Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.

A SEMI-ANALYSIS METHOD OF DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS UNDER COMPLICATED BOUNDARY CONDITIONS

Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential equations with variable coefficients. On most occasions and due to the nonuniformity nature, nonlinearity property can cause the equations of the kinds. Using the model, the satisfactory valuable results with only a few units can be obtained.

INSTABILITY OF SOLUTION FOR THE FOURTH ORDER LINEAR DIFFERENTIAL EQUATION WITH VARIED COEFFICIENT

In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.

EXACT ANALYTIC METHOD FG(?) SOLVING VARIABLE COEFFICIENT DIFFERENTIAL EQUATION

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation formal is obtained. Its convergence in proved. We can get analytic expressions which converge to exact solution and its higher order derivatives uniformy Four numerical examples are given, which indicate that satisfactory results can he obtanedby this method.

A General Converse Comparison Theorem for Backward Stochastic Differential Equation with Non-lipschitz Coefficient

In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.

DISCUSSION ON″THE BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OR SOLUTION DIFFERENTIAL SYSTEM OF SECOND-ORDER WITH VARIABLE COEFFICIENT" (App1ied Mathematics and Mechanics,Vo1.3,No.4,1982)

After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:

Thermodynamic Consistency and Thermomechanical Dynamics (TMD) for Nonequilibrium Irreversible Mechanism of Heat Engines

The irreversible mechanism of heat engines is studied in terms of thermodynamic consistency and thermomechanical dynamics (TMD) which is proposed for a method to study nonequilibrium irreversible thermodynamic systems. As an example, a water drinking bird (DB) known as one of the heat engines is specifically examined. The DB system suffices a rigorous experimental device for the theory of nonequilibrium irreversible thermodynamics. The DB nonlinear equation of motion proves explicitly that nonlinear differential equations with time-dependent coefficients must be classified as independent equations different from those of constant coefficients. The solutions of nonlinear differential equations with time-dependent coefficients can express emergent phenomena: nonequilibrium irreversible states. The couplings among mechanics, thermodynamics and time-evolution to nonequilibrium irreversible state are defined when the internal energy, thermodynamic work, temperature and entropy are integrated as a spontaneous thermodynamic process in the DB system. The physical meanings of the time-dependent entropy, T(t)dS(t), , internal energy, dƐ(t), and thermodynamic work, dW(t), are defined by the progress of time-dependent Gibbs relation to thermodynamic equilibrium. The thermomechanical dynamics (TMD) approach constitutes a method for the nonequilibrium irreversible thermodynamics and transport processes.

The theoretical analysis of dynamic response on cantilever beam of variable stiffness

The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable co- efficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last, the study obtains the solution of dy- namic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied, the dynamic characteristics are analyzed under several cases. The results dem- onstrate that the natural angular frequency reduces as the fiexural rigidity reduces. When the rigidity of clamped end is higher than that of free end, low-level mode contributes the larger displacement response to the total re- sponse. On the contrary, the contribution of low-level mode is lesser than that of hi^h-level mode.

Negative differential friction coefficients of two-dimensional commensurate contacts dominated by electronic phase transition

Friction force(f)usually increases with the normal load(N)macroscopically,according to the classic law of Da Vinci–Amontons(f=μN),with a positive and finite friction coefficient(μ).Herein near-zero and negative differential friction(ZNDF)coefficients are discovered in two-dimensional(2D)van der Waals(vdW)magnetic CrI_(3)commensurate contacts.It is identified that the ferromagnetic–antiferromagnetic phase transition of the interlayer couplings of the bilayer CrI_(3)can significantly reduce the interfacial sliding energy barriers and thus contribute to ZNDF.Moreover,phase transition between the in-plane(p_(x)and p_(y))and out-of-plane(p_(z))wave-functions dominates the sliding barrier evolutions,which is attributed to the delicate interplays among the interlayer vdW,electrostatic interactions,and the intralayer deformation of the CrI_(3)layers under external load.The present findings may motivate a new concept of slide-spintronics and are expected to play an instrumental role in design of novel magnetic solid lubricants applied in various spintronic nano-devices.

Microsatellite Marker-based Analysis of the Genetic Diversity of Yellow River Carp

To provide accurate base data about the genetic sourees of Yellow River carps, the genetic diversity in a^ficially bred population and wild population of Yellow River carps from Henan Province was analyzed with mierosatellite markers. The results showed that 16 alleles were detected at six microsateUite loci in each population. The average effective number of alleles （Ne） was 2. 350 in artificially bred population and 2. 085 in wild population. The observed heterozygosity （Ho） of artificially bred population, wild population and mixed population was 0. 614, 0. 576 and 0. 601 ; and the unbiased expected heterozygesity （ He ） was 0. 569, 0.535 and 0.559 ; and the polymorphism infonnatian content （PlC） was 0.474, 0.428 and 0.468, respectively. The PIC of the six loci ranged from 0.304 to 0. 864. The analysis of the genetic differentiation for the six microsatellitc loci in the two populations showed that the genetic differentiation coefficient （ F,, ） at only one microsatellite locus HLJ483 was greater than 0.05, and that at five rnicrosatellite loci were less than 0.05, which was consistent with the standard of non- genetic differentiation between populations （F,, = 0 -0.05）. The average F,, at the six loci was 0.02, and the gene flow value （Nm） at all loci was greater than 1 and the average of Nm was 12.202. The results indicate that there is relatively abundant genetic diversity in Yellow River carps.

ON A SYSTEM OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH PERIODIC IMPULSE COEFFICIENTS

A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be applied to one with other periodic impulse coefficients.

APPLICATIONS OF MALLIAVIN CALCULUS TO STOCHASTIS DIFFERENTIAL EQUATIONS WITH TIME-DEPENDENT COEFFICIENTS

In this paper, we apply Malliavin calculus to discuss when the solutions of stochastic differen-tial equations (SDE's) with time-dependent coefficients have smooth density. Under Hormander'scondition,we conclude that the solutions of the SDE's have smooth density. As a consequence,we get the hypoellipticity for inhomogeneous differential operators.

Fuzzy Modeling, Tracking Control and Synchronization of the Rossler's Chaotic System

In this paper, a novel method to model, track control and synchronize the Rossler’s chaotic system is proposed. The fuzzy logical system is used so that the fuzzy inference rule is transferred into a type of variable coefficient nonlinear ordinary differential equation. Consequently the model of the chaotic system is obtained. Then a fuzzy tracking control and a fuzzy synchronization for chaotic systems is proposed as well. First, a known tracking control for the Rossler’s system is used in this paper. We represent the Rossler’s chaotic and control systems into fuzzy inference rules. Then the variable coefficient nonlinear ordinary differential equation is also got. Simulation results show that such an approach is effective and has a high precision.

TYPES AND CRITERIA OF NONOSCILLATORY SOLUTIONS FOR A CLASS OF SECOND ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONPOSITIVE COEFFICIENTS

In this paper we discuss the types and criteria of nonoscillatory solutions for the fol-lowing second order neutral functional differential equation with nonpositive coefficients