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Asymptotic initial-value method for second-order singular perturbation problems of reaction-diffusion type with discontinuous source term. (English) Zbl 1153.65076

The authors consider the boundary value problem (BVP)

$-\epsilon {u}^{\text{'}\text{'}}\left(x\right)+a\left(x\right)u\left(x\right)=-f\left(x\right),\phantom{\rule{1.em}{0ex}}x\in \left(0,d\right)\cup \left(d,1\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)=p,\phantom{\rule{1.em}{0ex}}u\left(1\right)=q,$

where $\epsilon$ is a small parameter, $a\left(x\right)\ge \alpha >0$ a smooth function, $f\left(x\right)$ is a sufficient smooth function on $\left[0,\phantom{\rule{0.277778em}{0ex}}d\right)$ and $\left(d,\phantom{\rule{0.277778em}{0ex}}1\right]$ which has a discontinuity at $d$. The solution of this singular BVP problem has, as a rule, two boundary and one interior layers. It is shown, that the solution can be approximated by the solution of the degenerated equation (with $\epsilon =0$), i.e. ${u}_{0}\left(x\right)=f\left(x\right)/a\left(x\right)$, and solutions of four auxiliary problems for equations of first order: two initial value problems on $\left(0,\phantom{\rule{0.277778em}{0ex}}d\right)$ and two terminal value problems on $\left(d,\phantom{\rule{0.277778em}{0ex}}1\right)$. For the numerical solution of the auxiliary problems the fitted mesh method is used. Two numerical examples are presented.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B05 Linear boundary value problems for ODE 65L12 Finite difference methods for ODE (numerical methods) 65L20 Stability and convergence of numerical methods for ODE 65L50 Mesh generation and refinement (ODE) 34E15 Asymptotic singular perturbations, general theory (ODE)
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