In this study, an effective numerical technique has been introduced for finding the solutions of the first-order integro-differential equations including neutral terms with variable delays. The problem has been defined by using the neutral integro-differential equations with initial value. Then, an alternative numerical method has been introduced for solving these type of problems. The method is expressed by fundamental matrices, Laguerre polynomials with their matrix forms. Besides, the solution has been obtained by using the collocation points with regard to the reduced system of algebraic equations and Laguerre series.

中文翻译：

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含时滞的中立型积分微分方程解的数值格式。

在这项研究中，引入了一种有效的数值技术来寻找一阶积分微分方程的解，其中包括具有可变时滞的中性项。通过使用具有初始值的中性积分微分方程定义了该问题。然后，引入了一种替代的数值方法来解决这些类型的问题。该方法由基本矩阵Laguerre多项式及其矩阵形式表示。此外，通过使用代数方程的简化系统和Laguerre级数的搭配点获得了解决方案。