Numerical analysis of singularly perturbed delay differential equations with layer behavior (Q1888502)

The problem under consideration is the singularly perturbed boundary value problem (BVP) for the delay differential equation \[ \varepsilon y''(x)+a(x)y'(x-\delta)+b(x)y(x)=f(x), \quad 0 < x < 1, \] under the boundary conditions \[ {y(x)}=\phi(x),\quad -\delta\leq x\leq 0, \quad y(1)=\gamma, \] where \(\varepsilon\) and \(\delta\) 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 \(y'(x-\delta)\) by its first order Taylor approximation \(y'(x)-\delta y''(x)\). 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.