Maximum Principle for the Space-Time Fractional Conformable Differential System Involving the Fractional Laplace Operator

In this paper, the authors consider a IBVP for the time-space fractional PDE with the fractional conformable derivative and the fractional Laplace operator. A fractional conformable extremum principle is presented and proved. Based on the extremum principle, a maximum principle for the fractional conformable Laplace system is established. Furthermore, the maximum principle is applied to the linear space-time fractional Laplace conformable differential system to obtain a new comparison theorem. Besides that, the uniqueness and continuous dependence of the solution of the above system are also proved

Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples

N-Doped Biochar from Lignocellulosic Biomass for Preparation of Adsorbent: Characterization, Kinetics and Application

Medulla tetrapanacis is composed of a lignocellulosic biopolymer and has a regular porous structure, which makes it a potential biomass material for preparing porous N-doped biochar. Herewith, N-doped Medulla tetrapanacis biochar (UBC) was successfully prepared by modification with urea and NaHCO3 under pyrolysis at 700 °C. The nitrogen-containing groups were efficiently introduced into biochar, and the micro-pore structures of the UBC were developed with sizeable specific surface area, which was loaded with massive adsorption sites. The adsorption kinetics and isotherms of the UBC conformed to pseudo-second-order and Langmuir model. The superior adsorption capacities of the UBC for methylene blue (MB) and congo red (CR) were 923.0 mg/g and 728.0 mg/g, and the capacities for Cu2+ and Pb2+ were 468.5 mg/g and 1466.5 mg/g, respectively. Moreover, the UBC had a stronger affinity for Cr3+ and Fe3+ in multiple metal ions and retained at a preferable adsorption performance for dyes and heavy metals after five cycles. Precipitation, complexation, and physical adsorption were the main mechanisms of the UBC-adsorbing metal ions and dyes. Thus, lignocellulosic biochar has great potential for removing dyes and heavy metals in aqueous solutions

Alkali and Alkaline Earth Hydrides-Driven N-2 Activation and Transformation over Mn Nitride Catalyst

Early 3d transition metals are not focal catalytic candidates for many chemical processes because they have strong affinities to O, N, C, or H, etc., which would hinder the conversion of those species to products. Metallic Mn, as a representative, undergoes nitridation under ammonia synthesis conditions forming bulk phase nitride and unfortunately exhibits negligible catalytic activity. Here we show that alkali or alkaline earth metal hydrides (i.e., LiH, NaH, KH, CaH2, and BaH2, AHs for short) promotes the catalytic activity of Mn nitride by orders of magnitude. The sequence of promotion is BaH2 > LiH > KH > CaH2 > NaH, which is different from the order observed in conventional oxide or hydroxide promoters. AHs, featured by bearing negatively charged hydrogen atoms, have chemical potentials in removing N from Mn nitride and thus lead to significant enhancement of N-2 activation and subsequent conversion to NH3. Detailed investigations on Mn-LiH catalytic system disclosed that the active phase and kinetic behavior depend strongly on reaction conditions. Based on the understanding of the synergy between AHs and Mn nitride, a strategy in the design and development of early transition metals as effective catalysts for ammonia synthesis and other chemical processes is proposed