A nonlinear reaction-diffusion two-point boundary value problem with multiple solutions is considered. Its second-order derivative is multiplied by a small positive parameter ε, which induces boundary layers. Using dynamical systems techniques, asymptotic properties of its discrete suband super-solutions are derived. These properties are used to investigate the accuracy of solutions of a standa...

In this paper, a parameter uniform numerical method based on Shishkin mesh is suggested to solve a system of second order singularly perturbed differential equations with a turning point exhibiting boundary layers. It is assumed that both equations have a turning point at the same point. An appropriate piecewise uniform mesh is considered and a classical finite difference scheme is applied on t...

In this article we propose an efficient numerical scheme based on a Shishkin mesh for a class of singularly perturbed parabolic convection-diffusion problems with boundary turning point and retarded arguments. The solution of the considered problem exhibit a boundary layer on the left side of the domain. The continuous problem is semidiscretized by means of backward Euler finite difference meth...

This paper develops a semi-analytic technique for generating smooth nonuniform grids for the numerical solution of singularly perturbed two-point boundary value problems. It is based on the usual idea of mapping a uniform grid to the desired nonuniform grid. We introduce the W-grid, which depends on the perturbation parameter 1. For problems on [0,1] with a boundary layer at one end point, the ...

A system of coupled singularly perturbed initial value problems with a small parameter is considered. The solution to the system have boundary layers. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform Shishkin mesh. On this mesh a hybrid finite difference scheme is proved to be almost second-order accurate, uniformly in the small parameter. Num...

A coupled system of two singularly perturbed ordinary differential equations of first order with the prescribed initial values are considered. The leading term of each equation is multiplied by a small positive parameter and the parameters may differ. The solution exhibits overlapping layers. A Shishkin mesh is constructed. A classical finite difference scheme applied on this mesh (which is pie...

We consider a system of two coupled reaction-diffusion equations. When the parameters multiplying the second-order derivatives in the equations are small, their solutions exhibit boundary layers. Moreover, when the parameters are of different magnitudes, two distinct but overlapping boundary layers are present. We study a finite element discretization on general layer-adapted meshes including t...

In this paper, singularly perturbed quasilinear boundary value problems are taken into account. With purpose, a finite difference scheme is proposed on Shishkin-type mesh (S-mesh). Quasilinearization technique and interpolating quadrature rules used to establish the numerical scheme. Then, an error estimate derived. A experiment demonstratedto verify theory.

In this paper, numerical solution for singularly perturbed problem with nonlocal boundary conditions is obtained. Finite difference method used to discretize on the Bakhvalov-Shishkin mesh. The some properties of exact are analyzed. error obtained first-order in discrete maximum norm. Finally, an example solved show advantages finite method.