Numerical approximations for singularly perturbed differential-difference BVPs with layer and oscillatory behavior (Q2882548)

The authors consider a boundary value problem (BVP) of a singularly perturbed differential-difference equation, which is a linear ordinary differential equation of second order including a delay in the derivative of first order. These problems have been investigated in several papers of the authors, for example, see [Appl. Math. Comput. 204, No. 1, 90--98 (2008; Zbl 1160.65043)].NEWLINENEWLINEIn the current paper, the authors apply a collocation method based on cubic splines again to solve the problem numerically. Now a specific mesh is constructed to discretise the delay argument. The authors prove an error bound, which shows that the method is convergent of second order with respect to the mesh size. Finally, four test examples are simulated, where the accuracy of the method is compared for different perturbation parameters. Moreover, the behaviour of the boundary layers is investigated, where the boundary layer disappears and becomes a periodic solution in one of the examples.