Approximate solution of a non-linear boundary value problem with a small parameter for the highest-order differential

Robust Numerical Methods for Singularly Perturbed Differential Equations

“…A related construction appears in [Bog84], where Boglaev considers a reaction-diffusion problem −ε 2 u + c(x)u = f with c(x) ≥ γ 2 > 0 and takes τ = (2/γ)ε| ln ε| then constructs the mesh using a variant of (2.135). This choice appears also in the A-mesh of Remarks III.2.12 and III.3.122.…”

Uniform Numerical Method for a Quasilinear System With Boundary Layer

“…Thus we will consider the initial boundary value problem of the form. u2(0) = A, 2~2(1) = B, (1)(2)(3)(4) where 0 < E < 1 is a small parameter, u*, A , B are given constants. f1(x, u1, UZ), f2(2, u1,ug) are given smooth functions satisfying certain regularity conditions.…”

A uniformly convergent difference scheme on a modified Shishkin mesh for the singularly perturbed reaction-diffusion boundary value problem

“…This method was first presented by Boglaev [2], where the discretisation of the problem (1)-(3) on a modified Bakhvalov mesh was analysed and first order uniform convergence with respect to ϵ was demonstrated. Afterwards, Boglaev [3] also analysed the analogous 2D problem.…”

“…We therefore demand that the numerical solution y converges to y for every value of the perturbation parameter in the domain 0 < ϵ < 1 with respect to the discrete maximum norm ∥·∥ . The problem (1)- (2) has been researched by many authors with various assumptions on f (x, y). Various different difference schemes have been constructed which are uniformly convergent on equidistant meshes as well as schemes on specially constructed, mostly Shishkin and Bakvhvalov-type meshes, where ϵ-uniform convergence of second order has been demonstrated, see e.g.…”

“…Afterwards, Boglaev [3] also analysed the analogous 2D problem. We therefore aim to construct an ϵ-uniformly convergent difference scheme on a modified Shishkin mesh, using the results presented in [2].…”