Anomalous diffusion with ballistic scaling: A new fractional derivative

Anomalous diffusion with ballistic scaling is characterized by a linear spreading rate with respect to time that scales like pure advection. Ballistic scaling may be modeled with a symmetric Riesz derivative if the spreading is symmetric. However, ballistic scaling coupled with a skewness is observed in many applications, including hydrology, nuclear physics, viscoelasticity, and acoustics. The goal of this paper is to find a governing equation for anomalous diffusion with ballistic scaling and arbitrary skewness. To address this problem, we propose a new operator called the Zolotarev derivative, which is valid for all orders 0 < α ≤ 2. The Fourier symbol of this operator is related to the characteristic function of a stable random variable in the Zolotarev M parameterization. In the symmetric case, the Zolotarev derivative reduces to the well-known Riesz derivative. For α ̸= 1, the Zolotarev derivative is a linear combination of Riemann-Liouville fractional derivatives and a first derivative. For α = 1, the Zolotarev derivative is a non-local operator that models ballistic anomalous diffusion. We prove that this operator is continuous with respect to α. We derive generator, Caputo, and Riemann-Liouville forms of this operator and provide two examples. The solution of diffusion equations utilizing the Zolotarev derivative with an impulse initial condition are shifted and scaled stable densities in the M parameterization.