Solution Of Kinetic Equation Research Articles

Solution of Fractional Kinetic Equations by Using Pα-Transform

The authors study a generalized form of fractional kinetic equations (FKE). The main results have been presented in the form of two theorems where in Pα-transform is used to obtain the solution of a generalized form of FKE containing τ -Gauss Hypergeometric functions and its derivatives. Some special cases of our main findings have also been recorded.

Fractional Kinetic Equation Involving Integral Transform

In this paper, we derive the solution of fractional kinetic equation with Laplace and Fourier transforms. Their respective solutions are given in terms of Mittag-Leffler function and their generalization, which can also be represented as Fox’s H-function. The result proved in this paper is application to wide range of engineering, astrophysics and physical science.

Improved kinetic description of fast relaxation of cylindrical micelles

On the basis of the linearized analytical and numerical kinetic description of stepwise aggregation of surfactant aggregates, the hierarchical relaxation times have been found for a polydisperse micellar system close and above the critical micellar concentration. The description was based on the difference and differential Becker–Döring kinetic equations with using a specific boundary condition and improved models for the attachment rates of surfactant monomers to cylindrical aggregates. Two models have been considered: the linear model for cylindrical aggregates and the attachment rate to elongated spheroidal aggregates. The rate of attachment of monomers to an elongated spheroidal aggregate was found explicitly as a function of the aggregation number. With applying the truncation techniques, the analytical solution of differential kinetic equations for fast relaxation of polydisperse micellar systems has been obtained for a linear model of the aggregation rate. In the case of the attachment rate for an elongated spheroidal aggregate, the semi-analytical solution has been found.

Stability analysis of the solutions of kinetic equations for modelling of gas flow at arbitrary heat ratio

In the presented paper the stability of the solutions of kinetic equations, which formed the models of ideal compressible gases at arbitrary heat ratio, is investigated. Models are widely used in applications of the lattice Boltzmann method, presented in the literature. The stability analysis is based on the considering of linear approximation and on the transition to the system of ordinary differential equations with complex coefficients. After the computations the stability criteria of the models are formulated for the cases of fixed values of relaxation time. The stability analysis is an important stage of the modelling process, because it may realize some help in choosing of the proper model for using in practical simulations.

On the solution of the neutron diffusion kinetic equation in planar geometry free of stiffness with convergence analysis

In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution. To this end we present error estimates and a convergence and stability analysis.

On fractional kinetic equations [formula omitted]-Struve functions based solutions

In the present research article, we investigate the solutions for fractional kinetic equations, involving k-Struve functions, some of the salient properties of which we present. The method used is Laplace transform based. The methodology and results can be adopted and extended to various related fractional problems in mathematical physics.

Influence of the conditions of methane decomposition on gas-jet synthesis of diamond structures

The present work is devoted to a numerical and experimental investigation of the effect of methane decomposition conditions on the growth of diamond structures during gas-jet deposition.The results of experiments on the diamond gas-jet synthesis from methane and hydrogen mixture flows are presented. The direct simulation Monte Carlo method in cylindrical geometry for numerical analysis of these experiments was applied. A one-dimensional approach based on the solution of equilibrium chemical kinetics equations was used to analyze gas-phase methane decomposition. The conducted researches have shown that at decrease in time of decomposition of methane in absence of high temperatures of the fed mixture it is possible to receive higher rates of growth of diamond structures. The obtained results can be useful for optimization of gas-dynamic sources of activated gas diamond synthesis.

Mathematical Modeling of Ethane Cracking Furnace of Olefin Plant with Coke Formation Approach

Abstract In this study, ethylene furnace modeling is carried out by ethane pyrolysis (thermal cracking or hydrocracking) method in Arya Sasol Petrochemical Company (ninth olefin unit, Assaluyeh, Iran), which includes the solution of kinetic equations and transfer phenomena, by the forward finite difference method in the MATLAB. Due to study and compare coke formation, a specific time period has been selected in equal segments and equations have been solved. It means that in a length segment of coil (Δz), momentum, energy as well as mass equations are solved, then the amount of precipitated coke in each length segment is achieved. With new efficient coil diameter calculating all mentioned approach will be repeated for next time segment. The results of this model have been compared with actual data and deviation has been reported. It was found that modeling approach is more capable to define the parameters of coke formation equations. The model has a good agreement between the values of prediction and experimental of in most cases.

On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function

The aim of the present paper is to develop a new generalized form of the fractional kinetic equation involving a generalized k-Mittag-Leffler function E^{gamma,rho}_{k,zeta,eta}(cdot). The solutions of fractional kinetic equations are discussed in terms of the Mittag-Leffler function. Further, numerical values of the results and their graphical interpretation is interpreted to study the behavior of these solutions. The results established here are quite general in nature and capable of yielding both known and new results.

