Atangana-Baleanu Fractional Operators Research Articles

Application of the modern trend of fractional differentiation to the MHD flow of a generalized Casson fluid in a microchannel: Modelling and solution⋆

The motivation for writing this article is based on the contributions of Atangana and Baleanu to the fractal and fractional differentiation. Recently, Atangana and Baleanu launched a new fractional operator, namely, the Atangana-Baleanu fractional operator with the Mettang-Leffler function as the kernel of integration. This new operator is an efficient tool to model complex and real-world problems. This article deals with modeling and solution of the generalized magnetohydrodynamic (MHD) flow of a Casson fluid in a microchannel. This 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 the Atnagana-Baleanu time fractional operator along with physical initial and boundary conditions. The partial differential equations are transformed to ordinary differential equations via the fractional Laplace transformation and are solved for an exact solution using the transformed conditions. To explore the physical significance of various pertinent parameters affecting the flow, the numerical techniques of Zakian and Tzou are utilized for inversion of the Laplace transformation. The results obtained here may have useful industrial and engineering applications.

Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel

In this paper, time-fractional partial differential equations (FPDEs) involving singular and non-singular kernel are considered. We have obtained the approximate analytical solution for linear and nonlinear FPDEs using the Laplace perturbation method (LPM) defined with the Liouville-Caputo (LC) and Atangana-Baleanu (AB) fractional operators. The AB fractional derivative is defined with the Mittag-Leffler function and has all the properties of a classical fractional derivative. In addition, the AB operator is crucial when utilizing the Laplace transform (LT) to get solutions of some illustrative problems with initial condition. We show that the mentioned method is a rather effective and powerful technique for solving FPDEs. Besides, we show the solution graphs for different values of fractional order $\alpha$ , distance term x and time value t . The classical integer-order features are fully recovered if $\alpha$ is equal to 1.

Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo–Fabrizio and Atangana–Baleanu fractional derivatives embedded in porous medium

This manuscript investigates the temperature difference versus temperature or time and the effects of newly introduced fractional operators, namely Caputo–Fabrizio and Atangana–Baleanu fractional derivatives, on the magnetohydrodynamic flow of nanofluid in a porous medium. Three different types of nanoparticles are suspended in ethylene glycol, namely titanium oxide, copper and aluminum oxide. The mathematical modeling of the governing equations is developed by the modern fractional derivatives. The general solutions for velocity field and temperature distribution have been established by invoking Laplace transforms, and obtained solutions are expressed in terms of special functions, namely Fox-H function $${\mathbf{H}}_{\upalpha ,\upbeta + 1}^{1,\alpha } \left( F \right)$$ and $${\mathbf{M}}_{\upbeta ,\upgamma }^{\alpha } \left( F \right)$$ Mittag-Leffler functions. Dual solutions have been analyzed by graphical illustrations for the influence of pertinent parameters on the motion of a fluid. The base fluid and three different types of nanoparticles have intersecting similarities and differences in the heat transfer and fluid flows. The results show the reciprocal behavior of different types of nanoparticles which are suspended in ethylene glycol via Caputo–Fabrizio and Atangana–Baleanu fractional operators.

Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena

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-Gomez 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.

Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators

The model of Ebola spread within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced by Atangana and Baleanu. It is expected that the proposed model will show better approximation than the models established before. The existence and uniqueness of solutions for the spread of Ebola disease model is given via the Picard-Lindelof method. Finally, numerical solutions for the model are given by using different parameter values.

A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives

This research article is analyzed for the comparative study of RL and RC electrical circuits by employing newly presented Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. The governing ordinary differential equations of RL and RC electrical circuits have been fractionalized in terms of fractional operators in the range of $0 \leq \xi \leq 1$ and $0 \leq \eta \leq 1$ . The analytic solutions of fractional differential equations for RL and RC electrical circuits have been solved by using the Laplace transform with its inversions. General solutions have been investigated for periodic and exponential sources by implementing the Atangana-Baleanu and Caputo-Fabrizio fractional operators separately. The investigated solutions have been expressed in terms of simple elementary functions with convolution product. On the basis of newly fractional derivatives with and without singular kernel, the voltage and current have interesting behavior with several similarities and differences for the periodic and exponential sources.

Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media

This work presents alternative solutions of the fractional waves in dielectric media using the Atangana–Baleanu fractional operator in Liouville-Caputo and Riemann–Liouville sense, the order considered is for the space-time domain. We solved analytically the fractional equation using the properties of Laplace transform operator together with the convolution theorem. New results are presented and the analytical solutions are given in terms of the multivariate Mittag-Leffler functions preserving the physical units of the system studied. Numerical simulations were performed for different values of fractional order.