Fractional Derivatives with Mittag-Leffler Kernel

Atangana and Baleanu introduced a derivative with fractional order to answer some outstanding questions that were posed by many investigators within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. Therefore, we apply the reproducing kernel method to fractional differential equations with non-local and non-singular kernel. In this work, a new method has been developed for the newly established fractional differentiation. Examples are given to illustrate the numerical effectiveness of the reproducing kernel method when properly applied in the reproducing kernel space. The comparison of approximate and exact solutions leaves no doubt believing that the reproducing kernel method is very efficient and converges toward exact solution very rapidly.

Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this chapter, the necessary and sufficient optimality conditions for systems involving Atangana–Baleanu’s derivatives are discussed. The fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems that contains a fractional Atangana–Baleanu’s derivatives are investigated. The fractional contains both the fractional derivatives and the fractional integrals in the sense of Atangana–Baleanu. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP.

We introduce new fractional operators of variable order in isolated time scales with Mittag–Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler–Lagrange type.

In this chapter, a fractional epidemic model for the leptospirosis disease with Atangana–Baleanu (AB) derivative is formulated. Initially, we present the model equilibria and basic reproduction number. The local stability of disease free equilibrium point is proved using fractional Routh Harwitz criteria. The Picard–Lindelof method is applied to show the existence and uniqueness of solutions for the model. A numerical scheme using Adams–Bashforth method for solving the proposed fractional model involving the AB derivative is presented. Finally, numerical simulations are performed in order to validate the importance of the arbitrary order derivative. The numerical result shows that the fractional order plays an important role to better understand the dynamics of disease.

The concentration of alcohol in blood differs with vessel diameter (arterial diameter). In case of arteries having thinner diameter, alcohol concentrates around their walls because of Fahraeus–Lindqvist effect. The fluctuating concentration of alcohol in blood directly affects normal human body functions causing peptic ulcer and hypertension. In this work, we made the comparative analysis of blood alcohol model via Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. The governing ordinary differential equations of blood alcohol model have been converted in terms of non-integers order derivatives. The analytic calculations of the concentrations of alcohol in stomach \((C_1(t))\) and the concentrations of alcohol in the blood \((C_2(t))\) have been investigated by applying Laplace transform method. The general solutions of the concentrations of alcohol in stomach \((C_1(t))\) and the concentrations of alcohol in the blood \((C_2(t))\) are expressed in the terms of wright function \(\varPhi (a,b;c)\). The graphs of both types of concentrations are depicted on the basis of fractional parameters of Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Finally, the comparative analysis of both fractional types of concentration of alcohol level in blood decay faster for higher fractional order.

In this chapter, a meta-heuristic optimization algorithm, called cuckoo search algorithm is applied to determine the optimal parameters of the fractional Gompertz model via Liouville–Caputo and Atangana–Baleanu–Caputo fractional derivatives. The numerical solutions of the proposed models were obtained using the Adams method. The proposed methodology is tested on epidemiological examples. In the interval considered, the fractional models had the best fit for the epidemiological data considered. The effectiveness of the methodology is shown by a comparison with the classical models. A comparison between the fractional models and the classical models was carried out to show the effectiveness of our methodology.

The Atangana–Baleanu fractional differential and integral operators have been used in this chapter to describe the crossover behavior of a chaotic complex system. The existing model was extended and modified by replacing the conventional time local operator by the fractional differential operator with non-local and non-singular kernel. We established the conditions under which the existence of a uniquely exact solution can be found. A newly established numerical scheme was used to solve the modified model and numerical solutions are displayed for different values of fractional order.

Until the neurologists J.L. Hindmarsh and R.M. Rose improved the Hodgkin–Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh–Rose neurons with external current input. Combining with fractional differentiation, the model is generalized with introduction of an additional parameter, the non-integer order of the derivative \(\sigma \) and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamic show that in the standard case the control parameter is \(\sigma =1,\) the nerve cell’s behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as \(\sigma \) decreases (\(\sigma =0.8\) and \(\sigma =0.5\)) with the pole of attraction becoming chaotic.

