Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine learning, mechanics, economics, electrodynamics and so on. Besides, special classes of functional differential equations have been investigated in many researches. In this study, a numerical investigation of retarded type of these models together with initial conditions are introduced. The technique is based on a polynomial approach along with collocation points which maintains an approximated solutions to the problem. Besides, an error analysis of the approximate solutions is given. Accuracy of the method is shown by the results. Consequently, illustrative examples are considered and detailed analysis of the problem is acquired. Consequently, the future outlook is discussed in conclusion.

翻译：延迟差异方程式在科学、工程、医学和生物模型中非常重要,包括时间延迟现象,有助于确定机器学习、机械学、经济学、电子动力学等方面的现实应用。此外,在许多研究中也调查了功能差异方程式的特殊类别。在本研究中,对这些模型的迟钝类型和初始条件进行了数字调查。该技术以多元法和合用点为基础,维持了问题的近似解决办法。此外,对近似解决办法进行了错误分析,结果显示了方法的准确性。因此,考虑了示例,并获得了对问题的详细分析。因此,对未来前景进行了总结讨论。