In this work we consider a parabolic system of two linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms. The small values of the diffusion parameters, in general, cause that the solution has boundary layers at the ends of the spatial domain. To obtain an efficient approximation of the solution we propose a numerical method combining the Crank-Nicolson m...

This work develops an ε-uniform finite element method for singularly perturbed two-point boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value probl...

Pointwise accurate numerical methods are constructed and analysed for three classes of singularly perturbed first order transport problems. The involve piecewise-uniform Shishkin meshes the approximations shown to be parameter-uniformly convergent in maximum norm. A problem from modelling fluid–particle interaction is formulated used as a test these methods. Numerical results presented illustra...

This work develops an ε-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By inserting one node arbitrarily in any element, the new finite element solution always intersects with the original one at fixed points, and the errors at those points converge at the same rate as regular boundary value problems (witho...

In this article, we analyze a fully discrete $\varepsilon-$uniformly convergent finite element method for singularly perturbed convection-diffusion-reaction boundary-value problems, on piecewise-uniform meshes. Here, choose $L-$splines as basis functions. We will concentrate the convergence analysis of which employ $L-$spline functions instead their continuous counterparts. The are approximated...

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...

A singularly perturbed convection–diffusion problem, posed on the unit square in $${\mathbb {R}}^2$$ , is studied; its solution has both exponential and characteristic boundary layers. The problem solved numerically using local discontinuous Galerkin (LDG) method Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $$k>0$$ each variable, error between LDG true proved to...

in this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer. a fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. thomas algorithm is used to solve the tri-diagonal system. the stability of the algorithm is investigated. it ...

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which ...

In this article, a singularly perturbed convection-diffusion problem with small time lag is examined. Because of the appearance perturbation parameter, boundary layer observed in solution problem. A hybrid scheme has been constructed, which combination cubic spline method region and midpoint upwind outer on piecewise Shishkin mesh spatial direction. For discretization derivative, Crank-Nicolson...