As the Riemann–Liouville derivative is a derivative of a convolution of a function and the power law, the fractal–fractional derivative of a function is the fractal derivative of a convolution of that function with the power law or exponential decay. In order to further open new doors on ongoing investigations with field of partial differential equations with non-conventional differential operators, we introduce in this paper new Cauchy problems with fractal–fractional differential operators. We consider two cases, when the operator is constructed with power law and when it is constructed with exponential decay law with Delta-Dirac property. For each case, we present the conditions under which the exact solution exists and is unique. We suggest a suitable and accurate numerical scheme that can be used to solve such differential equation numerically. We present illustrative examples where an application to a partial differential equation and to a model of groundwater flow within the confined aquifer are done with numerical simulations provided. The clear variation of water level shows the impact of the fractal–fractional derivative on the dynamics.

中文翻译：

---

分形-分数算子的柯西问题及其在地下水动力学中的应用

由于 Riemann-Liouville 导数是函数与幂律卷积的导数，因此函数的分形-分数导数是该函数与幂律或指数衰减卷积的分形导数。为了进一步打开具有非常规微分算子的偏微分方程领域正在进行的研究的新大门，我们在本文中引入了分形-分数微分算子的新柯西问题。我们考虑两种情况，一种是使用幂律构造算子，另一种是使用具有 Delta-Dirac 属性的指数衰减律构造算子。对于每种情况，我们都给出了精确解存在且唯一的条件。我们提出了一种合适且准确的数值方案，可用于数值求解此类微分方程。我们提供了说明性示例，其中通过提供的数值模拟完成了对偏微分方程和承压含水层内地下水流动模型的应用。水位的明显变化显示了分形-分数导数对动力学的影响。