On the fractional derivatives at extrema points

We correct a recent result concerning the fractional derivative at extrema points. We then establish new results for the Caputo and Riemann-Liouville fractional derivatives at extrema points

Bounds and critical parameters for a combustion problem

AbstractA model from combustion theory consisting of a nonlinear elliptic equation and boundary conditions of Dirichlet type, is considered. Upper and lower solutions for the problem are obtained by solving linear elliptic equations. These solutions are used to obtain analytical bounds for the extinction and ignition limits. Numerical results are presented for the slab, cylindrical and spherical geometries. Results compare very well with existing ones in the literature

Dynamic Traffic Light System to Reduce The Waiting Time of Emergency Vehicles at Intersections within IoT Environment

Traditional traffic light system, which works based on fixed cycle can be a main reason for traffic jam, due to lack of adaptation to road conditions. Traffic jam has a bad impact on drivers and road users due to the time delay it causes for road users to reach their destinations. This delay can cause a life threat in case of emergency vehicles, such as ambulance vehicles and police cars. One key solution to solve traffic jam on intersections is the dynamic traffic lights, where traffic light operation adapts based on the intersection traffic conditions. Since few of researches projects in the literature interested in solving traffic jam problem for emergency vehicles, the contribution of this paper is to introduces a novel approach to operate traffic light system. The new approach consists of two algorithms which are pure operation mode and hybrid operation mode. These operation modes aim to reduce the waiting time of emergency vehicles on traffic intersections. They assume that there is a smart infrastructure system uses Internet of Things (IoT) that can detect emergency vehicles arrival to an intersection. The smart infrastructure system switches traffic light operation from fixed cycle mode to dynamic mode. The dynamic mode manages traffic lights at intersections to reduce the waiting time of emergency vehicles. The paper presents a simulation of the proposed algorithms, highlights their advantages. In order to evaluate the efficiency of the new technique, we compared our approach with Wen algorithm in the literature and the Traditional traffic light system. Our evaluation study indicated that the proposed algorithms outperformed Wen technique and the Traditional system under different traffic scenario

A maximum principle for a fractional boundary value problem with convection term and applications

We consider a fractional boundary value problem with Caputo-Fabrizio fractional derivative of order 1 < α < 2 We prove a maximum principle for a general linear fractional boundary value problem. The proof is based on an estimate of the fractional derivative at extreme points and under certain assumption on the boundary conditions. A prior norm estimate of solutions of the linear fractional boundary value problem and a uniqueness result of the nonlinear problem have been established. Several comparison principles are derived for the linear and nonlinear fractional problems. First Published Online: 21 Nov 201

Upper and lower bounds for a reactive-diffuse system with Arrhenius kinetics

Comparison arguments are used to study a problem in combustion theory consisting of a nonlinear parabolic equation together with initial and boundary conditions. Upper and lower bounds for the problem are constructed. The lower solutions are used to determine whether the solution of the problem is increasing in time for certain initial condition. Numerical results are presented for the slab, infinite cylinder, and unit sphere. The bounds are compared with the existing ones in the literature for the slab geometry

An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed