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.

Vertex solution theorem for the upper and lower bounds on the dynamic response of structures with uncertain-but-bounded parameters

The aim of this paper is to evaluate the effects of uncertain-but-bounded parameters on the dynamic response of structures. By combining the interval mathematics and the finite element analysis, the mass matrix, damping matrix, stiffness matrix and the external loads are represented as interval matrices and vector. With the help of the optimization theory, we present the vertex solution theorem for determining both the exact upper bounds or maximum values and the exact lower bounds or minimum values of the dynamic response of structures, in which these parameters reach their extreme values on the boundary of the interval mass, damping, stiffness matrices and the interval extemal loads vector. Three examples are used to illustrate the computational aspects of the presented vertex solution theorem.

WAVELET-BASED ESTIMATOR FOR THE HURST PARAMETERS OF FRACTIONAL BROWNIAN SHEET

It is proposed a class of statistical estimators H = （H1,… ,Hd） for the Hurst parameters H = （H1,… ,Hd） of fractional Brownian field via multi-dimensional wavelet analysis and least squares, which are asymptotically normal. These estimators can be used to detect self-similarity and long-range dependence in multi-dimensional signals, which is important in texture classification and improvement of diffusion tensor imaging （DTI） of nuclear magnetic resonance （NMR）. Some fractional Brownian sheets will be simulated and the simulated data are used to validate these estimators. We find that when Hi ≥ 1/2, the estimators are accurate, and when Hi 〈 1/2, there are some bias.

Analytical Solutions of System of Non-Linear Differential Equations in the Single-Enzyme, Single-Substrate Reaction with Non-Mechanism-Based Enzyme Inactivation

A closed form of an analytical expression of concentration in the single-enzyme, single-substrate system for the full range of enzyme activities has been derived. The time dependent analytical solution for substrate, enzyme-substrate complex and product concentrations are presented by solving system of non-linear differential equation. We employ He’s Homotopy perturbation method to solve the coupled non-linear differential equations containing a non-linear term related to basic enzymatic reaction. The time dependent simple analytical expressions for substrate, enzyme-substrate and free enzyme concentrations have been derived in terms of dimensionless reaction diffusion parameters ε, λ1, λ2 and λ3 using perturbation method. The numerical solution of the problem is also reported using SCILAB software program. The analytical results are compared with our numerical results. An excellent agreement with simulation data is noted. The obtained results are valid for the whole solution domain.

Darboux Transformation and New Multi-Soliton Solutions of the Whitham-Broer-Kaup System

Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.

New Exact Solutions of the CDGSK Equation Related to a Non-local Symmetry

A non-local symmetry of the Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK)equation has been used for finding exact solution in two different ways.Firstly,using the standard prolongation approach,we obtain the finite Lie Bäcklund transformation and the single soliton solution.Secondly,combining some local symmetries and the nonlocal symmetry,we get the group invariant solution which is described by the Weierstrass elliptic function and is deduced to the so-called interacting soliton for a special parameter.

Some Remarks on the Method of Fundamental Solutions for Two- Dimensional Elasticity

In this paper,some remarks for more efficient analysis of two-dimensional elastostatic problems using the method of fundamental solutions are made.First,the effects of the distance between pseudo and main boundaries on the solution are investigated and by a numerical study a lower bound for the distance of each source point to the main boundary is suggested.In some cases,the resulting system of equations becomes ill-conditioned for which,the truncated singular value decomposition with a criterion based on the accuracy of the imposition of boundary conditions is used.Moreover,a procedure for normalizing the shear modulus is presented that significantly reduces the condition number of the system of equations.By solving two example problems with stress concentration,the effectiveness of the proposed methods is demonstrated.

Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions

The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.

GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.

The universal characteristic water content of aqueous solutions

Raman measurements at room temperature reveal a characteristic concentration for a series of aqueous solutions of electrolytes, through which O–H stretching vibration of H2 O or dilute HDO obviously changes their concentration dependence. This characteristic concentration is very consistent with another, through which the solutions undergo an abrupt change in their glass-forming ability. Interestingly, the molar ratio of water to solute at these two consistent concentration points is almost solute-type independent and about twice the hydration number of solutes. We suggest that just when the concentration increases above this characteristic concentration, bulk-like free water disappears in aqueous solutions and all water molecules among closely-packed hydrated solutes exhibit the characteristics of confined water.

Theoretical analysis of microscopic oil displacement mechanism by viscoelastic polymer solution

The microscopic oil displacement mechanism in viscoelastic polymer flooding is theoretically analyzed with mechanical method.The effects of viscoelasticity of polymer solution on such three kinds of residual oil as in pore throat,in sudden expansion pore path,and in dead end are analyzed.Results show that the critical radius of mobile residual oil for viscoelastic polymer solution is larger than that for viscous polymer solution,which makes the oil that is immobile in viscous polymer flooding displaced under the condition of viscoelastic polymer solution.The viscous polymer solution hardly displaces the oil in dead ends.However,when the effect of viscoelasticity is considered,the residual oil in sudden expansion pore paths and dead ends can be partly displaced. A dimensionless parameter is suggested to denote the relative dominance of gravity and capillary pressure.The larger the dimensionless parameter,the more accurate the increment expressions.

