StructuRal Derivative Based on Inverse Mittag-Leffler Function for Modeling Ultraslow Diffusion

Abstract

This paper proposes a novel structural derivative approach to tackle the perplexing modeling problem of ultraslow diffusion. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space fabric of physical systems. Ultraslow diffusion has been observed in numerous lab experiments and field observations, whose behaviors deviate dramatically from the standard anomalous diffusion models characterizing power function of time. The logarithmic diffusion model has since been used to describe bizarre process of ultraslow diffusion but with very limited success. This study applies the inverse Mittag-Leffler function as the structural function in the structural derivative modeling ultraslow diffusion of a random system of two interacting particles. It is observed that the dynamics of two interacting particles are respectively the ballistic motion at the short time scale and the Sinai ultraslow diffusion at the long time scale. Compared with the logarithmic diffusion model, the inverse Mittag-Leffler diffusion model has higher accuracy and manifests clearer physical mechanism. Numerical experiments show that the structural derivative is a feasible mathematical tool to model the ultraslow diffusion using the inverse Mittag-Leffler function as its structural function.

A spatial structural derivative model for ultraslow diffusion

“…In recent years, the Mittage-Leffler function has widely been used in the fractal dynamics, anomalous diffusion and fractal random field [27][28][29]. In addition, the inverse Mittage-Leffler function as has also been applied to describe ultraslow diffusion [25]. In further study, we will try to investigate different structural functions with clear physical mechanism, such as Mittag-Leffler function and its inverse function, to construct both local and global structural derivative diffusion equation in modeling non-Gaussian motion.…”

Relative Entropy Method Applied for the Fatigue Life Distribution of Carbon Fiber/Epoxy Composites

“…In this study, we would like to investigate the feasibility of the M-L distribution in fitting the fatigue data based on the relative entropy method [ 17 ]. Nowadays, the M-L distribution has been applied as a novelty statistical tool to describe non-exponential statistical phenomena in diverse fields [ 18 , 19 ], such as bridge fatigue life assessment [ 12 ] and modeling of an anomalous diffusion with hereditary effects for the importance of the M-L function in the fractional calculus [ 20 , 21 ]. We choose the M-L distribution as a tool to describe the distribution of fatigue data, since it has an apparent hereditary effect and power decay or heavy-tailed traits [ 22 , 23 , 24 , 25 , 26 , 27 ].…”

Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media

“…However, many theoretical, experimental and field results show that the Einstein relationship is not valid for particles diffusion in heterogeneous porous media. The MSD becomes a nonlinear function, such as power law form used to describe the anomalous diffusion [9], the logarithmic form [10] and the inverse Mittag-Leffler form used to characterize the ultraslow diffusion [11], i.e., superslow diffusion [12] and more complicated cases with more than one exponent to describe multi-scaling behaviors [13][14].…”

“…The distributed order fractional derivative diffusion model is also a kind of popular models to depict multi-scale non-Fickian transport dynamics, and its order is a time or space derivative order independent function [19]. The structural derivative diffusion model is proposed to explore the physical mechanism of the ultraslow diffusion, which diffuses even more slowly than the sub-diffusion [11]. The structural function used in the definition of the structural derivative diffusion equation determines the ultraslow dynamics evolution.…”

“…The fundamental solution of the Hausdorff derivative diffusion model is a stretched Gaussian distribution [22]. It should be pointed out that the Hausdorff derivative is a special case of the local structural derivative, when the structural function is a power law function [11]. In this study, the distributed order Hausdorff derivative model is proposed to describe the tracer transport in porous media.…”