A Second Order Convergence Method for Differential Difference Equation with Mixed Shifts using Mixed Non-Polynomial Spline

Abstract

A proposed numerical approximation method is presented for solving a singularly perturbed second-order differential-difference equation with both the delay and advance shifts. The algorithm utilises a nonpolynomial spline with a fitting factor finite difference scheme. The application of finite difference approximations for higher order derivatives leads to the derivation of a tri-diagonal system. To efficiently solve this system of equations, an algorithm based on discrete invariant imbedding is employed and the stability of the method is analysed. An assessment of the applicability and efficiency of the proposed scheme is conducted by performing three numerical experiments and comparing the results with other methods. The maximum absolute errors are used as the basis for comparison. The impact of minor shifts on the boundary layer behaviour of the solution is illustrated using plotted graphs featuring different degrees of shifts. The method is theoretically and numerically analysed using uniformly convergent solutions with quadric convergence rate.