Mittag-Leffler Law Research Articles

New numerical analysis of Riemann-Liouville time-fractional Schrödinger with power, exponential decay, and Mittag-Leffler laws

New numerical analysis of Riemann-Liouville time-fractional Schrödinger with power, exponential decay, and Mittag-Leffler laws

Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system

New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.

Modelling groundwater fractal flow with fractional differentiation via Mittag-Leffler law

Modelling the flow of groundwater within a network of fractures is perhaps one of the most difficult exercises within the field of geohydrology. This physical problem has attracted the attention of several scientists across the globe. Already two different types of differentiations have been used to attempt modelling this problem including the classical and the fractional differentiation. In this paper, we employed the most recent concept of differentiation based on the non-local and non-singular kernel called the generalized Mittag-Leffler function, to reshape the model of groundwater fractal flow. We presented the existence of positive solution of the new model. Using the fixed-point approach, we established the uniqueness of the positive solution. We solve the new model with three different numerical schemes including implicit, explicit and Crank-Nicholson numerical methods. Experimental data collected from four constant discharge tests conducted in a typical fractured crystalline rock aquifer of the Northern Limb (Bushveld Complex) in the Limpopo Province (South Africa) are compared with the numerical solutions. It is worth noting that the four boreholes (BPAC1, BPAC2, BPAC3, and BPAC4) are located on Faults.

Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws

The nature is very complex to model with mathematical equations. Some physical problems found in nature could follow the power law; other could follow the Mittag–Leffler law and other the exponential decay law. On the other hand one could observe in nature a physical problem that combines both, it is therefore important to provide a new fractional operator that could possibly be used to model such physical problem. In this paper, we suggest a fractional operator exponential-Mittag–Leffler kernel with two fractional orders. Some very useful properties are obtained. Numerical solutions were obtained for three examples proposed.

New insight in fractional differentiation: power, exponential decay and Mittag-Leffler laws and applications

Some physical problems found in nature can follow the power law; others can follow the Mittag-Leffler law and others the exponential decay law. On the other hand, one can observe in nature a physical problem that combines the three laws, it is therefore important to provide a new fractional operator that could possibly be used to model such physical problem. In this paper, we suggest a fractional operator power-law-exponential-Mittag-Leffler kernel with three fractional orders. Some very useful properties are obtained. Numerical solutions were obtained for three examples proposed. The results show that the new fractional operators are powerful mathematical tools to model complex problems.

Generalized groundwater plume with degradation and rate-limited sorption model with Mittag-Leffler law

Abstract The concept of differentiation with the generalized Mittag-Leffler law is used in this paper to construct the model of movement of groundwater pollution with degradation and limited sorption. The fractional differentiation used in the model is in Riemann-Liouville sense. The new model is solved analytically using the Green Laplace transform approach. A numerical scheme is used to obtain the numerical solution of the modified model.

Moment convergence of first-passage times in renewal theory

Let ξ1,ξ2,… be independent copies of a positive random variable ξ, S0=0, and Sk=ξ1+…+ξk, k∈N. Define N(t)=inf{k∈N:Sk>t} for t≥0. The process (N(t))t≥0 is the first-passage time process associated with (Sk)k≥0. It is known that if the law of ξ belongs to the domain of attraction of a stable law or P(ξ>t) varies slowly at ∞, then N(t), suitably shifted and scaled, converges in distribution as t→∞ to a random variable W with a stable law or a Mittag-Leffler law. We investigate whether there is convergence of the power and exponential moments to the corresponding moments of W. Further, the analogous problem for first-passage times of subordinators is considered.

Number of first-passage times as a measurement of information for weakly chaotic systems.

We consider a general class of maps of the interval having Lyapunov subexponential instability |δxt|∼|δx0|exp[Λt(x0)ζ(t)], where ζ(t) grows sublinearly as t→∞. We outline here a scheme [J. Stat. Phys. 154, 988 (2014)] whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate ζ(t). This matching approach is based on the infinite measure property of such systems. We show that the average information that is necessary to record without ambiguity a trajectory of the system tends to 〈Λ〉ζ(t), suitably extending the Kolmogorov-Sinai entropy and Pesin's identity. For such systems, information behaves like a random variable for random initial conditions, its statistics obeying a universal Mittag-Leffler law. We show that, for individual trajectories, information can be accurately inferred by the number of first-passage times through a given turbulent phase-space cell. This enables us to calculate far more efficiently Lyapunov exponents for such systems. Lastly, we also show that the usual renewal description of jumps to the turbulent cell, usually employed in the literature, does not provide the real number of entrances there. Our results are supported by exhaustive numerical simulations.

Thermal expansion effects and heat conduction in granular materials

In this paper, we report results and analysis on a simulation study of the effects of thermal expansion in granular systems. We show that these effects impact the force distribution inside a two-dimensional system of disks that are subject to thermal heating under two different boundary conditions. A significant increase in the average force is observed for steel particles confined within a box with fixed walls at temperature rises of 50 degrees C and 100 degrees C, respectively. As previously noted in the literature, thermal expansion also induces compaction. The results show that a systematic and controllable increase in granular packing can be induced by simply raising and then lowering the temperature, without the input of mechanical energy in agreement with previous experimental observations. We find that the evolution of the packing fraction is well described by a fractional relaxation model, which follows the Mittag-Leffler law.

On the Geometric Mittag-Leffler Distributions

As a generalization of geometric exponential distribution, geometric Mittag-Leffler distribution is studied. It is shown that the geometric Mittag- Leffler is the limit of geometric sums of Quasi Factorial gamma random variables. It can he seen that the geometric Mittag-Leffier distribution is geometrically infinitely divisible. As a generalization of the geometric Mittag-Leffler distribution, geometric Quasi Factorial gamma distributions are introduced and studied. It is shown that geometric sum of geometric Mittag-Leffler has geometric Quasi Factorial gamma distribution. Autoregressive process with geometric Mittag-Leffler margiuals are introduced and studied. Generalization to higher order autoregressive process is also done. A new class of discrete distributions related to discrete Mittag-Leffler laws, namely discrete geometric Mittag-Leffler distribution is introduced and its properties are studied.