## An improved error estimate for a numerical method for a system of coupled singularly perturbed reaction-diffusion equations.(English)Zbl 1040.65066

This paper deals with the following system of two coupled singularly perturbed equations $-\varepsilon^2 u^{\prime\prime}_1+ a_{11}(x) u_1+ a_{12}(x) u_2= f(x),$
$-\mu^2 u^{\prime\prime}_2+ a_{21}(x) u_1+ a_{22}(x) u_2= f_1(x),$ for $$x\in (0,1)$$, $$\varepsilon,\mu$$-small constants, subject to homogeneous Dirichlet boundary conditions. The aim of the present study is to improve the theoretical error bound to almost second order for the case $$0< \varepsilon= \mu\ll 1$$. A finite difference scheme on the Shiskin mesh is presented. The results of two numerical examples are given.

### MSC:

 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 34E15 Singular perturbations for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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