Zoubir Dahmani

Inequalities in Fractional Integrals

In this paper, we use the Riemann-Liouville fractional integral to present recent results on fractional integral inequalities. By considering the extended Chebyshev functional in the case of synchronous functions, we establish two main results. The first one deals with some inequalities using one fractional parameter. The second result concerns others inequalities using two fractional parameters.

Boundary Value Problems for Fractional Differential Equations

In this paper, we establish sufficient conditions for the existence of solutions for a boundary value problem for fractional differential equations of order 0 < α < 1 in Banach spaces. These results are obtained using Banach contraction fixed point theorem and Scheafer fixed point theorem.

New Fractional Results for a Boundary Value Problem with Caputo Derivative

In this paper, we consider three point boundary value problem for fractional dierential equations of order 1 < < 2. We establish new conditions for the existence and uniqueness of solutions by using Banach xed point theorem. We also generate other existence results using Scheafer and Krasnoselskii xed point theorems.

The Riemann-liouville operator to generate some integral inequalities

Abstract In this paper, the Riemann-Liouville fractional integral operator is used to establish some new integral inequalities of Hermite-Hadamard type.

On a three point boundary value problem of arbitrary order

Abstract In this paper, we study a three point boundary value problem of nonlinear fractional differential equations of order α, 2 < α < 3. New existence and uniqueness results are established using Banach fixed point theorem. Other existence results are obtained using Schaefer and Krasnoselskii theorems. Some illustrative examples are also presented.

Two Numerical Methods for Solving the Fractional Thomas-Fermi Equation

Abstract In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM) for solving the Fractional Thomas-Fermi Equation (FTFE). Further, we give comparative remarks for the obtained results.

Solutions of the Reaction-Diffusion Brusselator with Fractional Derivatives

Abstract By introducing the fractional derivative in the sense of Caputo, we apply the Adomian decomposition method for the Reaction-Diffusion Brusselator Model with time and space-fractional derivative. As a result, numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The behavior of Adomian solutions are shown graphically for some examples.

New integral results using Pólya-Szegö inequality

In this paper, we use the Riemann-Liouville fractional integral to present recent integral results using new inequalities of Polya-Szego type.

The variational iteration method for solving the fractional coupled lotka-volterra equation

Abstract In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Heís Variational Iteration Method (VIM) for the Coupled Lotka-Volterra Equation (CLVE) with time-and space-fractional derivative. The results show that the VIM method is very efficient and convinient and can be applied to a large class of problems. Numerical solutions are obtained for the fractional Coupled Lotka Volterra Equation to show the nature of solution as the fractional derivative parameters are changed.

A new problem of singular fractional differential equations

Abstract In this paper, we introduce a new class of nonlinear singular fractional differential equations. We axe interested in the singularity by using the contraction mapping principle and Schauder fixed point theorem. We present new results on the existence and uniqueness of solutions. Moreover, we investigate the Ulam-Hyers stability and the generalized Ulam-Hyers stability for this fractional equations. Some examples are provided to illustrate the application of our results.