The aim of this paper is to apply the Taylor expansion method to solve the first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: the FPA and the DSA. In order to apply the DSA, we use the CESTAC method and the CADNA library. Using this method, we can find the optimal step of the method, the optimal approximation, the optimal error, and some of numerical instabilities. They are the main novelties of the DSA in comparison with the FPA. The error analysis of the method is proved. Furthermore, the main theorem of the CESTAC method is presented. Using this theorem we can apply a new termination criterion instead of the traditional absolute error. Several examples are approximated based on the FPA and the DSA. The numerical results show the applications and advantages of the DSA than the FPA.

中文翻译：

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离散随机算术验证泰勒展开法求解广义阿贝尔积分方程结果的优势

本文的目的是应用泰勒展开法求解具有Abel核的第一类和第二类Volterra积分方程。本研究侧重于两种主要算法：FPA 和 DSA。为了应用 DSA，我们使用 CESTAC 方法和 CADNA 库。使用这种方法，我们可以找到该方法的最优步长、最优逼近、最优误差和一些数值不稳定性。与 FPA 相比，它们是 DSA 的主要创新点。证明了该方法的误差分析。此外，还介绍了 CESTAC 方法的主要定理。使用这个定理，我们可以应用一个新的终止标准，而不是传统的绝对误差。几个示例是基于 FPA 和 DSA 的近似值。数值结果显示了 DSA 比 FPA 的应用和优势。