The mathematical description of the bulk fluid flow and that of the content impurity dispersion, obtained by replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective

Abstract

In the field of fractional calculus applications, there is a tendency to admit that “integer-order derivatives cannot simply be replaced by fractional-order derivatives to develop fractional-order theories”. There are different arguments for that: initialization problem, inconsistency, use of nonsingular or singular kernels, loss of objectivity. In this paper it is shown that the mathematical description of the bulk fluid flow and that of the content impurity spread replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. More precisely, it is proven that, the mathematical description of the bulk fluid 2D flow and that of the content impurity spread, in a horizontal unconfined aquifer, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. It is also proven that, the mathematical description of a Newtonian, incompressible, viscous bulk fluid 3D flow and that of the contained impurity dispersion, obtained replacing integer order temporal derivatives with general temporal Caputo or general temporal Riemann-Liouville fractional order derivatives, are objective. The obtained results show the compatibility of the general temporal Caputo and general temporal Riemann-Liouville fractional order derivatives with the understanding of the “measured time” evolution. In the same time these results reveal that, the objectivity violation, when integer order temporal derivatives are replaced by classic temporal Caputo or classic temporal Riemann-Liouville fractional order derivatives, is originated in the incompatibility of the classic fractional order derivatives, with the understanding of the “measured time” evolution.