ε-Uniform fitted mesh method for singularly perturbed differential-difference equations:: Mixed type of shifts with layer behavior

An epsilon-uniform fitted mesh method is presented to solve boundary-value problems for singularly perturbed differential-difference equations containing negative as well as positive shifts with layer behavior. Such types of BVPs arise at various places in the literature such as the variational problem in control theory and in the determination of the expected time for the generation of action potentials in nerve cells. The method consists of the standard upwind finite difference operator on a special type of mesh. Here, we consider a piecewise uniform fitted mesh, which turns out to be sufficient for the construction of epsilon-uniform method. One may use some more complicated meshes, but the simplicity of the piecewise uniform mesh is supposed to be one of their major attractions. The error estimate is established which shows that the method is epsilon-uniform. Several numerical examples are solved to show the effect of small shifts on the boundary layer solution. Numerical results in terms of maximum errors are tabulated and graphs of the solution are drawn to demonstrate the efficiency of the method.