In this paper, we introduce a new fractional integro-differential equation involving newly introduced differential and integral operators so-called fractal-fractional derivatives and integrals. We present a numerical scheme that is convenient for obtaining solution of such equations. We give the general conditions for the existence and uniqueness of the solution of the considered equation using Banach fixed-point theorem. Both the suggested new equation and new numerical scheme will considerably contribute for our readers in theory and applications.

中文翻译：

---

具有分形分数算子的新型 volterra 积分微分方程：存在性、唯一性和数值格式

在本文中，我们引入了一个新的分数阶积分-微分方程，其中包含新引入的微分和积分算子，即所谓的分形-分数阶导数和积分。我们提出了一种便于求解此类方程的数值方案。我们使用 Banach 不动点定理给出了所考虑方程解的存在唯一性的一般条件。建议的新方程和新数值方案都将为我们的读者在理论和应用方面做出巨大贡献。