A class of singularly perturbed two point Boundary Value Problems (BVPs) of reaction-diffusion type for third order Ordinary Differential Equations (ODEs) with a small positive parameter (ε) multiplying the highest derivative and a discontinuous source term is considered. The BVP is reduced to a weakly coupled system consisting of one first order ordinary differential equation with a suitable i...

Singularly perturbed systems of ordinary differential equations arise in many biological, physical and chemical systems. We present an example of a singularly perturbed system of ordinary differential equations that arises as a model of the electrical potential across the cell membrane of a neuron. We describe two periodic solutions of this example that were numerically computed using continuat...

Nonlinear singularly perturbed interior layer problems are examined. Numerical results are presented for a numerical method consisting of a monotone scheme on a Shishkin mesh refined around the approximate location of the interior layer. keywords: Singular Perturbation, Shishkin mesh, Nonlinear, Interior Layer

Singularly perturbed problems of convection-diiusion type are dif-cult to solve using standard numerical methods. This shows that it is desirable to construct robust methods that converge uniformly in the singular perturbation parameter. In this paper we derive new conditions that schemes of this type must satisfy, with particular reference to the case of parabolic layers.

We considered finite difference methods of higher order for semilinear singularly perturbed boundary value problems, consisted of constructing difference schemes on nonuniform meshes. Construction of schemes is presented and convergence uniform in perturbation parameter for one method is shown on Bakhvalov’s type of mesh. Numerical experiments demonstrated influence of different meshes on devel...

One of directions in the development Lomov’s regularization method is approach related to holomorphic singularly perturbed problems, which allows one construct solutions such problems form series powers a small parameter that converge usual sense. For boundary-value problem pseudo-holomorphic continuation very urgent. In this paper, we examine for Tikhonov system and give conditions existence i...

A kind of singularly perturbed boundary value problems is under consideration. In order to obtain its approximation, simple upwind difference discretization is applied. We use a moving mesh iterative algorithm based on equi-distributing of the arc-length function of the current computed piecewise linear solution. First, a maximum norm a posteriori error estimate on an arbitrary mesh is derived ...

In this paper, a computational method is presented for solving a class of nonlinear singularly perturbed two-point boundary value problems with a boundary layer at the left of the underlying interval. First a zeroth order asymptotic expansion for the solution of the given singularly perturbed boundary value problem is constructed. Then the reduced terminal value problem is solved analytically u...

This paper presents the numerical analysis of the Nonlinear Subgrid Scale (NSGS) model for approximating singularly perturbed transport models. The NSGS is a free parameter subgrid stabilizing method that introduces an extra stability only onto the subgrid scales. This new feature comes from the local control yielded by decomposing the velocity field into the resolved and unresolved scales. Suc...

Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction–diffusion pr...