Many mathematical models describing dynamics that occur in autocatalytic reaction networks have been proved to be chaotic, exhibiting orbits with unpredictable outcomes. Is it always possible to modulate that chaos? We use Haar wavelet numerical method to investigate a fractional system modeling autocatalytic reaction networks, where particular attention is made on biochemical systems of four-component networks. The convergence of the method is detailed through error analysis. Graphical representations reveal that the dynamic of the whole system is characterized by limit-cycles followed by period-doubling bifurcations that culminate with chaos, depending on the change of the total concentration of cofactor. The behavior of the system becomes more unpredictable as the concentration of cofactor increases, but the phenomenon is shown to be regulated by an additional parameter, the order of the fractional derivative \(\gamma ,\) which plays an important role in triggering and controlling the appearance of chaos. Moreover, the chaotic behavior observed in the cascade diagram of the pure fractional case is proven to appear earlier, showing that the parameter \(\gamma \) is a valuable tool to regulate the chaos observed in some biochemical systems.

After the finding of the compound structure for standard chaotic attractors, the main concern was related to how to regulate such a fascinating dynamic. Hence, the question about the existence of a compound structure for chaotic attractors generated by fractional systems was raised. In this work, we investigate the existence of compound structure of a chaotic attractor generated from a Atangana–Baleanu fractional system where two cases are studied: the integer case and the fractional one. The model is first solved numerically thanks to the Haar Wavelets scheme whose convergence is proved via error analysis. Numerical simulations are performed and clearly reveal the existence of the desired compound structure in both cases and characterized by the generation of a left-attractor seen as the reflection of a right attractor through the mirror operation. Moreover, those two simple attractors can always be combined together to form the resulting chaotic attractor. The mechanism of forming those simple attractors is shown and leads to a bounded partial attractor. Furthermore, that same mechanism appears to be strongly dependent on two parameters, the model parameter u and the Atangana–Baleanu derivative with order \(\alpha ,\) important in controlling the systems. It is observed that, in the fractional case (\(\alpha =0.9\)), the period-doubling bifurcations start at a higher value of u compared to the integer case (\(\alpha =1\)).

The constructions of physically adequate forms of the diffusion equation with implementation of the Atangana–Baleanu derivative with Mittag-Leffler exponential kernel have been discussed. The specific form of the corresponding Atangana–Baleanu integral relates it directly to the fading memory concept, following the Boltzmann linear superposition principle with the standard Riemann-Liouville integral as the time-fading term. This approach relates the new fractional operators with non-singular kernel to the classical Riemann-Liouville integral. Using the concept of the fading memory and the specific form of the Atangana–Baleanu integral three forms of the diffusion equation have been investigated. The adequate definition of the flux to gradient relationship has been the main focus of the study resulting in two physically adequate formulations of the diffusion equation. The direct (formalistic) fractionalization of the classical diffusion equation results in physically inadequate relationships.

Nowadays, a lot of researchers have challenged the use of classical diffusion equation to model real life situations. To circumvent some of the up-roaring challenges, time and space fractional derivatives have been proposed as alternative to model some anomalous diffusion or related processes where a particle plume spreads at inconsistent rate with the classical Brownian motion model. In this work, we shall consider the general diffusion equations with fractional order derivatives which describe the diffusion in complex systems. Fractional diffusion equation is obtained by allowing the exponent order \(\alpha \) to vary in the intervals (0, 1) and (1, 2) which correspond to subdiffusion and superdiffusion special cases. For the numerical approximations, we propose to use the newly correct version of the Adams-Bashforth scheme which takes into account the nonlinearity of the kernels such as the Mittag-Leffler law for the Atangana-Baleanu case, the power law for the Riemann-Liouville and Caputo derivatives. The efficiency and accuracy of the numerical schemes based on these operators will be justified by reporting their norm infinity and norm relative errors. The complexity of the dynamics in the equations will be discussed theoretically by examining their local and global stability analysis. Our numerical experiment results are expected to give a new direction into pattern formation process in fractional diffusion-like scenarios.

At the end of 2016, Atangana and Baleanu introduced a new definition for fractional derivatives, namely Atangana–Baleanu fractional derivatives with the non-singular and non-local kernel. The idea of Atangana–Baleanu was used by several authors for various types of fractional problems. However, for heat transfer problem, this idea is rarely used in particular when nanofluid is considered. Based on this motivation, this chapter aims to study the flow of ethylene glycol based Molybdenum disulfide generalized nanofluid (EGMDGN) over an isothermal vertical plate. A fractional model with non-singular and non-local kernel, Atangana–Baleanu fractional derivatives is developed in the form of partial differential equations along with appropriate initial and boundary conditions. Molybdenum disulfide nanoparticles of spherical shape are suspended in Ethylene Glycol (EG) taken as conventional base fluid. The exact solutions are developed for velocity and temperature profiles via the Laplace transform technique. In a limiting sense, the obtained solutions are reduced to fractional Newtonian \((\beta \rightarrow \infty )\), classical Casson fluid \((\alpha \rightarrow 1)\) and classical Newtonian nanofluids. The influence of various pertinent parameters is analyzed in various plots and discussed physically.

We apply the new Atangana–Baleanu derivative in Caputo sense to study gas dynamics equations of arbitrary order using modified homotopy analysis transform method (MHATM). Atangana and Baleanu suggested an interesting fractional operator in 2016 which is based on the exponential kernel. An alternative framework of MHATM with Atangana–Baleanu derivative is presented and the modified Gas dynamics equations are solved numerically and analytically using aforesaid the method. Illustrative examples are included to demonstrate the validity and applicability of the presented technique with new Atangana–Baleanu derivative.

This book chapter higlights a new direction of Atangana–Baleanu fractional derivative to channel flow of non-Newtonian fluids. Because the idea to apply fractional derivatives with Mittag-Leffler kernel is a quite new direction for non-Newtonian fluids when flow is in a parallel plate channel. This new and inreresting fractional derivative launched by Atangana and Baleanu with a new fractional operator namely, Atangana–Baleanu fractional operator with Mittag-Leffler function as the kernel of integration has attracted the interest of the researchers. Because this new operator is an efficient tool to model complex and real-world problems. Therefore, this chapter deals with modeling and solution of generalized magnetohydrodynamic (MHD) flow of Casson fluid in a microchannel. The microchannel is taken of infinite length in the vertical direction and of finite width in the horizontal direction. The flow is modeled in terms of a set of partial differential equations involving Atangana–Baleanu time fractional operator with physical initial and boundary conditions. The partial differential equations are transformed to ordinary differential equations via fractional Laplace transformation and solved for exact solutions. To explore the physical significance of various pertinent parameters, the solutions are numerically computed and plotted in different graphs with a physical explanation. The results obtained here may have useful industrial and engineering applications.

In this chapter, we present the solution for a Liénard type model of a pipeline expressed by Liouville–Caputo and Atangana-Baleanu-Caputo fractional order derivatives. For this model, new approximated analytical solutions are derived by using the Laplace homotopy perturbation method and the modified homotopy analysis transform method. Both the efficiency and the accuracy of the method are verified by comparing the obtained solutions versus the exact analytical solution.

In this chapter, the approximate analytical solutions of a new reaction-diffusion fractional time model are studied. For this analysis is used the p-homotopy transform method based on different kernels (power, exponential and Mittag-Leffler). The system nonlinearities are addressed by the Adomian polynomials. The system convergence is studied by determining the interval of the convergence by \(\hbar \)-curves, as well as, searching for the optimal value of \(\hbar \) which minimize the residual error. Therefore, the optimal \(\hbar \) value is calculated to estimate the order \(\beta \) error. At the end of the chapter, we explained the obtained behavior by plotting the solutions in 3D. The results are accurate.

Modelling groundwater transport in fractured aquifer systems is complex due to the uncertainty associated with delineating the specific fractures along which water and potential contaminants could be transported. The resulting uncertainty in modelled contaminant movement has implications for the protection of the environment, where inadequate mitigation or remediation measures could be employed. To improve the governing equation for groundwater transport modelling, the Atangana–Baleanu in Caputo sense (ABC) fractional derivative is applied to the advection-dispersion equation with a focus on the advection term to account for anomalous advection. Boundedness, existence and uniqueness for the developed advection-focused transport equation is presented. In addition, a semi-discretisation analysis is performed to demonstrate the equation stability in time. Augmented upwind schemes are investigated as they have been found to address stability problems when solute transport is advection-dominated. The upwind-based schemes are developed, and stability analysis conducted, to facilitate the solution of the complex equation. The numerical stability analysis found the upwind Crank–Nicolson to be the most stable, and is thus recommended for use with the ABC fractional advection-dispersion equation.