Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion

Abstract.

There have been numbers of conflicting and confusing situations, but also uniformity, in the application of the two most popular fractional derivatives, namely the classic Riemann-Liouville and Caputo fractional derivatives. The range of these issues is wide, including the initialization with the Caputo derivative and its observed difficulties compared to the Riemann-Liouville initialization conditions. In this paper, being aware of these issues and reacting to the newly introduced Caputo-Fabrizio fractional derivative (CFFD) without singular kernel, we introduce a new definition of fractional derivative called the new Riemann-Liouville fractional derivative (NRLFD) without singular kernel. The filtering property of the NRLFD is pointed out by showing it as the derivative of a convolution and contrary to the CFFD, it matches with the function when the order is zero. We also explore various scientific situations that may be conflicting and confusing in the applicability of both new derivatives. In particular, we show that both definitions still have some basic similarities, like not obeying the traditional chain rule. Furthermore, we provide the explicit formula for the Laplace transform of the NRLFD and we prove that, contrary to the CFFD, the NRLFD requires non-constant initial conditions and does not require the function f to be continuous or differentiable. Some simulations for the NRLFD are presented for different values of the derivative order. In the second part of this work, numerical approximations for the first- and second-order NRLFD are developped followed by a concrete application to diffusion. The stability of the numerical scheme is proved and numerical simulations are performed for different values of the derivative order $\alpha$. They exhibit similar behavior for closed values of $\alpha$.