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Numerical analysis of singularly perturbed delay differential equations with layer behavior. (English) Zbl 1069.65086

The problem under consideration is the singularly perturbed boundary value problem (BVP) for the delay differential equation

εy '' (x)+a(x)y ' (x-δ)+b(x)y(x)=f(x),0<x<1,

under the boundary conditions

y(x)=φ(x),-δx0,y(1)=γ,

where ε and δ are small positive parameters.

The stated BVP for the delay differential equation is approximated by one for the ordinary differential equation (ODE), created by replacing the retarded term y ' (x-δ) by its first order Taylor approximation y ' (x)-δy '' (x). The approximate BVP for the ODE is approximated by a standard three points difference scheme. The stability and convergence of the method is discussed for two cases corresponding to the location the boundary layer, on the left side (when a(x)>0) and on the right (when a(x)<0). Numerical examples are presented.

MSC:
65L10Boundary value problems for ODE (numerical methods)
34K28Numerical approximation of solutions of functional-differential equations
65L20Stability and convergence of numerical methods for ODE
34K26Singular perturbations of functional-differential equations
34K10Boundary value problems for functional-differential equations