We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems without turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and L norm error bounds. These bound...

Abstract In this paper, we have developed a fifth order compact difference method for a class of singularly perturbed singular two-point boundary value problems. To avoid the singularity at zero a terminal boundary condition in the implicit form is derived. Using this condition as one of the boundary condition we solve the singularly perturbed singular two-point boundary value problem by the fi...

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This paper deals with the asymptotic study of the so-called canard solutions, which arise in the study of real singularly perturbed ODEs. Starting near an attracting branch of the ”slow curve”, those solutions are crossing a turning point before following for a while a repelling branch of the ”slow curve”. Assuming that the turning point is degenerate (or non-generic), we apply a correspondence...

Abstract In this paper, we presented a domain decomposition method via exponential splines for solving singularly perturbed two-point boundary value problems with the boundary layer at one end (left or right) point. The method is distinguished by the following fact: The original singularly perturbed two-point boundary value problem is divided into two problems, namely inner and outer region pro...

* Correspondence: [email protected] School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, People’s Republic of China Abstract Based on the method of differential inequalities, by constructing the upper ad lower solutions suitably, delayed phenomenon of loss of stability of solutions in a secondorder quasi-linear singularly perturbed Dirichlet boundary value problem ...

A numerical algorithm for solving stiff boundary value problems with turning points is presented. The stiff systems are characterized as singularly perturbed differential equations. The numerical method is derived by appropriately discretizing the boundary layer and connection theory for such systems. Numerical results demonstrate the effectiveness of the method. In many cases the calculation p...

In this note, recent developements of the Complex WKB method allowing to compute the S-matrix naturally associated with a singularly perturbed three-dimensional system of linear diierential equations without turning point on the real axis are reviewed. It is shown that for a fairly large class of examples, the Complex WKB method gives results far better than what is proven under generic circums...