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Uniform numerical method for singularly perturbed delay differential equations. (English) Zbl 1134.65042
The problem under consideration is the singular delay initial value problem $$ \varepsilon u'(t)+a(t)u(t)+b(t)u(t-r)=f(t), \; 0<t\leq T, \; u(t)=\varphi(t), \; -r<t\leq 0,$$ where $a(t)\geq \alpha>0$, $b(t), f(t), \varphi(t)$ are given smooth functions, $r$ is a constant delay, $T=mr$. A finite difference scheme is constructed for this problem which involves an appropriate piecewise-uniform mesh on each sequent subinterval $I_p =((p-1)r<t\leq pr]$, $p=1,\dots,m$. The scheme is shown to converge to the solution uniformly with respect to the perturbation parameter $\varepsilon$. An error estimate and numerical experiments are presented.

65L05Initial value problems for ODE (numerical methods)
65L12Finite difference methods for ODE (numerical methods)
65L50Mesh generation and refinement (ODE)
34K28Numerical approximation of solutions of functional-differential equations
65L20Stability and convergence of numerical methods for ODE
34K06Linear functional-differential equations
Full Text: DOI
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