A New Method for Solving a Neutral Functional-Differential Equation with Proportional Delays

Abstract

This study presents and implements a new hybrid technique that combines the Sawi transform (ST) and Homotopy perturbation method (HPM) to solve neutral functional-differential equations with proportional delays. Some of the important properties of the method are established and validated. We start the method by first applying ST to obtain the recurrence relation. We, next, implement HPM to find convergent series solutions of the recurrence relation. The series is free of assumptions and restrictions, highlighting its adaptability and robustness. Moreover, the convergence of the method is established through convincing proof. To demonstrate its effectiveness and applicability, we provide five examples. The method yields accurate approximate solutions, or in some cases exact solutions, with a few number of iterations, reinforcing its reliability and validity. Moreover, the performance of the method is compared with some available methods and demonstrates its superiority and efficiency.