In the last decade, theoretical and applied studies were done in order to provide a suitable definition of fractional derivative, which meets all the requirement of a derivative in its primary sense. ...It was concluded by some eminent researchers that the Riemann‐Liouville version was the most suitable. However, many numerical approximation of fractional derivative were done with Caputo version. This paper addresses the numerical approximation of fractional differentiation based on the Riemann‐Liouville definition, from power‐law kernel to generalized Mittag‐Leffler‐law via exponential‐decay‐law.
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To answer some issues raised about the concept of fractional differentiation and integration based on the exponential and Mittag-Leffler laws, we present, in this paper, fundamental differences ...between the power law, exponential decay, Mittag-Leffler law and their possible applications in nature. We demonstrate the failure of the semi-group principle in modeling real-world problems. We use natural phenomena to illustrate the importance of non-commutative and non-associative operators under which the Caputo-Fabrizio and Atangana-Baleanu fractional operators fall. We present statistical properties of generator for each fractional derivative, including Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu ones. The Atangana-Baleanu and Caputo-Fabrizio fractional derivatives show crossover properties for the mean-square displacement, while the Riemann-Liouville is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the Caputo Fabrizio kernel has a steady state between the transition. Only the Atangana-Baleanu kernel is a crossover for the waiting time distribution from stretched exponential to power law. A new criterion was suggested, namely the Atangana-Gómez fractional bracket, that helps describe the energy needed by a fractional derivative to characterize a 2-pletic manifold. Based on these properties, we classified fractional derivatives in three categories: weak, mild and strong fractional differential and integral operators. We presented some applications of fractional differential operators to describe real-world problems and we proved, with numerical simulations, that the Riemann-Liouville power-law derivative provides a description of real-world problems with much additional information, that can be seen as noise or error due to specific memory properties of its power-law kernel. The Caputo-Fabrizio derivative is less noisy while the Atangana-Baleanu fractional derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state. The study suggests that the properties of associativity and commutativity or the semi-group principle are just irrelevant in fractional calculus. Properties of classical derivatives were established for the ordinary calculus with no memory effect and it is a failure of mathematical investigation to attempt to describe more complex natural phenomena using the same notions.
•Semigroup principle failures to capture natural phenomena.•The future of modeling real world problem relies on fractional differential operators with non-index law property.•Atangana–Baleanu ...fractional differential operators are convolution of Riemann–Liouville–Caputo derivative with the Mittag–Leffler function.•Index law is not valid in fractional differentiation.
Recently fractional differential operators with non-index law properties have being recognized to have brought new weapons to accurately model real world problems particularly those with non-Markovian processes. This present paper has two double aims, the first was to prove the inadequacy and failure of index law fractional calculus and secondly to show the application of fractional differential operators with no index law properties to statistic and dynamical systems. To achieve this, we presented the historical construction of the concept of fractional differential operators from Leibniz to date. Using a matrix based on the fractional differential operators, we proved that, fractional operators obeying index law cannot model real world problems taking place in two states, more precisely they cannot describe phenomena taking place beyond their boundaries, as they are scaling invariant, more precisely our results show that, mathematical models based on these differential operators are not able to describe the inverse memory, meaning the full history of a physical problem cannot be described accurately using these derivatives with index law properties. On the other hand, we proved that, differential operators with no index-law properties are scaling variant, thus can describe situations taking place in different states and are able to localize the frontiers between two states. We present the renewal process properties included in differential equation build out of the Atangana–Baleanu fractional derivative and counting process, which is connected to its inter-arrival time distribution Mittag–Leffler distribution which is the kernel of these derivatives. We presented the connection of each derivative to a statistical family, for instance Riemann–Liouville–Caputo derivatives are connected to the Pareto statistic, which has no well-defined average when alpha is less than 1 corresponding to the interval where fractional operators mostly defined. We established new properties and theorem for the Atangana–Baleanu derivative of an analytic function, in particular we proved that, they are convolution of the Mittag–Leffler function with the Riemann–Liouville–Caputo derivatives. To see the accuracy of the non-index law derivative to in modeling real chaotic problems, 4 examples were considered, including the nine-term 3-D novel chaotic system, King Cobra chaotic system, the Ikeda delay system and chaotic chameleon system. The numerical simulations show very interesting and novel attractors. The king cobra system with the Atangana–Baleanu presented a very novel attractor where at the earlier time we observed a random walk and latter time we observed the real sharp of the cobra. The Ikeda model with Atangana–Baleanu presented different attractors for each value of fractional order, in particular we obtain a square and circular explosions. The results obtained in this paper show that, the future of modeling real world problem relies on fractional differential operators with non-index law property. Our numerical results showed that, to not model physical problems with fractional differential operators with non-singular kernel and imposing index law in fractional calculus is rightfully living with closed eyes without ever taking a risk to open them.
This paper is devoted to investigation of the fractional order fuzzy dynamical system, in our case, modeling the recent pandemic due to corona virus (COVID-19). The considered model is analyzed for ...exactness and uniqueness of solution by using fixed point theory approach. We have also provided the numerical solution of the nonlinear dynamical system with the help of some iterative method applying Caputo as well as Attangana-Baleanu and Caputo fractional type derivative. Also, random COVID-19 model described by a system of random differential equations was presented. At the end we have given some numerical approximation to illustrate the proposed method by applying different fractional values corresponding to uncertainty.
