Inverse Mittag-Leffler Function Research Articles

Extensions of fractional cumulative residual entropy with applications

Recently, Zhan and Shang (2021) proposed a modification of fractional entropy and proved some properties based on the inverse Mittag-Leffler function (MLF). In this article, we introduce extensions of fractional cumulative residual entropy (FCRE). Our results contain bivariate version of extended FCRE, linear transformation, bounds, stochastic ordering, and some properties of its dynamic version. We also study on the fractional cumulative residual mutual information and the conditional extended FCRE. Finally, we propose an estimator of extended FCRE using empirical approach. We establish a central limit theorem for the empirical extended FCRE under the exponential distribution. Additionally, the validity of this new measure is supported by numerical simulations on logistic map.

Ultrafast dynamics modeling via fractional Brownian motion run with Mittag-Leffler clock in porous media

Ultrafast diffusion process characterized by unusually large diffusivities is often occurs on porous media and the mean square displacement grows exponentially in time. This paper clarifies the characteristics of ultrafast diffusion and tackles this perplexing problem using the fractional Brownian motion run with a nonlinear clock model. We employ the Mittag-Leffler function as the nonlinear clock, the increments are dependent and obey Gaussian distribution, and the derived corresponding mean square displacement is more widely than the exponential function. A comparison between the power law model and the proposed model with respect to available experimental data verifies that the proposed model is more effective and accurate. Ultraslow diffusion is also studied with the inverse Mittag-Leffler function as the nonlinear clock. The results show that it can capture the ultraslow diffusion process better than the case of the logarithmic model. As the generalization of fractional Brownian motion, fractional Brownian motion run with a nonlinear clock is an alternative model method for extreme anomalous diffusion in complex systems.

Characterizing the Ultra-Slow Creep in Concrete Based on the Non-Local Structural Derivative Maxwell Model

Creep of concrete can last for decades, which displays the ultra-slow rheological phenomena. As an empirical formula, the logarithmic law is usually used to describe the ultra-slow creep. However, the logarithmic law does not always work well especially for the long-term creep. And its corresponding relaxation response cannot be obtained analytically. It is known that the Mittag-Leffler and the inverse Mittag-Leffler functions are generalized from the exponential and the logarithmic functions, respectively. And the inverse Mittag-Leffler function is much slower and generalized than the logarithmic function. In this paper, we use the non-local structural derivative to establish a new Maxwell-type viscoelastic model, in which the inverse Mittag-Leffler function is employed as the structural function to capture the ultra-slow rheology. The viscoelastic responses of the generalized Maxwell model are analytically derived, including creep and relaxation. Moreover, the proposed model is tested by three long-term creep experimental data of concrete. Compared with the standard linear solid model, the Lomnitz logarithmic creep model and the fractional Maxwell model, the proposed generalized Maxwell model agrees better with the experimental data. Numerical results indicate that the non-local structural derivative involving the inverse Mittag-Leffler function is feasible to capture the ultra-slow creep in concrete.

Non-local structural derivative Maxwell model for characterizing ultra-slow rheology in concrete

Ultra-slow rheological phenomena have widely been observed in engineering materials. The logarithmic law is normally used to describe the slow rheology, but it does not work well for the long-term ultra-slow rheology. In this paper, we devise a new Maxwell-type viscoelastic model to capture the ultra-slow rheology by using the non-local structural derivative, where the inverse Mittag-Leffler (ML) structural function is adopted. The viscoelastic responses of the ultra-slow Maxwell model are analytically derived, including creep and relaxation. The logarithmic creep law can be regarded as a special case of the ultra-slow Maxwell model. In addition, the proposed model is tested by several experimental data of concrete. Compared with the existed models, the present ultra-slow Maxwell model shows the reasonable accuracy. The derived results indicate that the non-local structural derivative involving the inverse ML function is feasible to capture the ultra-slow rheology of concrete.

Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function

Abstract This study analyzes the complexity of ultraslow diffusion process using both the classical Shannon entropy and its general case with inverse Mittag-Leffler function in conjunction with the structural derivative. To further describe the observation process with information loss in ultraslow diffusion, e.g., some defects in the observation process, two definitions of fractional entropy are proposed by using the inverse Mittag-Leffler function, in which the Pade approximation technique is employed to numerically estimate the diffusion entropy. The results reveal that the inverse Mittag-Leffler tail in the propagator of the ultraslow diffusion equation model adds more information to the original distribution with larger entropy. Smaller value of α in the inverse Mittag-Leffler function indicates more complicated of the underlying ultraslow diffusion and corresponds to higher value of entropy. The proposed definitions of fractional entropy can serve as candidates to capture the information loss in ultraslow diffusion.