THE ANALYTICAL SOLUTION WITH RESPECT TO CHARACTERISTICS OF ELEMENTS' CROSS-SECTION AS VARIABLES OF THE PLANE FRAME

Taking the sectional area and the bending moment of inertia as variables for each beam element, the plane frame will possess a stiffness matrix containing parameters.In terms of the symbolic computation software, the inverse matrix is solved to obtain the new analytical solution with respect ic characteristics of elements cross-section.The general program is coded in the microcomputer and corresponding exmpales are computed.

Types of focal mechanism solutions and parameter consistency of the sub-blocks in Sichuan and Yunnan Provinces

Based on P- and S-wave amplitudes and some clear initial P-wave motion data, we calculated focal mechanism solutions of 928 M≥2.5 earthquakes （1994-2005） in four sub-blocks of Sichuan and Yunnan Provinces, namely Sichuan-Qinghai, Yajiang, Central Sichuan and Central Yunnan blocks. Combining these calculation results with those of the focal mechanism solutions of moderately strong earthquakes, we analyzed the stress field characteristics and dislocation types of seismogenic faults that are distributed in the four sub-blocks. The orientation of principal compressive stress for each block is： EW in Sichuan-Qinghai, ESE or SE in Yajiang, Central Sichuan and Central Yunnan blocks. Based on a great deal of focal mechanism data, we designed a program and calculated the directions of the principal stress tensors, σ1, σ2 and σ3, for the four blocks. Meanwhile, we estimated the difference （also referred to as consistency parameter θ^- ） between the force axis direction of focal mechanism solution and the direction of the mean stress tensor of each block. Then we further analyzed the variation of θ^- versus time and the dislocation types of seismogenic faults. Through determination of focal mechanism solutions for each block, we present information on the variation in θ^- value and dislocation types of seismogenic faults.

Asymptotic Formulas of the Solutions and the Trace Formulas for the Polynomial Pencil of the Sturm-Liouville Operators

This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.

A Method for the Solution of the 2D-Oswatitsch Equations

Corresponding to Oswatitsch’s Mach number independence principle the Mach number of hypersonic inviscid flows, , does not affect components of various non-dimensional formulations such as velocity and density, pressure coefficients and Mach number behind a strong shock. On this account, the principle is significant in the development process for hypersonic vehicles. Oswatitsch deduced a system of partial differential equations which describes hypersonic flow. These equations are the basic gasdynamic equations as well as Crocco’s theorem which are reduced for the case of very high Mach numbers, . Their numerical solution can not only result in simplified algorithms prospectively utilized to describe the flow around bodies flying mainly in the lower stratosphere with very high Mach numbers. It also offers a deeper understanding of similarity effects for hypersonic flows. In this paper, a solution method for Oswatisch’s equations for perfect gas, based on a 4-step Runge-Kutta-algorithm, is presented including a fast shock-fitting procedure. An analysis of numerical stability is followed by a detailed comparison of results for different Mach numbers and ratios of the specific heats.

DECAY ESTIMATE OF THE SOLUTIONS TO THE MAGNETO-SEMICONDUCTOR EQUATIONS

In this paper,the relationship between the time-dependent solutions andsteady-state solutions of the semiconductor equations affected by magnetic field isconsidered.A decay estimate between the time-dependent solutions and steady-statesolutions is proved by a series of estimetes on the solutions under some conditions.

GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation（δ2 gij）/δt2＋μ/（（1 ＋ t）λ）（δ gij）/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ ＋ 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R（t, x） of the solution metric gij remains uniformly bounded.

Analytical Solution of the Flow Field Induced by the Uniformly Moving Double Helical Vortex Filaments

The analytical expressions was deduced for the inviscid flow field induced by the double vortex filaments that move uniformly and rigidly without change of its form in a cylindrical tube, where the vortex filaments rotate around its axial with a constant angular velocity and translates along its axial with a constant transferal velocity. It is a key of solving problem to set up a moving cylindrical coordinate system together with the vortex filaments motion, in which the relative velocity field is presumed to be time-independent and with helical symmetry. The result shows that the absolute velocity field and pressure field are all time-periodic functions, and may degenerate into a time-independent field when the helical vortex filaments slip along the filaments themselves or is immobile. The calculation results at the location of pressure peaks and valleys on pipe wall are accordant with experimental results. When the cylindrical pipe radius tends to infinitely large quantity, it is also concluded that the double helical vortex filaments induce flow field in an unbound space.

Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.

Fermi Energy-Incorporated Generalized BCS Equations for the Temperature-Dependent Critical Current Density and the Related Parameters of a Superconductor for All <i>T</i>≤ <i>Tc</i>and Their Application to Aluminium Strips

Presented here are the Generalized BCS Equations incorporating Fermi Energy for the study of the {Δ, Tc, jc(T)} values of both elemental and composite superconductors (SCs) for all T ≤ Tc, where Δ, Tc and jc(T) denote, respectively, one of the gap values, the critical temperature and the T-dependent critical current density. This framework, which extends our earlier study that dealt with the {Δ0, Tc, jc(0)} values of an SC, is also shown to lead to T-dependent values of several other related parameters such as the effective mass of electrons, their number density, critical velocity, Fermi velocity (VF), coherence length and the London penetration depth. The extended framework is applied to the jc(T) data reported by Romijn et al. for superconducting Aluminium strips and is shown not only to provide an alternative to the explanation given by them, but also to some novel features such as the role of the Sommerfeld coefficient γ(T) in the context of jc(T) and the role of VF(T) in the context of a recent finding by Plumb et al. about the superconductivity of Bi-2212.