Theoretical analysis of a model of fluid flow in a reservoir with the Caputo–Fabrizio operator

A Fast Compact Finite Difference Method for Fractional Cattaneo Equation Based on Caputo–Fabrizio Derivative

“…Inspired by the work [43], the topic about modelling and numerical solutions of porous media flow equipped with fractional derivatives is very interesting and challenging and will be our main research direction in the future.…”

“…Furthermore, Hristov [42] indicated that the CF operator is not applicable for explaining the physical examples in [37,40]; instead, he suggested that the CF operator can be used for the analysis of materials that do not follow a power-law behavior. e authors of [43] believe that models with CF operators produce a better representation of physical behaviors than do integer-order models, providing a way to model the intermediate (between elliptic and parabolic or between parabolic and hyperbolic) behaviors.…”

“…We will consider the time-fractional Cattaneo equation equipped with the Caputo-Fabrizio derivative. Vivas-Cruz et al [43] gave the theoretical analysis of a model of fluid flow in a reservoir with the Caputo-Fabrizio operator. ey proved that this model cannot be used to describe nonlocal processes since it can be represented as an equivalent differential equation with a finite number of integer-order derivatives.…”

“…In this paper, we will establish a high-order finite difference scheme and propose a procedure to reduce the computational cost. In [43], the authors proposed a recurrence formula of discretized CF operator and obtained an algorithm which can be considered a stencil with a one-step expression without the need of integrals over the whole history. It seems that the procedure in our paper and the algorithm in [43] are different in approach but equally satisfactory in result.…”

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

“…It has been possible to model a wide range of complex systems that exhibit fractional behaviors [7,8]. In this field of research, many advances have been achieved, such that the models with FC capture behaviors that classical models with integer derivatives cannot explain [9][10][11]. Until now, one of the challenging challenges is to attribute a concrete physical meaning to a differential equation involving derivatives of non-integer order [12].…”