Solution Of Kinetic Equation Research Articles

On the matrix versions of the k analog of ℑ‐incomplete Gauss hypergeometric functions and associated fractional calculus

In this paper, our aim is to introduce a new definition of ( ‐incomplete Wright hypergeometric matrix functions ( ‐IWHMFs) using the ‐incomplete Pochhammer matrix symbol. First, we define the ‐incomplete gamma matrix function and introduce the ‐incomplete Pochhammer matrix symbols. Furthermore, we present differential formulas and integral representation related to these ‐IWHMFs. We have also obtained some results regarding the ‐fractional calculus operators of these ‐IWHMFs. Finally, we investigate the solutions of fractional kinetic equations (FKEs) involving the ‐IWHMFs.

Analytic solutions of fractional kinetic equations involving incomplete ℵ-function

This paper expressed the fractional kinetic equation (KE) in terms of the incomplete [Formula: see text]-function, I-function, Fox H-function, and Meijer’s G-function. It assesses the importance of fractional KE in diverse scientific and engineering contexts, highlighting its relevance across various scenarios. In this paper, we investigate the results of Katugampola’s kinetic fractional equations with the product of the generalized M-series and incomplete [Formula: see text]-function. The [Formula: see text]-Laplace transform is applied for obtaining the desired results. Utilizing an M-series, the universality of this series is harnessed to deduce solutions for a fractional KE. Furthermore, a graphical representation of the obtained solutions’ behaviors is presented, enhancing the impact of the findings.

Sensitivity Analysis and Uncertainty Quantification on Point Defect Kinetics Equations with Perturbation Analysis

The concentration of radiation-induced point defects in general materials under irradiation is commonly described by the point defect kinetics equations based on rate theory. However, the parametric uncertainty in describing the rate constants of competing physical processes, such as recombination and loss to sinks, can lead to a large uncertainty in predicting the time-evolving point defect concentrations. Here, based on perturbation theory, we derive up to the third-order correction to the solution of point defect kinetics equations. This new set of equations enables a full description of continuously changing rate constants and can accurately predict the solution up to 50% deviation in these rate constants. These analyses can also be applied to reveal the sensitivity of the solution to input parameters and aggregated uncertainty from multiple rate constants.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

Wasserstein-penalized Entropy closure: A use case for stochastic particle methods

We introduce a framework for generating samples of a distribution given a finite number of its moments, targeted at particle-based solutions of kinetic equations and rarefied gas flow simulations. Our model, referred to as the Wasserstein-Entropy distribution (WE), couples a physically-motivated Wasserstein penalty term to the traditional maximum-entropy distribution (MED) function, which serves to regularize the latter. The penalty term becomes negligible near the local equilibrium, reducing the proposed model to the MED, known to reproduce the hydrodynamic limit. However, in contrast to the standard MED, the proposed WE closure can cover the entire physically realizable moment space, including the so-called Junk line. We also propose an efficient Monte Carlo algorithm for generating samples of the unknown distribution which is expected to outperform traditional non-linear optimization approaches used to solve the MED problem. Numerical tests demonstrate that given moments up to the heat flux—that is, information equivalent to that contained in the Chapman-Enskog distribution—the proposed methodology provides a reliable closure in the collision-dominated and early transition regimes. Applications to greater rarefaction demand information from higher-order moments, which can be incorporated within the proposed closure.

The kinetic theory of prompt redeposition in the case of thin Debye sheath

Prompt redeposition is a process in which a neutral atom sputtered from a divertor target or the first wall of a tokamak returns on the surface right after ionization, before experiencing any collisions with the background plasma. This paper presents analytical solutions of kinetic equations for sputtered neutrals and resulting ions. Using obtained distribution functions, the redeposition coefficient and average impact angle are derived under assumptions of thin Debye sheath. Our expression for the prompt redeposition coefficient explicitly depends on the angular and energy distributions of sputtered atoms and, therefore, is more universal than often used Fussmann equation valid only for cosine distribution. The effect of the angular distributions of sputtered particles on the prompt redeposition efficiency and parameters of redeposited ions is analyzed.

