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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.

MSC:
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
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References:
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