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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

εu ' (t)+a(t)u(t)+b(t)u(t-r)=f(t),0<tT,u(t)=φ(t),-r<t0,

where a(t)α>0, b(t),f(t),φ(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<tpr], p=1,,m. The scheme is shown to converge to the solution uniformly with respect to the perturbation parameter ε. 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