Robust and conservative dynamical low-rank methods for the Vlasov equation via a novel macro-micro decomposition

Dynamical low-rank (DLR) approximation has gained interest in recent years as a viable solution to the curse of dimensionality in the numerical solution of kinetic equations including the Boltzmann and Vlasov equations. These methods include the projector-splitting and Basis-update & Galerkin (BUG) DLR integrators, and have shown promise at greatly improving the computational efficiency of kinetic solutions. However, this often comes at the cost of conservation of charge, current and energy. In this work we show how a novel macro-micro decomposition may be used to separate the distribution function into two components, one of which carries the conserved quantities, and the other of which is orthogonal to them. We apply DLR approximation to the latter, and thereby achieve a clean and extensible approach to a conservative DLR scheme which retains the computational advantages of the base scheme. Moreover, our approach requires no change to the mechanics of the DLR approximation, so it is compatible with both the BUG family of integrators and the projector-splitting integrator which we use here. We describe a first-order integrator which can exactly conserve charge and either current or energy, as well as an integrator which exactly conserves charge and energy and exhibits second-order accuracy on our test problems. To highlight the flexibility of the proposed macro-micro decomposition, we implement a pair of velocity space discretizations, and verify the claimed accuracy and conservation properties on a suite of plasma benchmark problems.

Computational analysis of fractional kinetic equations having Bessel–Maitland function

This paper’s objective is to arrange fractional active conditions while taking into account the Bessel–Maitland capability reported by Khan et al. The solution’s final form is expressed in terms of the Wright hypergeometric function, which allows us to define numerous additional fractional kinetic equation solutions. Generally, fractional kinetic condition is applied to many problems in astronomy and material science.

Peculiarities of charged particle kinetics in spherical plasma

We describe transient kinetic simulations of partially ionized weakly-collisional plasma around spherical bodies absorbing or emitting charged particles. Numerical solutions of kinetic equations for electrons and ions in 1D2V phase space are coupled to an electrostatic solver using the Poisson equation or quasineutrality condition for small Debye lengths. The formation of particle groups and their contributions to electric current flow and screening of charged bodies by plasma are discussed for applications to Langmuir probes and solar wind.

Solution of Fractional Kinetic Equations Involving Laguerre Polynomials via Sumudu Transform

This study focuses on calculating solutions for fractional kinetic equations involving Laguerre polynomials and their fractional derivatives. By leveraging the Sumudu transform technique, we derive these solutions in the form of the Mittag‐Leffler function. Our investigation includes graphical representations generated using MATLAB to illustrate the behavior of these solutions under varying parametric conditions. It is essential to note that the results obtained in this study are exceptionally versatile and have the potential to yield both established and potentially novel findings in this field of research.

On solution of fractional kinetic equation involving Riemann xi function via Sumudu transform

Several significant questions of mathematics and mathematical physics have been effectively explained and answered through the use of fractional kinetic equations containing special functions. Due to the high importance of arbitrary-order kinetic equations, the aim of this work is to obtain the solution of a new arbitrary-order kinetic equation related to the Riemann xi function. The Sumudu transform technique is used to solve it. The findings are plotted with the help of MATLAB R2016a. The outcomes of the paper are in the form of an infinite series representation of the Riemann xi function and in terms of the Mittag-Leffler function.

On some new inequalities and fractional kinetic equations associated with extended gauss hypergeometric and confluent hypergeometric function

Fractional kinetic equations are of immense importance in describing and solving numerous intriguing problems of physics and astrophysics. Inequalities are important topics in special functions. In this paper, we studied the monotonicity of the extended Gauss and confluent hypergeometric function that are derived by using the inequalities on generalized beta function involving Appell series and Lauricella function. We also establish generalized fractional kinetic equation involving extended hypergeometric and confluent hypergeometric functions. The solutions of generalized fractional kinetic equation is derived and studied as an application of extended hypergeometric and confluent hypergeometric function using the General integral transform. The results obtained here are general and can be used to derive many new solutions of fractional kinetic equations involving various types of special functions.

Evolution of the Shape of a Gas Cloud during Pulsed Laser Evaporation into Vacuum: Direct Simulation Monte Carlo and the Solution of a Model Equation

The dynamics of gas expansion during nanosecond laser evaporation into vacuum is studied. The problem is considered in an axisymmetric formulation for a wide range of parameters: the number of evaporated monolayers and the size of the evaporation spot. To obtain a reliable numerical solution, two different kinetic approaches are used—the direct simulation Monte Carlo method and solution of the BGK model kinetic equation. The change in the shape of the cloud of evaporated substance during the expansion process is analyzed. The strong influence of the degree of rarefaction on the shape of the forming cloud is shown. When a large number of monolayers evaporate, good agreement with the continuum solution is observed.

Higher harmonics of the intensity modulated Photocurrent/Photovoltage spectroscopy response - a tool for studying photoelectrochemical nonlinearities

In this work, a higher harmonic analysis (HHA) of the intensity modulated photocurrent/photovoltage (IMPS/IMVS) spectroscopy data is proposed as a potent tool for studying nonlinear phenomena in photoelectrochemical and photovoltaic systems. Analytical solutions of kinetic equations were constructed for cases of single and double resonance accounting for various sources of higher harmonics. These sources correspond to the physical sources of nonlinear effects in the response including intensity-dependent generation current, intensity-dependent recombination, and second-order recombination. Due to the fact that the solutions for those cases are different, a qualitative methodology for analysis of harmonics is proposed. The methodology is illustrated by two experimental examples – Si photodiode for IMVS and TiO2 nanotubes for IMPS. It was capable of distinguishing second order recombination in the first case and intensity-dependent transport rate for the latter.

Theory of Anhysteretic Remanent Magnetization of Single-Domain Grains

Abstract—A new approach to the solution of kinetic equations describing the process of formation of anhysteretic remanent magnetization (ARM) is proposed, which made it possible to accelerate the numerical calculation of the process of formation of ARM by two orders of magnitude for uniaxial oriented non-interacting single-domain particles, while practically not yielding in accuracy to a strict numerical solution. It follows from the calculation results that the susceptibility of ARM is entirely determined by the magnitude of the particle’s coercivity parameter. The data of the previous approximate calculations of ARM value are compared with the exact results presented here and it is shown that the discrepancy between the exact data and the approximate estimates increases with the growth of g, but remains relatively small, within 23%. The proposed algorithm for the rapid calculation of kinetic equations allows us to analyze with physical rigor the method of pseudo-Thellier estimation of paleointensivity for an ensemble of single-domain particles, which is supposed to be done in subsequent works.

Kinetics of Chemisorption on the Surface of Nanodispersed SnO2–PdOx and Selective Determination of CO and H2 in Air

In this work, the kinetics and mechanisms of the interaction of carbon monoxide and hydrogen with the surface of a nanosized SnO2-PdOx metal oxide material in air is studied. Non-stationary temperature regimes make it possible to better identify the individual characteristics of target gases and increase the selectivity of the analysis. Recently, chemometric methods (PCA, PLS, ANN, etc.) are often used to interpret multidimensional data obtained in non-stationary temperature regimes, but the analytical solution of kinetic equations can be no less effective. In this regard, we studied the kinetics of the interaction of carbon monoxide and hydrogen with atmospheric oxygen on the surface of SnO2-PdOx using semiconductor metal oxide sensors under conditions as close as possible to classical gas analysis. An analysis of the influence of catalytic surface temperature on the mechanisms of chemisorption processes allowed us to correctly interpret and mathematically describe the electrophysical characteristics of the sensor in the selective determination of carbon monoxide and hydrogen under nonstationary temperature conditions. The reaction mechanism is applied as well to the analysis of the operation scheme of the CO sensor TGS 2442 of Figaro Inc.

