Numerical technique for solving multi-delay systems

Our paper presents a new class of θ−methods for solving both linear and nonlinear delay differential equations (DDEs) and their systems, which may involve multiple delays. These methods are developed by combining iterative techniques with implicit numerical approaches. Importantly, the methods, as outlined in the Butcher tableau, differ from traditional Runge-Kutta methods. We perform a stability analysis of these new methods alongside an investigation of their convergence properties. Our findings demonstrate that the proposed methods are more efficient than existing alternatives across a variety of applications. The tables and figures included in the paper illustrate the accuracy and practicality of our approach. The primary advantages of this technique include straightforward coding, a clear algorithm, and ease of implementation. The successive approximations method exhibits high accuracy and quick convergence. Its effectiveness is validated through various applications involving DDEs. This method is demonstrated through a number of illustrative examples using Maple 2019.