an object-parachute system

We propose a method of finding the solutions of the equations of perturbed motion of an object-parachute system in analytic form. We perform an analysis of the roots of the characteristic equation of the linearized system. On the basis of the analysis we determine all the solutions of the equations of perturbed motion of the object-parachute system. We exhibit conditions under which the unperturbed motion (descent with slipping) is asymptotically stable.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 41–46.     

On an asymptotic solution of a singularly perturbed system with two small parameters

An initial value problem for a singularly perturbed system of two nonlinear ordinary differential equations with two small parameters is considered. An asymptotic expansion of a solution of this problem is constructed under the assumption that the parameters tend to zero independently of each other.     

Capacity solution for a perturbed nonlinear coupled system

We shall give the existence of a capacity solution to a nonlinear elliptic coupled system, whose unknowns are the temperature inside a semiconductor material, u, and the electric potential, $$\varphi $$, the model problem we refer to is $$\begin{aligned} \left\{ \begin{array}{l} \Delta _p u+g(x,u)= \rho (u)|\nabla \varphi |^2 \quad \mathrm{in} \quad \Omega ,\\ {{\,\mathrm{div}\,}}(\rho (u)\nabla \varphi ) =0 \quad \mathrm{in} \quad \Omega ,\\ \varphi =\varphi _0 \quad \text{ on } \quad {\partial \Omega },\\ u=0 \quad \mathrm{on} \quad {\partial \Omega }, \end{array} \right. \end{aligned}$$where $$\Omega \subset \mathbb {R}^N$$, $$N\ge 2$$ and $$\Delta _p u=-{\text {div}}\left( |\nabla u|^{p-2} \nabla u\right) $$ is the so-called p-Laplacian operator, and g a nonlinearity which satisfies the sign condition but without any restriction on its growth. This problem may be regarded as a generalization of the so-called thermistor problem, where we consider the case of the elliptic equation is non-uniformly elliptic.     

Asymptotic solution of the cauchy problem for a singularly perturbed linear system

We construct the asymptotics of the solution of the Cauchy problem for a degenerate singularly perturbed linear system in the case of multiple spectrum of the principal operator. Nezhin Pedagogic University, Nezhin. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1126–1128, August, 1999,     

On the construction of an asymptotic solution of a perturbed Bretherton equation

We consider the application of the asymptotic method of nonlinear mechanics to the construction of the first and second approximations of a solution of the Bremerton equation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 58–71, January, 1998.     

Error estimates for the finite-element solution of an elliptic singularly perturbed problem

e-mail:ang{at}am.uni-erlangen.de The paper deals with an upwind discretization of finite-volume-typefor singularly perturbed elliptic boundary value problems. Stability,inverse monotonicity and convergence are considered. The mainobjective is to pursue the dependence of the error estimateon the perturbation parameter.     

Effective solution of a singularly perturbed system of nonlinear differential equations

The authors consider a boundary-value problem for a singularly-perturbed system of three nonlinear ordinary differential equations arising in modeling the interaction of physical fields in a semiconducting diode. They suggest a higher-accuracy method for solution of this problem based on the combination of the Newton operator method and the continuation method with respect to a parameter. In this case, the Frechét derivative, the Green function of the corresponding differential equation, and the initial approximation are found in explicit analytical form. The numerical implementation confirms the higher effectivity and the superexponential rate of convergence of the proposed method. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Stability of a singularly perturbed problem

We prove the stability of the mixed problem for a system of telegraph equations under a perturbation of one of the boundary conditions by a sum of a singular perturbation (a small parameter multiplying the highest derivative) and a small regular perturbation. The solution of the problem consists of the current and voltage in a segment of a telegraph line. One of its ends is short-circuited, and a capacitor of small capacity, together with a nonlinear resistance whose volt-ampere characteristic is perturbed by a small term, is connected to the other end. We prove the convergence of the solution of the problem to the unique continuous piecewise continuously differentiable solution of the unperturbed problem bifurcating at some instant of time from its unique classical solution.     

Asymptotics of the solution to a singularly perturbed system of parabolic equations in the critical case

A formal asymptotic representation is constructed for the solution of an initial-boundary value problem for a singularly perturbed system of parabolic equations. A specific feature of the problem is that its solution has an internal transition layer.     

Asymptotic solution method for the control of the minimal force for a linear singularly perturbed system

The optimization of a transient process in a linear singularly perturbed system is considered. The goal is to find a (multidimensional) control of minimal intensity. An algorithm for constructing asymptotic approximations to the solution of the problem is proposed. The main advantage of the algorithm is that the original problem decomposes into optimal control problems of lower dimension.     

一类奇摄动椭圆方程在摄动区域上的渐近解

研究了一类具有摄动边界的非线性椭圆方程摄动问题.经过极坐标变换,在适当的条件下,通过构建近似解以及校正项,利用上下解方法和微分不等式理论得到了解的渐近性态,并通过实际例子进行了验证.