Integral Inequalities and Differential Equations via Fractional Calculus

In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order n for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any α ≥ 1 , β ≥ 1 . For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed

A THREE FRACTIONAL ORDER JERK EQUATION WITH ANTI PERIODIC CONDITIONS

We study a new Jerk equation involving three fractional derivatives and anti periodic conditions. By Banach contraction principle, we present an existence and uniqueness result for the considered problem. Utilizing Krasnoselskii fixed point theorem we prove another existence result governing at least one solution. We provide an illustrative example to claim our established results. At the end, an approximation for Caputo derivative is proposed and some chaotic behaviours are discussed by means of the Runge Kutta 4th order method

Solvability for a Differential System of Duffing Type Via Caputo-Hadamard Approach

In this work, we investigate a new sequential coupled differential system of Duffing type. The considered system involves Caputo Hadamard derivatives. Based on both Banach contraction principle and Scheafer fixed point theorems, we establish two results on the existence and uniqueness of solutions for the introduced problem. Some examples are presented to show the validity of our results. To give more interpretation to the examples, we establish a new approximation of Caputo-Hadamard derivative for the case 1 \u3c β \u3c 2. Then, we plot the dynamics of one of the examples in terms of time and space coordinates

On a fractional problem of Lane-Emden type: Ulam type stabilities and numerical behaviors

In this work, we study some types of Ulam stability for a nonlinear fractional differential equation of Lane-Emden type with anti periodic conditions. Then, by using a numerical approach for the Caputo derivative, we investigate behaviors of the considered problem.WOS:0006715596000012-s2.0-8510973506