Fractional Reaction Order Research Articles

Study of two-dimensional nonlinear coupled time-space fractional order reaction advection diffusion equations using shifted Legendre-Gauss-Lobatto collocation method

In this article, the nonlinear coupled two-dimensional space-time fractional order reaction-advection–diffusion equations (2D-STFRADEs) with initial and boundary conditions is solved by using Shifted Legendre-Gauss-Lobatto Collocation method (SLGLCM) with fractional derivative defined in Caputo sense. The SLGLC scheme is used to discretize the coupled nonlinear 2D-STFRADEs into the shifted Legendre polynomial roots to convert it to a system of algebraic equations. The efficiency and efficacy of the scheme are confirmed through error analysis while applying the scheme on two existing problems having exact solutions. The impact of advection and reaction terms on the solution profiles for various space and time fractional order derivatives are shown graphically for different particular cases. A drive has been made to study the convergence of the proposed scheme, which has been applied on the proposed mathematical model.

Hydrogen Oxidation and Evolution Reaction on Platinum in Alkaline Electrolyte: Fixed Reference vs Overpotential and the Effect of H2 Concentration

Understanding the kinetics of the electrochemical hydrogen oxidation and evolution reaction (HOR/HER) in alkaline media is of great interest, but the underlying reaction mechanism and the dependency on the hydrogen partial pressure are still debated. In this work, we investigated the HOR/HER kinetics on a polycrystalline platinum electrode in alkaline electrolyte using the rotating disk electrode (RDE) technique. Unlike typically done, we rigorously performed data treatment and analysis on a fixed potential scale with a fixed reference potential, which has implications for the use and assessment of RDE mass-transport corrections and the definition of reaction orders. Analyzing the kinetic current curves on a fixed potential scale, we find reaction orders with respect to the hydrogen partial pressure of essentially one and zero for the HOR and HER, respectively, at larger overpotentials. In contrast, fitting the kinetic current curves in a potential range around equilibrium with a generalized Butler-Volmer model yields differing kinetic parameters and fractional reaction orders. Our findings can be explained with a potential-dependent transition in the HOR mechanism from Tafel–Volmer with a rate-determining Volmer step at small overpotentials to Heyrovsky–Volmer with a rate-determining Heyrovsky step at more positive overpotentials, with a concomitant change in the H2 reaction order from fractional to unity, thus reconciling previous propositions in the debate on the alkaline HOR/HER mechanism. Our results demonstrate the strength and advantages of using a fixed potential scale rather than the commonly employed overpotential, and we expect this approach to be beneficial also for kinetic studies of other electrocatalytic reactions.

Negative Reaction Order for CO during CO2 Electroreduction on Au.

The potential-dependent negative fractional reaction orders with respect to the CO partial pressures were measured for CO2 electroreduction (CO2R) on Au under mass-transfer-controlled conditions using a rotating ring-disk electrode setup. At high overpotentials, the CO reaction order approaches -1 due to enhanced CO adsorption on Au, which is supported by kinetic analysis and density functional theory (DFT) simulations. This work illustrates that the CO site-blocking effect cannot be ignored, even on a weak CO-binding metal such as Au in the electrochemical environment. The CO site-blocking effect can greatly hamper the activity and the selectivity of the CO2R to CO. This observation enriches the current mechanistic understanding of the CO2R and could have significant implications not only in the theoretical modeling of the CO2R but also in the evaluation of intrinsic CO2R activity at practical current density and high conversion conditions.

Adaptive synchronization of quaternion-valued neural networks with reaction–diffusion and fractional order

This paper is dedicated to the study of adaptive finite-time synchronization (FTS) for generalized delayed fractional-order reaction–diffusion quaternion-valued neural networks (GDFORDQVNN). Utilizing the suitable Lyapunov functional, Green’s formula, and inequalities skills, testable algebraic criteria for ensuring the FTS of GDFORDQVNN are established on the basis of two adaptive controllers. Moreover, the numerical examples validate that the obtained results are feasible. Furthermore, they are also verified in image encryption as the application.

Stability and synchronization of fractional-order reaction–diffusion inertial time-delayed neural networks with parameters perturbation

This study is centered around the dynamic behaviors observed in a class of fractional-order generalized reaction–diffusion inertial neural networks (FGRDINNs) with time delays. These networks are characterized by differential equations involving two distinct fractional derivatives of the state. The global uniform stability of FGRDINNs with time delays is explored utilizing Lyapunov comparison principles. Furthermore, global synchronization conditions for FGRDINNs with time delays are derived through the Lyapunov direct method, with consideration given to various feedback control strategies and parameter perturbations. The effectiveness of the theoretical findings is demonstrated through three numerical examples, and the impact of controller parameters on the error system is further investigated.

Fractional-Order Degn–Harrison Reaction–Diffusion Model: Finite-Time Dynamics of Stability and Synchronization

This study aims to address the topic of finite-time synchronization within a specific subset of fractional-order Degn–Harrison reaction–diffusion systems. To achieve this goal, we begin with the introduction of a novel lemma specific for finite-time stability analysis. Diverging from existing criteria, this lemma represents a significant extension of prior findings, laying the groundwork for subsequent investigations. Building upon this foundation, we proceed to develop efficient dependent linear controllers designed to orchestrate finite-time synchronization. Leveraging the power of a Lyapunov function, we derive new, robust conditions that ensure the attainment of synchronization within a predefined time frame. This innovative approach not only enhances our understanding of finite-time synchronization, but also offers practical solutions for its realization in complex systems. To validate the efficacy and applicability of our proposed methodology, extensive numerical simulations are conducted. Through this comprehensive analysis, we aim to contribute valuable insights to the field of fractional-order reaction–diffusion systems while paving the way for practical implementations in real-world applications.

Sliding mode control for uncertain fractional-order reaction–diffusion memristor neural networks with time delays

This paper investigates a sliding mode control method for a class of uncertain delayed fractional-order reaction–diffusion memristor neural networks. Different from most existing literature on sliding mode control for fractional-order reaction–diffusion systems, this study constructs a linear sliding mode switching function and designs the corresponding sliding mode control law. The sufficient theory for the globally asymptotic stability of the sliding mode dynamics are provided, and it is proven that the sliding mode surface is finite-time reachable under the proposed control law, with an estimate of the maximum reaching time. Finally, a numerical test is presented to validate the effectiveness of the theoretical analysis.

True Order Determination of Sonochemical Degradation Kinetics of Organic Contaminants in Water Using the Half-lives Method

In the literature, the modeling of the sonolytic decomposition kinetics of organic contaminants in water has been carried out using the pseudo-first and pseudo-second order equations. The results obtained with this method are totally biased because it requires the assumption of the order of the degradation reaction, which is a real limitation, and the sonochemical oxidation mechanism is a very complex phenomenon that cannot be described by such simple models. In this paper, for the first time, the true order of the sonochemical degradation kinetics of organic pollutants in aqueous solutions at various initial substrate concentrations was determined using the half-lives method combined with the general rate law model, i.e., the pseudo-nth order equation. Linear and nonlinear regression techniques were used to determine the order of the sonolytic destruction reaction. The half-life technique for the general rate law model can adequately describe the experimental results of the sono-destruction of the investigated pollutants. For rhodamine B and naphthol blue black, the sono-destruction mechanism for both contaminants does not change as the initial contaminant concentration varies over the range examined, while for 4-isopropylphenol and furosemide, the sono-destruction mechanism changes as the initial substrate concentration varies. A fractional order of the sonolytic destruction reaction was determined for all the pollutants tested. These fractional orders of sono-destruction reactions indicate the complex nature and behavior of the reactions, often involving multiple steps or mechanisms. They also reveal a more complex interaction between pollutant concentration and sonolytic destruction rate. In a word, the method developed in this paper should be used to determine the real order of the sono-oxidation reaction of nonvolatile organic pollutants in aqueous phase.

Global Asymptotic Stability and Synchronization of Fractional-Order Reaction–Diffusion Fuzzy BAM Neural Networks with Distributed Delays via Hybrid Feedback Controllers

In this paper, the global asymptotic stability and global Mittag–Leffler stability of a class of fractional-order fuzzy bidirectional associative memory (BAM) neural networks with distributed delays is investigated. Necessary conditions are obtained by means of the Lyapunov functional method and inequality techniques. The hybrid feedback controllers are then developed to ensure the global asymptotic synchronization of these neural networks, resulting in two additional synchronization criteria. The derived conditions are applied to check the fractional-order fuzzy BAM neural network’s Mittag–Leffler stability and synchronization. Three examples are given to demonstrate the effectiveness of the achieved results.

Computational study for the Caputo sub-diffusive and Riesz super-diffusive processes with a fractional order reaction–diffusion equation

A Numerical solution of the Caputo-time and Riesz-space fractional reaction–diffusion model is considered in this paper. Based on finite difference schemes, we formulate both second-order and fourth-order numerical methods for the approximation of the Riesz space fractional reaction–diffusion-like equation of Fisher type. In the experiment, it was observed that the fourth-order scheme has better accuracy than the second-order method when applied to solve the fractional diffusion equation. It should be mentioned that the lower order scheme computes rapidly and save more computational time as displayed in the table of results. Finally, some simulation results are presented to justify the effectiveness and applicability of the numerical methods. The one- two- and three-dimensional results obtained for some instances of fractional order (α,β) depict some amazing complex and spatiotemporal patterns which are applicable in applied sciences and engineering.

Fractional-order reaction–diffusion model to study the dysregulatory impacts of superdiffusion and memory on neuronal calcium and IP3 dynamics

Fractional-order reaction–diffusion model to study the dysregulatory impacts of superdiffusion and memory on neuronal calcium and IP3 dynamics

Event-triggered impulsive control for synchronization in finite time of fractional-order reaction–diffusion complex networks

This paper is concerned with the finite-time synchronization (FTS) issue for fractional-order reaction–diffusion complex networks (RDCNs). A finite-time stability principle, which plays a key role in the synchronization analysis later, is developed for fractional-order nonlinear impulsive system. A novel hybrid controller, which consists of a event-triggered controller and a impulsive controller, is designed to realize the global FTS objective for the considered network. By applying the Lyapunov stability theory and the fractional calculus, the global FTS conditions are addressed in the form of algebraic inequalities. In addition, the exclusion of Zeno behavior are proved for the designed event-triggered strategy. Finally, a numerical example is provided to illustrate the feasibility of the proposed control approach and the correctness of the theoretical results.

The FitzHugh–Nagumo Model Described by Fractional Difference Equations: Stability and Numerical Simulation

The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results.

Elucidating the Influence of Intercalated Anions in NiFe LDH on the Electrocatalytic Behavior of OER: A Kinetic Study

AbstractThe oxygen evolution reaction (OER) as one half‐cell reaction of electrochemical water splitting has a fundamental impact on water splitting efficiency and thus on the competitiveness of electrochemically generated hydrogen in the energy market. Nickel‐iron layered double hydroxides (NiFe LDH) are among the most promising electrocatalysts for efficient OER under alkaline conditions. Despite intensive research, correlations of the material properties and the resulting kinetically limiting surface processes are poorly investigated. This work focuses on the kinetic behavior of NiFe LDH catalysts containing different anions in the basal spacing in alkaline OER. Steady‐state Tafel plots, impedance measurements as well as reaction order plots were used to elucidate differences in the catalytic performance. All catalysts showed a dual Tafel behavior and fractional reaction orders. For kinetic modelling, the physisorbed hydrogen peroxide mechanism and Temkin adsorption model were adopted to fit experimental data. Our study showed that the intercalated anions affect the kinetics of rate determining steps. The hypophosphite intercalated LDH possessed the highest OER activity and the first step as rate determining. While for both carbonate and borate intercalated NiFe LDH, the second step proved to be rate determining in the low Tafel region, while the first step was found to be rate‐limiting in the high Tafel region.

Uniqueness of traveling fronts in premixed flames with stepwise ignition-temperature kinetics and fractional reaction order

In this paper, we consider a reaction–diffusion system describing the propagation of flames under the assumption of ignition-temperature kinetics and fractional reaction order. It was shown in Brauner et al. (2021) that this system admits a traveling front solution. In the present work, we show that this traveling front is unique up to translations. We also study some qualitative properties of this solution using the combination of formal asymptotics and numerics. Our findings allow conjecture that the velocity of the propagation of the flame front is a decreasing function of all of the parameters of the problem: ignition temperature, reaction order and an inverse of the Lewis number.

Fractional order reaction diffusion of calcium regulating NFAT production in T Lymphocyte

[Formula: see text] signals have a crucial role in the immune system for cell functions such as proliferation, differentiation, apoptosis and gene transcription as well as the activation of Nuclear factor of activated T Cell (NFAT) which modulates the immune response of the cells. In this study, a fractional order spatiotemporal calcium dynamics model is formulated using the reaction-diffusion equation incorporating parameters like buffers, source influx, [Formula: see text] and [Formula: see text]. Grünwald estimation is employed to obtain the solution and stability analysis has been performed. The analysis of numerical results provides novel insights about the role of Brownian motion, super diffusion, source influx, buffers, etc., in the regulation of calcium and NFAT concentration levels in T cell and the conditions which might lead to disordered immune responses causing diseases such as HINI, HIV and HBV.

