Lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation

A lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation is proposed. By using multi-scale technique and the Chapman–Enskog expansion on complex lattice Boltzmann equation, we obtain a series of complex partial differential equations, complex equilibrium distribution function and its complex moments. Then, the complex reaction–diffusion equation is recovered with higher-order accuracy of the truncation error. This equation can be used to describe the bimolecular autocatalytic reaction–diffusion systems, in which a rich variety of behaviors have been observed. Based on this model, the Fitzhugh–Nagumo model and the Gray–Scott model are simulated. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex reaction–diffusion equation.     

Discontinuous Shock Structure in a Reacting Mixture Modelled by Grad 13 Moment Approximation

A preliminary analysis on the possible occurrence of sub-shocks into a gas mixture is carried out. The mixture, undergoing a reversible bimolecular reaction, is described by macroscopic equations obtained by Grad 13 moment approximation of the reactive Boltzmann equation.     

Solvability of a finite system of parabolic and ordinary differential equations

We generalize a mathematical model of a bimolecular heterogeneous chemical reaction for the case of more than two reactants. The model is described by a finite system of parabolic and ordinary differential equations for which the existence and uniqueness theorem of a classical solution is proved.     

Nonlinear orientational dynamics of a molecular chain

We investigate the nonlinear rotational dynamics of a molecular chain with quadrupole interaction in both the discrete and the continuous cases. Based on a system of nonlinear differential-difference equations, we obtain approximate equations describing the chain excitations and preserving the initial symmetry. We introduce an effective potential and normal coordinates, using which allows decoupling the system into linear and nonlinear parts. As a result of a strong anisotropy of the potential, narrow “valleys” occur in the angle plane. Motion along a valley corresponds to a softer interaction (nonlinear equations). Linear equations describe motion across a valley (hard interaction). We consider cases where the derived nonlinear equations reduce to the sine-Gordon equation. We find integrals of motion and exact solutions of our approximate equations. We uniformly describe the energy interval encompassing the domains of order, of orientational melting, and of rotational motion of the molecules in the chain.     

Two-time temperature Green’s functions in kinetic theory and molecular hydrodynamics: II. Equations for pair-interaction systems

We consider two approaches to the calculation of correlation functions for a system of particles with direct pair interaction. The first is based on a chain of equations that determines a Boltzmann-type kinetic equation; the second is based on a chain of molecular hydrodynamic equations. We demonstrate that the two approaches are equivalent in the sense that they completely describe the system under consideration. We discuss the advantages of the approach based on the molecular hydrodynamic equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 142–166, April, 1999.     

Linear independence of root equations forM/G/1 type Markov chains

There is a classical technique for determining the equilibrium probabilities ofM/G/1 type Markov chains. After transforming the equilibrium balance equations of the chain, one obtains an equivalent system of equations in analytic functions to be solved. This method requires finding all singularities of a given matrix function in the unit disk and then using them to obtain a set of linear equations in the finite number of unknown boundary probabilities. The remaining probabilities and other measures of interest are then computed from the boundary probabilities. Under certain technical assumptions, the linear independence of the resulting equations is established by a direct argument involving only elementary results from matrix theory and complex analysis. Simple conditions for the ergodicity and nonergodicity of the chain are also given.     

Fluid-dynamic equations for reacting gas mixtures

Starting from the Grad 13-moment equations for a bimolecular chemical reaction, Navier-Stokes-type equations are derived by asymptotic procedure in the limit of small mean paths. Two physical situations of slow and fast reactions, with their different hydrodynamic variables and conservation equations, are considered separately, yielding different limiting results.This work was performed in the frame of the activities sponsored by MIUR (Project Mathematical Problems of Kinetic Theories), by INdAM, by GNFM, by the University of Parma (Italy), and by the European TMR Network Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis.This revised version was published online in April 2005 with a corrected issue number.     

Structure equations and constraint manifolds on Lorentz plane

Calculating the structure equation of a chain is important to represent the position of the end link on the chain. Furthermore, the structure equation helps to determine the constraint manifold of the chain. The constraint manifold satisfies to make geometric interpretations about the form that is obtained. What is more, the constraint forced on the positions of the end link by the rest of the chain is represented by the manifold. In Lorentz space, the structure equations change according to the causal characters of the first link. In this paper, we attain the structure equations of a planar open chain in terms of the causal character of the first link in this space. Later, the constraint manifolds of the chain by using these equations are given. Some geometric comments about these manifolds are explained.     

