Numerical Methods for SPDDE's

In general, any differential equation in which the highest order derivative is multiplied by a small positive parameter ¿ (0<¿<<1) is called the Singular Perturbation Problem. A differential equation in which the highest order derivative is multiplied by a small positive parameter and has at least one shift term (delay or advance) is called singularly perturbed differential-difference equation (SPDDE). Here the negative shift is used for the delay, and a positive shift is used for advance. When we apply the existing standard numerical methods to this SPDDE, we get oscillatory/unsatisfactorily results when step size h is greater than the value of the perturbation parameter ¿. As a result of this, finding solutions for SPDDE has become the most exciting and challenging task. Thus it is of considerable scientific interest to the researchers to develop simple and efficient computational methods for singularly perturbed differential-difference equations.