A singularly perturbed parabolic equation of convection-diffusion type with an interior layer in the initial condition is studied. The solution is decomposed into a discontinuous regular component, a continuous outflow boundary layer component and a discontinuous interior layer component. A priori parameter-explicit bounds are derived on the derivatives of these three components. Based on these...

In this paper, we propose and analyze a high-order uniform method for solving boundary value problems (BVPs) for singularly perturbed nonlinear delay differential equations with small shifts (delay and advance). Such types of BVPs play an important role in the modeling of various real life phenomena, such as the variational problem in control theory and in the determination of the expected time...

In this work, the bilinear nite element method on a Shishkin mesh for convection-diiusion problems is analyzed in the two-dimensional setting. A su-perconvergence rate O(N ?2 ln 2 N + N ?1:5 ln N) in a discrete-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter. Numerical tests in...

In this work, the bilinear nite element method on a Shishkin mesh for convection-diiusion problems is analyzed in the two-dimensional setting. A su-perconvergent rate O(N ?2 ln 2 N + N ?3=2) in a discrete-weighted energy norm is established under certain regularity assumption. This convergent rate is uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate t...

In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate O(N−2 ln N + N−1.5 lnN) in a discrete -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter . Numerical tests ind...

ABSTRACT. In this paper, a numerical method is constructed for solving one-dimensional time dependent modified Burgers’ equation for various values of Reynolds number. At high Reynolds number, an inviscid boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem involving a sma...

We consider a spline difference scheme on a piecewise uniform Shishkin mesh for a singularly perturbed boundary value problem with two parameters. We show that the discrete minimum principle holds for a suitably chosen collocation points. Furthermore, bounds on the discrete counterparts of the layer functions are given. Numerical results indicate uniform convergence. AMS Mathematics Subject Cla...

A linear singularly perturbed time dependent convection–diffusion problem is examined. The initial condition is designed to have steep gradients in the vicinity of the inflow point, which are transported in time, thus creating a moving interior shock layer. The location of this interior layer is tracked by the characteristics of the reduced first order problem. A numerical method is designed an...

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem. A Shishkin piecewise–uniform mesh is constructed, which is used, in conjunct...

A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are not necessarily equal. The components of the solution exhibit overlapping layers. A Shishkin piecewise– uniform mesh is constructed, which is used, in conjunction with a cl...