Synchronization of fractional-order reaction–diffusion neural networks with Markov parameter jumping: Asynchronous boundary quantization control

This paper tries to study the synchronization problem of a kind of fractional-order reaction–diffusion neural networks (FRDNNs) with Markov parameter jumping. Considering the spatial and parametric characteristics of the proposed Markov FRDNNs, asynchronous boundary quantization control, is applied to achieve the driven-response synchronization of the systems. In this control scheme, one actuator is placed at the spatial boundary, and two types of quantizers is used to further reduce the control energy consumption, which is economical and easier to implement than the distributed control strategies. Besides, the parameter jumping rules of the controller and system are subject to two different Markov chains, which is more general and practical. Moreover, for fractional-order Markovian systems, the continuous frequency distributed model of fractional integrator is introduced to deal with synchronization issue of the considered systems, the corresponding synchronization criteria are obtained with the help of indirect Lyapunov functionals method. Last but not least, a numerical simulation is carried out to support the proposed control methods and theoretical results.

Kinetics of Catalytic Oxidation of Carbenicillin: A Degradation Approach for Penicillanic Acid Derivatives (PADs)

Diperiodatocuprate [DPC (III)] was selected to predict the kinetic studies for carbenicillin (CRBC) oxidation in the basic media. The investigation was completed in the presence of CoCl3 as a catalyst by using a UV/Visible spectrophotometer at 298K temperature and 0.01 mol-dm-3 ionic strength confirming a 1:4 stoichiometry between CRBC and DPC (III). Both spectral and elemental analysis was used to identify the final products. Monoperiodatocuperate [MPC (III)] was found to be the primary active species of DPC (III). Pseudo-first order reaction was declared for DPC (III), while fractional order reactions were noticed in the case of CRBC (substrate), Co (III) catalyst as well as KOH (alkali). However, the reaction was determined to be in negative fractional order for periodate. Spectral evidence, determination of various rate constants, and both activation, as well as thermodynamic parameters, were used to predict plausible mechanisms.

Кинетика растворения триоксида молибдена в щелочной среде

The paper presents experimental study results of the MoO3 powder samples dissolution kinetics in the aqueous ammonia solution at various pH concentrations and values. Molybdenum ions concentration in the filtrate samples was determined spectrophotometrically. Kinetic characteristics were obtained, and kinetic parameters (specific dissolution rate, reaction order with respect to the H+ ion) were calculated. It was established that the dissolution rate was increasing in the ammonia solution concentration range of 0.02--1.26 mol/l, and with the growing pH it passed through the maximum. Taking into account the acid-base equilibria constants, the dissolution process was simulated, and its stage-by-stage nature was established. Fractional reaction order with respect to the H+ ions calculated from the curves plotted in the α--t/t0.5 coordinates (affine transformation method) indicated the adsorption mechanism of dissolution. It was shown that the MoO3 dissolution proceeded with formation of the intermediate adsorption complexes. Due to the HMoO4- low concentration in the MoO3 concentration within the studied pH range by the surface-active particle, on which groups of ions were adsorbed, the MoO4- could be considered. The results obtained are an addition to the data possessed on the molybdenum oxide phase and other transitional metals behavior. They could be applied in the practical applications associated with dissolving molybdenum in the alkaline electrolytes

Active disturbance rejection control to stabilization of coupled delayed time fractional-order reaction–advection–diffusion systems with boundary disturbances and spatially varying coefficients

In this paper we consider the boundary asymptotic stabilization of a coupled time fractional-order reaction–advection–diffusion (FRAD) system with boundary input disturbances, spatially varying coefficients and time varying delays. The complex system composition makes it difficult to analyze system property directly. For this issue, we here address it by introducing two transformations to convert the original system into the one where theories of fractional evolution equations and operator semigroup are easily applicable. The active disturbance rejection control (ADRC) and backstepping control are also utilized in investigation. Using ADRC, the disturbance is first estimated by constructing two auxiliary systems and then canceled through an approximation in the feedback-loop. For such auxiliary systems, one is used to take disturbances from the original system and to put them into a stable system. Another is used to estimate disturbances. In the second part, we use backstepping to develop a boundary state feedback control law with disturbance approximation to ’eliminate’ the disturbances and to achieve the closed-loop stability. The well-posedness and disturbance estimation are established by theories of fractional evolution equations. With fractional Halanay’s inequality, sufficient conditions are obtained to make the controlled system asymptotically stable and the auxiliary system bounded. Fractional simulation scheme is constructed to test the proposed synthesis generated by ADRC when the explicit solution of kernel equations does not exist.