CAUCHY PROBLEMS WITH FRACTAL–FRACTIONAL OPERATORS AND APPLICATIONS TO GROUNDWATER DYNAMICS

Abstract

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.