An Efficient Approach for Mixed Neutral Delay Differential Equations

In this paper, neutral delay differential equations, which contain constant and proportional terms, termed mixed neutral delay differential equations, are solved numerically. Moreover, an efficient numerical approach is introduced (a combination of the method of steps and the Haar wavelet collocation method) to solve mixed neutral delay differential equations. Furthermore, we prove the existence and uniqueness theorem using successive approximation methods. Three numerical examples are presented to demonstrate the implementation of the proposed method. Furthermore, the precision and accuracy of the Haar wavelet collocation method are validated theoretically by proving that the error tends to zero as the resolution level increases, and numerically, by calculating the rate of convergence. The findings contribute to a broader application of the wavelet-based method to a more complex type of differential equation. This study offers a framework for the extension of the combination of both methods to be applied to potential real-world applications in control theory, biological models, and computational sciences.