On the numerical solution of a singularly-perturbed equation with a turning point

1984 ◽  
Vol 24 (6) ◽  
pp. 135-139 ◽  
Author(s):  
V.D. Liseikin
Keyword(s):  
Numerical Solution ◽  
Turning Point ◽  
Singularly Perturbed ◽  
Singularly Perturbed Equation ◽  
Perturbed Equation

scholarly journals Uniformly convergent schemes for singularly perturbed differential equations based on collocation methods

2000 ◽  
Vol 24 (5) ◽  
pp. 305-313
Author(s):  
Dialla Konate
Keyword(s):  
Boundary Layer ◽  
Collocation Method ◽  
Polynomial Approximation ◽  
Singularly Perturbed ◽  
Collocation Methods ◽  
Singularly Perturbed Equation ◽  
Approximation Solution ◽  
Repeated Use ◽  
Uniformly Convergent ◽  
Perturbed Equation

It is well known that a polynomial-based approximation scheme applied to a singularly perturbed equation is not uniformly convergent over the geometric domain of study. Such scheme results in a numerical solution, sayσwhich suffers from severe inaccuracies particularly in the boundary layer. What we say in the current paper is this: when one uses a grid which is not “too coarse” the resulted solution, even being nonuniformly convergent may be used in an iterated scheme to get a “good” approximation solution that is uniformly convergent over the whole geometric domain of study.In this paper, we use the collocation method as model of polynomial approximation. We start from a precise localization of the boundary layer then we decompose the domain of study, sayΩinto the boundary layer, sayΩϵand its complementaryΩ0. Next we go to the heart of our work which is to make a repeated use of the collocation method. We show that the second generation of polynomial approximation is convergent and it yields an improved error bound compared to those usually appearing in the literature.

scholarly journals Asymptotic expansion for the functional of Markovian evolution in Rd in the circuit of diffusion approximation

2005 ◽  
Vol 2005 (3) ◽  
pp. 247-257 ◽  
Author(s):  
I. V. Samoilenko
Keyword(s):  
Asymptotic Expansion ◽  
Diffusion Approximation ◽  
Singularly Perturbed ◽  
Singularly Perturbed Equation ◽  
Perturbed Equation

We study the asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in Rd. The view of regular and singular parts of solution is found.

scholarly journals Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2449
Author(s):  
Flaviano Battelli ◽  
Michal Fečkan
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Singularly Perturbed ◽  
Two Dimensional ◽  
Singularly Perturbed Equation ◽  
Discontinuous Differential Equations ◽  
Scalar Variable ◽  
Perturbed Equation

We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.

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