Closed-form solutions for the Lévy-stable distribution

The Lévy-stable distribution is the attractor of distributions which hold power laws with infinite variance. This distribution has been used in a variety of research areas; for example, in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport. This function does not have an explicit expression and no uniform solution has been proposed yet. This paper presents a uniform analytical approximation for the Lévy-stable distribution based on matching power series expansions. For this solution, the trans-stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the Lévy-stable distribution. Then, the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called "the inner solution" and "the outer solution." Finally, the results of analytical approximation are compared to the numerical results of the Lévy-stable distribution function, making this uniform solution valid to be applied as an analytical approximation.

The Solution of Quantum Kinetic Equation with Delta Potential and its Application for Information Technology

The Solution of Quantum Kinetic Equation with Delta Potential and its Application for Information Technology

Stochastic Hamiltonian flows with singular coefficients

In this paper we study the following stochastic Hamiltonian system in ${\mathbb R}^{2d}$ (a second order stochastic differential equation), $$ d \dot X_t=b(X_t,\dot X_t)d t+\sigma(X_t,\dot X_t)d W_t,\ \ (X_0,\dot X_0)=(x,v)\in{\mathbb R}^{2d}, $$ where $b(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d$ and $\sigma(x,v):{\mathbb R}^{2d}\to{\mathbb R}^d\otimes{\mathbb R}^d$ are two Borel measurable functions. We show that if $\sigma$ is bounded and uniformly non-degenerate, and $b\in H^{2/3,0}_p$ and $\nabla\sigma\in L^p$ for some $p>2(2d+1)$, where $H^{\alpha,\beta}_p$ is the Bessel potential space with differentiability indices $\alpha$ in $x$ and $\beta$ in $v$, then the above stochastic equation admits a unique strong solution so that $(x,v)\mapsto Z_t(x,v):=(X_t,\dot X_t)(x,v)$ forms a stochastic homeomorphism flow, and $(x,v)\mapsto Z_t(x,v)$ is weakly differentiable with ess.$\sup_{x,v}E\left(\sup_{t\in[0,T]}|\nabla Z_t(x,v)|^q\right)<\infty$ for all $q\geq 1$ and $T\geq 0$. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli \cite{Fi} and Trevisan \cite{Tre}.

English

English

Numerical Solution for Fuzzy Enzyme Kinetic Equations by the Runge–Kutta Method

One of the most important biochemical reactions is catalyzed by enzymes. A numerical method to solve nonlinear equations of enzyme kinetics, known as the Michaelis and Menten equations, together with fuzzy initial values is introduced. The numerical method is based on the fourth order Runge–Kutta method, which is generalized for a fuzzy system of differential equations. The convergence and stability of the method are also presented. The capability of the method in fuzzy enzyme kinetics is demonstrated by some numerical examples.

Study of laser cooling in deep optical lattice: two-level quantum model

We study a possibility of laser cooling of 24Mg atoms in deep optical lattice formed by intense off-resonant laser field in a presence of cooling field resonant to narrow (3s3s) 1S0 → (3s3p)3P1 (λ = 457 nm) optical transition. For description of laser cooling with taking into account quantum recoil effects we consider two quantum models. The first one is based on direct numerical solution of quantum kinetic equation for atom density matrix and the second one is simplified model based on decomposition of atom density matrix over vibration states in the lattice wells. We search cooling field intensity and detuning for minimum cooling energy and fast laser cooling.

Nuclear Electromagnetic Generator: Introduction in Neutron Algebra and Elements of Neutron Kinetics

In the work, the problem of neutron kinetics in general view taking into account the all groups of delayed neutrons is solved. Conception of the matrix of complete neutron kinetics is introduced and are determined its properties. In the task about a finding of overt form for the solution of kinetic equations, the separation method and the reducing method of integrodifferential task to integral task with the aid of matched theorem are used. Diffusion of neutrons in the toroidal nuclear electric generator (nuclegen) is investigated also. Important conclusion about the practically complete absence of the diffusion of neutrons and charged fission particles (ionites) through the external boundary under toroidal motion is substantiated when solving of corresponding diffusion equation with the aid of Fourier’s standard separation method.

Thermalization of Photons in a Microcavity

A theoretical interpretation of the spectral condensation of photons in a microcavity based on solution of kinetic equations describing photon-gas thermalization as a result of Raman scattering from an atomic-gas thermostat is proposed.

Решение сеточным методом пространственно неоднородного кинетического уравнения для электронов в квазинейтральном режиме

Решение сеточным методом пространственно неоднородного кинетического уравнения для электронов в квазинейтральном режиме

Comparison of some different ways used for approximate solution of kinetic equation and calculations of carrier mobility

Comparison of some different ways used for approximate solution of kinetic equation and calculations of carrier mobility

Dynamic k-Struve Sumudu solutions for fractional kinetic equations

In this present study, we investigate the solutions for fractional kinetic equations involving k-Struve function using the Sumudu transform. The graphical interpretations of the solutions involving k-Struve function and its comparison with generalized Bessel function are given. The methodology and results can be considered and applied to various related fractional problems in mathematical physics.