Control problems with time-lag. II

Consider a continuous time Markov chain with stationary transition probabilities. A function of the state is observed. A regular conditional probability distribution for the trajectory of the chain, given observations up to time t, is obtained. This distribution also corresponds to a Markov chain, but the conditional chain has nonstationary transition probabilities. In particular, computation of the conditional distribution of the state at time s is discussed. For s > t, we have prediction (extrapolation), while s < t corresponds to smoothing (interpolation). Equations for the conditional state distribution are given on matrix form and as recursive differential equations with varying s or t. These differential equations are closely related to Kolmogorov's forward and backward equations. Markov chains with one observed and one unobserved component are treated as a special case. In an example, the conditional distribution of the change-point is derived for a Poisson process with a changing intensity, given observations of the Poisson process.     

SINGULARLY PERTURBED REACTION‐DIFFUSION SYSTEMS IN CASES OF EXCHANGE OF STABILITIES

ABSTRACT. We study singularly perturbed elliptic and parabolic differential equations under the assumption that the associated equation has intersecting families of equilibria (exchange of stabilities). We prove by means of the method of asymptotic lower and upper solutions that the asymptotic behavior with respect to the small parameter changes near the curve of exchange of stabilities. The application of that result to systems modeling fast bimolecular reactions in a heterogeneous environment implies a transition layer (jumping behavior) of the reaction rate. This behavior has to be taken into account for identification problems in reaction systems.     

Zero-dispersion limit to the Korteweg-de Vries equation: a dressing chain approach

An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax-Levermore and Gurevich-Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat [1]. It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in x asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.      

Hirota’s difference equations

A review of selected topics for Hirota’s bilinear difference equation (HBDE) is given. This famous three-dimensional difference equation is known to provide a canonical integrable discretization for most of the important types of soliton equations. Similar to continuous theory, HBDE is a member of an infinite hierarchy. The central point of our paper is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to two-dimensional equations are considered, with discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain, and other examples. The article was written by the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 179–230, November, 1997.     

No limit cycle in two species second order kinetics

We give a new and direct proof of the nonexistence of limit cycle in a bimolecular system and the characterization of the unique bimolecular oscillator. The proof is an application of classification theorems on vector fields with homogeneous second degree polynomial perturbations.     

Default Times in a Continuous Time Markov Chain Economy

Abstract

A continuous time financial market is considered where randomness is modelled by a finite state Markov chain. Using the chain, a stochastic discount factor is defined. The probability distributions of default times are shown to be given by solutions of a system of coupled partial differential equations.     

Persistent currents in open spin chains

Nonparallel boundary magnetic fields can induce a longitudinal (spin) current in a quantum spin chain. We use functional approaches to formulate the spectral problem for the spin-1/2 Heisenberg model subject to a class of integrable boundary conditions in terms of an infinite hierarchy of nonlinear integral equations. From these equations, we compute finite-size corrections to the ground-state energy of the antiferromagnetic chain and the induced spin current for a certain range of boundary parameters.     

Three-Dimensional Maneuvering and Control of Flexible Robots

This paper develops a general approach to the three-dimensional maneuver and vibration control of a robot in the form of a chain of flexible links. The equations for the rigid-body maneuvering motions are derived by means of Lagrange equations in terms of quasi-coordinates and the equations for the elastic deformations by means of ordinary Lagrange equations. The equations of motion are derived for the full system simultaneously, using recursive equations to relate the motions of a given link to the motions of the preceding links in the chain. The maneuver is carried out by means of joint torques and the vibration is suppressed by means of point actuators dispersed throughout the links. The controls are designed by the Liapunov direct method. A numerical example demonstrates the theoretical developments.     

Optimal method for truncating a chain of equations for two-time Green’s functions

We use an example of a chain of equations describing a system of Bose particles with pairwise interaction to develop a method for decoupling the chain at its second element. We obtain an approximation of the interacting-modes type, which results in a system of nonlinear equations for one-, two-, and three-particle functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 503–510, June, 2006.     

Limit relation between toda chains and the elliptic SL(N, ?) top

We study a limit relation between the elliptic SL(N,?) top and Toda chains. We show that in the case of the nonautonomous SL(2, ?) top, whose equations of motion are related to the Painlevé VI equation, it turns out to be possible to modify the previously proposed procedure and in the limit obtain the nonautonomous Toda chain, whose equations of motion are equivalent to a particular case of the Painlevé III equation. We obtain the limit of the Lax pair for the elliptic SL(2, ?) top, which allows representing the equations of motion of the nonautonomous Toda chain as the equation for isomonodromic deformations.     

测度链上具有变号非线性项微分方程的正解(英文)

本文研究了测度链上具有变号非线性项微分方程的问题.利用拓扑方法,获得了此微分方程的正解存在性结果,推广和改进了一些文献中相应的结果.