@Article{IJNAMB-3-168,
author = {GRIGORY SHISHKIN AND LIDIA SHISHKINA},
title = {Scheme of the Solution Decomposition Method for a Singularly Perturbed Reaction-Diffusion Equation},
journal = {International Journal of Numerical Analysis Modeling Series B},
year = {2012},
volume = {3},
number = {2},
pages = {168--184},
abstract = {A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed to construct difference
schemes convergent uniformly with respect to a perturbation parameter &#949 for &#949 &#8712 (0, 1], i.e.,
&#949-uniformly. This approach is based on a decomposition of the discrete solution into the regular
and singular components which are solutions of discrete subproblems considered on uniform
grids. Using an asymptotic construction technique, a difference scheme of the solution decomposition
method is constructed that converges &#949-uniformly in the maximum norm at the rate
O(N^{-2}ln^2N), where N + 1 is the number of nodes in the grids used; for fixed values of the
parameter &#949, the scheme converges at the rate O(N^{-2}). For the constructed scheme, approximations
of the regular and singular components to the solution and their derivatives up to the
second order are studied. A modified scheme of the solution decomposition method is constructed
for which the regular component of the solution and its discrete derivatives converge $\varepsilon$-uniformly
in the maximum norm at the rate O(N^{-2}) for &#949 = o(ln^{-1} N).},
issn = {},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/ijnamb/276.html}
}