On the decomposition of a degenerate singularly perturbed linear system of differential equations

We study the problem of decomposition of degenerate singularly perturbed systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 401–415, July–September, 2006.     

Asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point

We consider a singularly perturbed system of differential equations with irregular singular point. We prove that, in the general case of multiple elementary divisors, the corresponding asymptotic expansions can be constructed in the form of double series in fractional powers of the parameter and the ratio of the independent variable to the parameter.     

On local asymptotic expansions of solutions of singularly perturbed systems of differential equations with degeneration at a point

We propose an algorithm for the construction of asymptotic solutions of singularly perturbed systems of differential equations in the case where the characteristic equation has simple roots. In contrast to previous investigations, in which the matrix multiplying the derivatives becomes degenerate on the entire interval, we study the case where degeneration occurs at a single point.     

On asymptotic properties of solutions of linear differential functional equations with linearly transformed argument

We establish new properties of C ¹[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 170–177, April–June, 2006.     

On the asymptotic properties of solutions of linear functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation {fx153-01} in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 147–150, April–June, 2008.     

On the asymptotic integration of a linear system of differential equations with a small parameter in the coefficients of some derivatives

Using a transformation matrix, we reduce a system of differential equations with a small parameter in the coefficients of some derivatives and a turning point to an integrable system of equations. We also study properties of the transformation matrix.     

Construction of asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point in the case of a multiple spectrum of the boundary matrix

We construct asymptotic solutions of a singularly perturbed linear system of differential equations with irregular singular point. We consider the case where the main matrix has a multiple eigenvalue associated with one or several elementary divisors of the same multiplicity. We establish that, in the case of multiple elementary divisors, the corresponding asymptotic expansions can be constructed in the form of double series in fractional powers of the parameter and the ratio of the independent variable to the parameter. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 128–144, January–March, 2007.     

On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.     

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 408–420, July–September, 2008.     

On asymptotic properties of solutions of functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f ₁(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 144–160, January–March, 2007.     

Transition layers for singularly perturbed delay differential equations with monotone nonlinearities

Transition layers arising from square-wave-like periodic solutions of a singularly perturbed delay differential equation are studied. Such transition layers correspond to heteroclinic orbits connecting a pair of equilibria of an associated system of transition layer equations. Assuming a monotonicity condition in the nonlinearity, we prove these transition layer equations possess a unique heteroclinic orbit, and that this orbit is monotone. The proof involves a global continuation for heteroclinic orbits.     

Boundary value problems for second order singularly perturbed differential equations

In this paper the existence and uniqueness of solutions of boundary value problems (which contains the Robin's problem) is discussed by using the upper and lower solution. In addition, the asymptotic estimation of the solution is given as well.     

Wexler inequality and almost periodic solutions of differential equations with deviating argument of mixed type

The paper is concerned with the existence and stability of an almost periodic solution of the system with deviating argument .The Wexler inequality for the Cauchy matrix is used. Conditions for the stability of the solution are given.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 295–301, July–September, 2004.     

Second-order accurate difference method for the singularly perturbed problem of fourth-order ordinary differential equations

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.     

The asymptotic expansions of singularly perturbed boundary value problems

In this paper we study the singularly perturbed boundary value problem:εy″=f(t,y,ε),y(0)=ξ(ε),y(1)=η(ε).where εis a positive small parameter.In the conditions:f_(?)(0,y,0)≥m_0 ,f_(?)(l,y,0)≥m_0 and f_(?)f(t,y,ε)≥0 ,we prove the existences,and uniformly valid asymptotic expansions of solutions for the given boundary value problems,and hence we improve the existing results.     

Averaging of a boundary-value problem for a multifrequency system with deviating argument

Using the averaging method, we investigate the solvability of a boundary-value problem for a multifrequency system with deviating argument and integral boundary conditions. We obtain an estimate for the difference between solutions of the original and averaged problems. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 519–527, October–December, 2007.     

Periodic solution to a kind of singularly perturbed differential equations of parabolic type in chemical kinetics

This paper deals with the problem on the periodic solution for the singularly perturbeddifferential equations of parabolic type originating from chemical kinetics in stratifiedmedia,A uniformly valid asymptotic solution is constructed and the related asymptoticestimate is given.     

Nonlinear singularly perturbed problems of ultra parabolic equations

A class of nonlinear singularly perturbed problem of ultra parabolic equations are considered. Using the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.     

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection-diffusion equations

In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.     

Conditions for the unique solvability of the initial-value problem for linear second-order differential equations with argument deviations

We establish new conditions under which the initial-value problem for a system of linear second-order differential equations with argument deviations has a unique solution, which depends monotonically on additive perturbations of the problem. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 535–547, October–December, 2006.