Continuous time random walk model with asymptotical probability density of waiting times via inverse Mittag-Leffler function

The mean squared displacement (MSD) of the traditional ultraslow diffusion is a logarithmic function of time. Recently, the continuous time random walk model is employed to characterize this ultraslow diffusion dynamics by connecting the heavy-tailed logarithmic function and its variation as the asymptotical waiting time density. In this study we investigate the limiting waiting time density of a general ultraslow diffusion model via the inverse Mittag-Leffler function, whose special case includes the traditional logarithmic ultraslow diffusion model. The MSD of the general ultraslow diffusion model is analytically derived as an inverse Mittag-Leffler function, and is observed to increase even more slowly than that of the logarithmic function model. The occurrence of very long waiting time in the case of the inverse Mittag-Leffler function has the largest probability compared with the power law model and the logarithmic function model. The Monte Carlo simulations of one dimensional sample path of a single particle are also performed. The results show that the inverse Mittag-Leffler waiting time density is effective in depicting the general ultraslow random motion.

New methodologies in fractional and fractal derivatives modeling

This paper surveys the latest advances of the first author's group on the three new methodologies of fractional and fractal derivatives modeling to meet the increasing and challenging demands in scientific and engineering communities. Firstly, the structural fractal was proposed as a generalization of the Euclidean distance. Using the structural metric, the structural derivative approach was derived as a significant extension of the global fractional calculus and the local fractional derivative approaches to tackle the perplexing modeling problems. The classical derivative describes the change rate of a certain physical variable with respect to time or space, which rarely takes into account the significant influence of mesoscopic time-space metric of a complex system on its physical behaviors. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space structural metric of physical systems. Secondly, we employed the fundamental solution or probability density function of statistical distribution which can describe the problem of interest to construct the implicit calculus governing equation. The ‘implicit’ suggests that the explicit calculus expression of this governing equation is difficult to derive and not required. The fundamental solution or potential function of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. We call this strategy the implicit calculus equation modeling. Thirdly, based on the implicit calculus equation modeling approach, we introduced the concept of fundamental solution on fractal and consequently defined the fractal differential operator to describe various mechanical behaviors of fractal materials. Fractal calculus operator significantly extends the application scope of the classical calculus modeling approach under the framework of continuum mechanics. This is also a step-forward advance of the fractal derivative proposed earlier by the first author. To demonstrate the structural derivative application, we applied the inverse Mittag-Leffler function as the structural function to model ultraslow diffusion of a random system of two interacting particles. On the other hand, this paper uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the power law behaviors of steady heat conduction in multiple phase material. Finally, by using the singular boundary method, we made numerical simulation of the fractal Laplacian equation for phenomenological modeling potential problems in fractal media. Numerical experiments show that all the three new methodologies are feasible mathematical tools to describe complex physical behaviors.

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.

Experimental evaluation of viscous damping coefficient in the fractional underdamped oscillator

In this article, we obtain the viscous damping coefficient β theoretically and experimentally in the spring–mass–viscodamper system. The calculation is performed to obtain the quasi-period τ. The influence of the viscosity of the fluid and the damping coefficient is analyzed using three fluids, water, edible oil, and gasoline engine oil SAE 10W-40. These processes exhibit temporal fractality and non-local behaviors. Our general non-local damping model incorporates fractional derivatives of Caputo type in the range [Formula: see text], and the viscous damping coefficients are determined in terms of the inverse Mittag-Leffler function. The classical models are recovered when the order of the fractional derivatives is equal to 1.

Finite Series Representation of the Inverse Mittag-Leffler Function

The inverse Mittag-Leffler functionEα,β-1zis valuable in determining the value of the argument of a Mittag-Leffler function given the value of the function and it is not an easy problem. A finite series representation of the inverse Mittag-Leffler function has been found for a range of the parametersαandβ; specifically,0<α<1/2forβ=1and forβ=2. This finite series representation of the inverse Mittag-Leffler function greatly expedites its evaluation and has been illustrated with a number of examples. This represents a significant advancement in the understanding of Mittag-Leffler functions.