Mathematical modelling of crystallization of polymer solutions

The processes of homogenization and crystallization of polymer solutions in cylindrical pipes are considered, which are described by the convective-diffusion equation with respect to the solution temperature and kinetic equations with respect to homogenization and crystallization of the polymer known as the thermokinetic nonlinear boundary value problem. A numerical-analytical iterative method for solving this problem is proposed, which consists of stepwise obtaining solutions of kinetic equations with respect to homogenization and crystallization of polymer solutions depending on the solution temperature and obtaining a solution of the convective-diffusion problem with respect to melt temperature. The accuracy of the obtained solution is determined by the norm of the difference between two adjacent iterations. The value of the crystallization coefficient, which is close to unity, determines the length of the dosing zone and the transition to the next zone – the flow of homogenized polymer into the distribution head of the extruder. The results of mathematical modelling are given.

INTEGRAL TRANSFORM AND FRACTIONAL KINETIC EQUATION

With the help of the Laplace and Fourier transforms, we arrive at the fractional kinetic equation's solution in this paper. Their respective solutions are given in terms of the Fox's H-function and the Mittag-Leffler function, which are also known as the generalisations and the Saigo-Maeda operator-based solution of the generalised fractional kinetic equation. The paper's findings have applications in a variety of engineering, astronomy, and physical scientific fields.

Unified gas-kinetic scheme with simplified multi-scale numerical flux for thermodynamic non-equilibrium flow in all flow regimes

In this paper, a BGK-type kinetic model for diatomic gases is proposed to describe the high-temperature thermodynamic non-equilibrium effect, which is a phenomenological relaxation model with continuous distribution modes of rotational and vibrational energies. In order to obtain the correct Prandtl number and reasonable relaxation rate of heat fluxes, the equilibrium distribution function is constructed by using a multi-dimensional Hermitian expansion around Maxwellian distribution. Based on this kinetic model equation, a unified gas-kinetic scheme (UGKS) with simplified multi-scale numerical flux is proposed for thermodynamic non-equilibrium flows involving the excitation of molecular vibrational degrees of freedom in all flow regimes. The present UGKS keeps the basic conservation laws of the macroscopic flow variables and microscopic gas distribution function in a discretized space. In order to improve the efficiency of UGKS, a simplified multi-scale numerical flux is directly constructed from the characteristic difference solution of complex kinetic model equation. Furthermore, the applications of unstructured discrete velocity space (DVS) and a simple integration error correction reduce the number of velocity mesh significantly and make the present method be rather efficient for flow simulation in all flow regimes. The new scheme is examined in a series of cases, such as Sod’s shock tube, high non-equilibrium shock structure, hypersonic flow around a circular cylinder with Knudsen (Kn) number Kn=0.01, supersonic rarefied flow over a flat plate with a sharp leading edge, and hypersonic rarefied flow past a blunt wedge. The present UGKS results agree well with the benchmark data of DSMC and other validated methods.

On the Theory of Unsteady-State Operation of Bulk Continuous Crystallization

Motivated by an important application in the chemical and pharmaceutical industries, we consider the non-stationary growth of a polydisperse ensemble of crystals in a continuous crystallizer. The mathematical model includes the effects of crystal nucleation and growth, fines dissolution, mass influx and withdrawal of product crystals. The steady- and unsteady-state solutions of kinetic and balance equations are analytically derived. The steady-state solution is found in an explicit form and describes the stationary operation mode maintained by the aforementioned effects. An approximate unsteady-state solution is found in a parametric form and describes a time-dependent crystallization scenario, which tends toward the steady-state mode when time increases. It is shown that the particle-size distribution contains kinks at the points of fines dissolution and product crystal withdrawal. Additionally, our calculations demonstrate that the unsteady-state crystal-size distribution has a bell-shaped profile that blurs with time due to the crystal growth and removal mechanisms. The analytical solutions found are the basis for investigating the dynamic stability of a continuous crystallizer.