Tension spline is a function that, for given partition x0 < x1 < . . . < xn, on each interval [xi, xi+1] satisfies differential equation (D − ρiD)u = 0, where ρi’s are prescribed nonnegative real numbers. In the literature, tension splines are used in collocation methods applied to two-points singularly perturbed boundary value problems with Dirichlet boundary conditions. In this paper, we adap...

The periodical in time problem for singularly perturbed second order linear ordinary differential equation is considered. The boundary layer behavior of the solution and its first and second derivatives have been established. An example supporting the theoretical analysis is presented.

In this series of three papers we study singularly perturbed (SP) boundary value problems for equations of elliptic and parabolic type. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converg...

In this paper, reproducing kernel method is proposed for solving singularly perturbed fourth order boundary value problem in 5 2 W [0,1] . The exact solutions are given in the form of series. By truncating the series, the app-roximate solutions is obtained. The errors of the approximate solutions are monotone decreasing with the increasing of nodal points. It is observed that our approach produ...

In this study, we implement a numerical method to solve singularly perturbed differential-difference equation with small shift. Taylor series is used deal the shift, and given problem converted into boundary value problem. To problem, fourth order finite difference approach used. The convergence of investigated. supported by results compared other in literature. Numerical experiments show how s...

We consider smooth systems limiting as ? ? 0 to piecewise-smooth (PWS) with a boundary-focus (BF) bifurcation. After deriving suitable local normal form, we study the dynamics for system sufficiently small but non-zero , using combination of geometric singular perturbation theory and blow-up . show that type BF bifurcation in PWS determines structure within an ?dependent domain which shrinks ze...

Boundary equilibria bifurcation (BEB) arises in piecewise-smooth (PWS) systems when an equilibrium collides with a discontinuity set under parameter variation. Singularly perturbed BEB refers to arising singular perturbation problems which limit as some ? 0 PWS undergo BEB. This work completes classification for codimension-1 singularly the plane initiated by present authors [19], using combina...

This paper considers the robust control problem for nonstandard singularly perturbed systems with time-varying norm-bounded parameter uncertainties in state equations. In order to obtain the controller such that both robust stability and a disturbance attenuation level larger than a arbitrary boundary value are achieved, irrespective of uncertainties, we must solve algebraic Riccati equation wi...

This paper deals with the singularly perturbed boundary value problem for a linear second order differential-difference equation of the convection-diffusion type with small delay parameter. A fourth order finite difference method is developed for solving singularly perturbed differential difference equations. To handle the delay argument, we construct a special type of mesh, so that the term co...

In this paper we demonstrate how hybrid mesh selection strategies based on conditioning can be used in codes designed for the numerical solution of singularly perturbed boundary value problems. The new mesh selection strategies are based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on standard estimates of the local discretization ...