This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).

This paper presents the Quartic Polynomial One-Step Method (QPOSM) based on quartic polynomial interpolating function for solving first order Ordinary Differential Equations (ODEs). The validity of the paper is analysed through consistency, order of convergence and stability. Also, the stability polynomial of this method is derived and the corresponding stability region is obtained. The applicability of the method has been demonstrated by COVID-19 model.