Numerical Scheme for Singularly Perturbed Differential Equations with Small Shifts Using Non-Polynomial Quartic Spline

In this paper, a non-polynomial quartic spline technique with a fitting parameter is applied to solve a second-order singularly perturbed differential-difference equation (SPDDE) having small shifts. Taylor series expansion is employed for the delayed and advanced terms in the considered problem to produce a singularly perturbed differential equation (SPDE), and then a non-polynomial quartic spline technique is applied. To manage the layer structure, a fitting parameter is introduced in the proposed computational method; based on the step size, this parameter is evaluated using the theory of singular perturbation theory. Two model examples with left-end boundary layer behavior are considered to theory validate the theoretical finding. The convergence method is analyzed, and the solutions are reported in terms of maximum absolute error with quadratic convergence rate using the fitting parameter. For comparison, solutions without the fitting parameter are reported for test problems. The graphs depict the layer profile for the values of perturbation and shift parameters using the fitting factor and the oscillations without it. The proposed scheme gives uniformly convergent and valid results.