$\varepsilon$-Uniform Numerical Technique for the Class of Time Dependent Singularly Perturbed Parabolic Problems With State Dependent Retarded Argument Arising from Generalised Stein’s Model of Neuronal Variability

Abstract

The motive of the present work is to develop a parameter robust numerical scheme for the class of problems involving singularly perturbed parabolic differential-difference equations with delay, which often arise in computational neuroscience. The numerical schemes developed prior to this work are restricted either to the case of small values of delay argument or linear convergence with restriction on the mesh generation. In practice, the delay argument can be of arbitrary size. Parameter $\varepsilon$ may take small enough values e.g., viscosity coefficient in Navier–Stokes equation for fluids with high Reynolds number. It is required to construct a higher order parameter robust numerical scheme without any restriction on the mesh generation for singularly perturbed parabolic differential-difference equations with state dependent delay of arbitrary size. A new class of non-standard finite difference method based on interpolation, $\theta$-method and Micken’s techniques is constructed to approximate the solution of singularly perturbed parabolic differential-difference equations with arbitrary values of delay. It is shown that proposed numerical scheme is parameter uniform convergent. It is proved that this method is unconditionally stable and is convergent for $\frac{1}{2} \le \theta \le 1,$ without having any restriction on the mesh. Some numerical experiments are provided to illustrate the performance of the method.