Fractional derivative defined by non-singular kernels to capture anomalous relaxation and diffusion

TL;DR

This paper introduces a new fractional derivative model with a stretched exponential kernel that better captures a wide range of anomalous relaxation and diffusion behaviors compared to previous models with exponential kernels.

Contribution

The study proposes a novel fractional derivative with a stretched exponential kernel, extending previous models and improving the characterization of complex anomalous diffusion processes.

Findings

The exponential kernel-based model cannot accurately describe non-exponential dynamics.

The stretched exponential kernel model can describe a broader spectrum of anomalous diffusion.

Numerical tests confirm the improved applicability of the new fractional derivative.

Abstract

Anomalous relaxation and diffusion processes have been widely characterized by fractional derivative models, where the definition of the fractional-order derivative remains a historical debate due to the singular memory kernel that challenges numerical calculations. This study first explores physical properties of relaxation and diffusion models where the fractional derivative was defined recently using an exponential kernel. Analytical analysis shows that the fractional derivative model with an exponential kernel cannot characterize non-exponential dynamics well-documented in anomalous relaxation and diffusion. A legitimate extension of the previous fractional derivative is then proposed by replacing the exponential kernel with a stretched exponential kernel. Numerical tests show that the fractional derivative model with the stretched exponential kernel can describe a much wider range of anomalous diffusion than the exponential kernel, implying the potential applicability of the new fractional derivative in quantifying real-world, anomalous relaxation and diffusion processes.