•A new fractional order learning algorithm is proposed in this paper.•The proposed fractional order neural network (neural network trained by the proposed fractional order learning algorithm) leads ...to accurate and simple system identification models.•On the three different systems that were identified, the proposed fractional order neural network reaches the best accuracy with less number of parameters.
Neural networks and fractional order calculus have shown to be powerful tools for system identification. In this paper we combine both approaches to propose a fractional order neural network (FONN) for system identification. The learning algorithm was generalized considering the Grünwald-Letnikov fractional derivative. This new black box modeling approach is validated by the identification of three different systems (two benchmark systems and a real system). Comparisons vs others approaches showed that the proposed FONN model reached better accuracy with less number of parameters.
This article investigates the effects of magnetite
Fe
3
O
4
nanoparticles on free convection flow of nanofluid with magnetohydrodynamics. The magnetite
Fe
3
O
4
nanoparticles that have been dispersed ...in water are taken as a conventional base fluid. In order to compare newly fractional derivatives, the governing equations have been fractionalized via Atangana–Baleanu and Caputo–Fabrizio fractional operators. The resulting partial differential equations are solved by employing Laplace transforms. Exact solutions have been investigated for temperature and velocity field via Atangana–Baleanu and Caputo–Fabrizio fractional operators and then expressed in Mittag–Leffler function
M
β
,
γ
y
W
and
M
-function
M
q
p
W
. The enhancement of heat transfer and effects in the natural convection flows are analyzed graphically by Atangana–Baleanu and Caputo–Fabrizio fractional operators. Graphical comparison has been depicted via Atangana–Baleanu and Caputo–Fabrizio derivatives for four types of models, i.e., (1) fractionalized nanofluid with magnetic field, (2) ordinary nanofluid with magnetic field, (3) fractionalized nanofluid without magnetic field and (4) ordinary nanofluid without magnetic field on fluid flows.
Amyotrophic lateral sclerosis (ALS) is the most serious form of degenerative motor neuron disease in adults, characterized by upper and lower motor neuron degeneration, skeletal muscle atrophy, ...paralysis, and death. High prevalence of malnutrition and weight loss adversely affect quality of life. Moreover, two thirds of patients develop a hypermetabolism of unknown cause, leading to increased resting energy expenditure. Inasmuch as lipids are the major source of energy for muscles, we determined the status of lipids in a population of patients with ALS and investigated whether lipid contents may have an impact on disease progression and survival.
Blood concentrations of triglycerides, cholesterol, low-density lipoprotein (LDL), and high-density lipoprotein (HDL) were measured in a cohort of 369 patients with ALS and compared to a control group of 286 healthy subjects. Postmortem histologic examination was performed on liver specimens from 59 other patients with ALS and 16 patients with Parkinson disease (PD).
The frequency of hyperlipidemia, as revealed by increased plasma levels of total cholesterol or LDL, was twofold higher in patients with ALS than in control subjects. As a result, steatosis of the liver was more pronounced in patients with ALS than in patients with PD. Correlation studies demonstrated that bearing an abnormally elevated LDL/HDL ratio significantly increased survival by more than 12 months.
Hyperlipidemia is a significant prognostic factor for survival of patients with amyotrophic lateral sclerosis. This finding highlights the importance of nutritional intervention strategies on disease progression and claims our attention when treating these patients with lipid-lowering drugs.
We introduce a novel anomaly search method based on (i) jet tagging to select interesting events, which are less likely to be produced by background processes; (ii) comparison of the untagged and ...tagged samples to single out features (such as bumps produced by the decay of new particles) in the latter. We demonstrate the usefulness of this method by applying it to a final state with two massive boosted jets: for the new physics benchmarks considered, the signal significance increases an order of magnitude, up to a factor of 40. We compare to other anomaly detection methods in the literature and discuss possible generalisations.
Neuronal networks are used in different fields of science and technology due to their capacity to approximate nonlinear functions through the synaptic weights optimization. This work shows a new form ...of optimization for neuronal networks based in fractional calculus. The fractional adaptation algorithm proposed was used to identify mechanical, electrical and biological systems. In each of the experiments a comparison between the proposed fractal-fractional model and the conventional model (with derivation order equal to one) was made.
In this paper, we obtain analytical solutions for the fractional cubic isothermal auto-catalytic chemical system with Caputo–Fabrizio and Atangana–Baleanu fractional time derivatives in ...Liouville–Caputo sense. We utilize the q-homotopy analysis transform method to compute the approximate solutions. We find the optimal values of h so we assure the convergence of the approximate solutions. Finally, we compare our results numerically with the finite difference method and excellent agreement is found.
•New Homotopy Analysis Transform Method (HATM) to compute the approximate solutions of the fractional cubic isothermal autocatalytic chemical system equation using the two fractional derivatives in Liouville–Caputo sense is applied.•The high accuracy and effectiveness are shown by computing the residual error function and average residual error.•The proposed method is efficient and general.
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