Fractional Differential Equations

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media

Numerical approach for MHD Al2O3-water nanofluid transportation inside a permeable medium using innovative computer method

In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario.The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional power α = 1. Numerical results are finally given to justify the effectiveness of the proposed schemes.

New numerical approach for fractional differential equations

In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.

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Active Control of a Chaotic Fractional Order Economic System

MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium

Thermal management of water based SWCNTs enclosed in a partially heated trapezoidal cavity via FEM

In this work, the well known invariant subspace method has been modified and extended to solve some partial differential 
equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods 
constructed for solving partial differential equations with CF and AB fractional derivatives.

A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

In this article, we construct new explicit solutions for a time-fractional nonlinear Calogero-Bogoyavlenskii-Schiff equation in $(2+1)$
 dimensions with conformable derivative with the help of the $\tan (\phi(\xi)/2)$
 -expansion method. The obtained solutions are expressed by different kinds of functions, such as exponential, hyperbolic and trigonometric ones. The solutions might be very useful in physics and engineering sciences.

Toufik Mekkaoui

Papers

Active Control of a Chaotic Fractional Order Economic System

MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium

Thermal management of water based SWCNTs enclosed in a partially heated trapezoidal cavity via FEM

A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

Optical solitons for the Calogero-Bogoyavlenskii-Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative