Relationship Between Soil Properties and Different Fractions of Soil Hg

Correlation and path analysis methods were used to study the relationship between soil properties and the distribution of different soil Hg fractions with nine representative soils from Chongqing, China. Results showed that clay (< 2 m) could increase water-soluble Hg (r = 0.700*). Soil organic matter (OM) could enhance the increase of elemental Hg (r = 0.674*). The higher the base saturation percentage (BSP), the more the residual Hg (T = 0.684*). Organic Hg, the sum of acid-soluble organic Hg and alkali-soluble Hg, was positively affected by silt (2-20 μm) but negatively affected by pH, with the direct path coefficients amounting to 1.0487 and 0.5121, respectively. The positive effect of OM and negative effect of BSP on organic Hg were the most significant, with the direct path coefficients being 0.7614 and -0.8527, respectively. The indirect effect of clay (< 2 μm) via BSP (path coefficient = 0.4186) was the highest, showing that the real influencing factor in the effect of clay (< 2 μm) on acid-soluble organic Hg was BSP. Since the available Hg fraction, water-soluble Hg, was positively affected by soil clay content, and the quite immobile and not bioavailable residual Hg by soil BSP, suitable reduction of clay content and increase of BSP would be of much help to reduce the Hg availability and Hg activity in Hg-contaminated soils.

Flotation kinetics performance of different coal size fractions with nanobubbles

The flotation kinetics of different size fractions of conventional and nanobubble(NB) flotation were compared to investigate the effect of NBs on the flotation performance of various coal particle sizes. Six flotation kinetics models were selected to fit the flotation data, and NBs were observed on a hydrophobic surface under hydrodynamic cavitation by atomic force microscope scanning. Flotation results indicated that the best flotation performance of size fraction at-0.125+0.074 mm can be obtained either in conventional or NB flotation. NBs increase the combustible recovery of almost all the size fractions, but they increase the product ash content of-0.25+0.074 mm and reduce the product ash content of-0.045 mm at the same time. The first-order models can be used to fit the flotation data in conventional and NB flotation, and the classical first-order model is the most suitable one. NBs considerably enhance flotation rate on coarse size fraction(-0.5+0.25 mm) but decrease the flotation rate of the medium size(-0.25+0.074 mm). The improvement of flotation speed on fine coal particles(-0.074 mm) is probably the reason for the improved performance of raw sample flotation.

A novel variable-order fractional chaotic map and its dynamics

In recent years,fractional-order chaotic maps have been paid more attention in publications because of the memory effect.This paper presents a novel variable-order fractional sine map(VFSM)based on the discrete fractional calculus.Specially,the order is defined as an iterative function that incorporates the current state of the system.By analyzing phase diagrams,time sequences,bifurcations,Lyapunov exponents and fuzzy entropy complexity,the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map.The results reveal that the variable order has a good effect on improving the chaotic performance,and it enlarges the range of available parameter values as well as reduces non-chaotic windows.Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values.Moreover,the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation,which proves the potential applications in the field of information security.

CONTROLLABILITY AND OPTIMALITY OF LINEAR TIME-INVARIANT NEUTRAL CONTROL SYSTEMS WITH DIFFERENT FRACTIONAL ORDERS

Control systems governed by linear time-invariant neutral equations with different fractional orders are considered. Sufficient and necessary conditions for the controllability of those systems are established. The existence of optimal controls for the systems is given. Finally, two examples are provided to show the application of our results.

The microstructural nanoindentation characteristics of different phases in precision annealed GCr15 steel

This study characterizes the mechanical properties and volume fractions of the different phases in precision annealed GCr15 steel using nanoindentation technology. Experimental results indicate that the nanoindentation hardness of cementite grains is between 14.15 GPa and 17.61 GPa,with a mean value of 15.40 GPa. This hardness is much higher than the hardness of ferrite grains. The nanoindentation hardness of ferrite is between 2.78 GPa and 4.89 GPa, with a mean value of 3.35 GPa. The volume fractions of the different phases were also determined using nanoindentation technology, and the volume fraction of cementite in the steel was identified as 15%.

Environmental change inferred from Rb and Sr of lacustrine sediments in Huangqihai Lake,Inner Mongolia

Based on the geochemical elements Rb and Sr in sediments with three different grain size fractions from profile H3 on the northern lacustrine bottomland 13 m above the Huangqihai Lake surface in 1986,the paper investigates the record of palaeolake stand state, sedimentary environmental evolution,and winter monsoon change.First,these samples are separated into three different grain size fractions,i.e.,total sediments,77-20μm and〈20μm. Second,the chemical elements-Rb and Sr-of the grain size separation were tested and analyzed systematically in this paper.Then the elements compositions of these samples are measured using VP-320 mode fluorescence spectrum instrument,respectively.The magnetic susceptibility of these samples is measured using Kappabridge KLY-3 mode instrument made in Czech AGICO Company.The results showed the elements and the ratios varied regularly with the grain size.But the ratio of Rb/Sr in the sediments〈20μm correlates positively with the magnetic susceptibility of these samples.Therefore,the ratio of Rb/Sr in the fraction〈20 μm from the lake sediments reflected the strengthening of the weathering in the deposition sites.It is a good indicator of the summer monsoon-induced weathering and pedogenesis fluctuations and can be used to reconstruct the conditions of the paleoclimate and paleoenvironment.

NUMERICAL METHOD OF MIXED FINITE VOLUME-MODIFIED UPWIND FRACTIONAL STEP DIFFERENCE FOR THREE-DIMENSIONAL SEMICONDUCTOR DEVICE TRANSIENT BEHAVIOR PROBLEMS

Transient behavior of three-dimensional semiconductor device with heat conduc- tion is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an ellip- tic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two con- centrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

THEORY AND APPLICATION OF FRACTIONAL STEP CHARACTERISTIC FINITE DIFFERENCE METHOD IN NUMERICAL SIMULATION OF SECOND ORDER ENHANCED OIL PRODUCTION

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced （chemical） oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l^2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results ate quite interesting and satisfactory.

ULAM-HYERS-RASSIAS STABILITY AND EXISTENCE OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENCE EQUATIONS WITH MULTIPOINT SUMMATION BOUNDARY CONDITION

The purpose of this study is to acquire some conditions that reveal existence and stability for solutions to a class of difference equations with non-integer orderμ∈(1,2].The required conditions are obtained by applying the technique of contraction principle for uniqueness and Schauder’s fixed point theorem for existence.Also,we establish some conditions under which the solution of the considered class of difference equations is generalized Ulam-Hyers-Rassias stable.Example for the illustration of results is given.

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.

Modified characteristic finite difference fractional step method for moving boundary value problem of nonlinear percolation system

A fractional step scheme with modified characteristic finite differences run- ning in a parallel arithmetic is presented to simulate a nonlinear percolation system of multilayer dynamics of fluids in a porous medium with moving boundary values. With the help of theoretical techniques including the change of regions, piecewise threefold quadratic interpolation, calculus of variations, multiplicative commutation rule of differ- ence operators, multiplicative commutation rule of difference operators, decomposition of high order difference operators, induction hypothesis, and prior estimates, an optimal order in 12 norm is displayed to complete the convergence analysis of the numerical algo- rithm. Some numerical results arising in the actual simulation of migration-accumulation of oil resources by this method are listed in the last section.

Finite Di erence Method for Riesz Space Fractional Advection-dispersion Equation with Fractional Robin Boundary Condition

In this paper,a Riesz space fractional advection-dispersion equation with fractional Robin boundary condition is considered.By applying the fractional central di erence formula and the weighted and shifted Grunwald-Letnikov formula,we derive a weighted implicit nite difference scheme with accuracy O(△t^2+h^2).The solvability,stability,and convergence of the proposed numerical scheme are proved.A numerical example is presented to confirm the accuracy and efficiency of the scheme.

CONVERGENCE ANALYSIS OF MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT FOR THREE-DIMENSIONAL CHEMICAL OIL-RECOVERY SEEPAGE COUPLED PROBLEM

The physical model is described by a seepage coupled system for simulating numerically three-dimensional chemical oil recovery, whose mathematical description includes three equations to interpret main concepts. The pressure equation is a nonlinear parabolic equation, the concentration is defined by a convection-diffusion equation and the saturations of different components are stated by nonlinear convection-diffusion equations. The transport pressure appears in the concentration equation and saturation equations in the form of Darcy velocity, and controls their processes. The flow equation is solved by the conservative mixed volume element and the accuracy is improved one order for approximating Darcy velocity. The method of characteristic mixed volume element is applied to solve the concentration, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computational stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. The scheme can adopt a large step while its numerical results have small time-truncation error and high order of accuracy. The mixed volume element method has the law of conservation on every element for the diffusion and it can obtain numerical solutions of the concentration and adjoint vectors. It is most important in numerical simulation to ensure the physical conservative nature. The saturation different components are obtained by the method of characteristic fractional step difference. The computational work is shortened greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and it is completed easily by using the algorithm of speedup. Using the theory and technique of a priori estimates of differential equations, we derive an optimal second order estimates in 12 norm. Numerical examples are given to show the effectiveness and practicability and the method is testified as a powerful tool to solve the important problems.

A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation

By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation,and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative,the fractional central difference operator to discritize the space Riesz fractional partial derivative,and the classical central difference formula to discretize the advection term,a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion equation.The stability and the convergence analysis of the numerical method are given.Numerical experiments show that the numerical method is effective.

An Efficient Approach for Solving One-Dimensional Fractional Heat Conduction Equation

Several researchers have dealt with the one-dimensional fractional heat conduction equation in the last decades,but as far as we know,no one has investigated such a problem from the perspective of developing suitable fractional-order methods.This has actually motivated us to address this problem by the way of establishing a proper fractional approach that involves employing a combination of a novel fractional difference formula to approximate the Caputo differentiator of orderαcoupled with the modified three-point fractional formula to approximate the Caputo differentiator of order 2α,where 0<α≤1.As a result,the fractional heat conduction equation is then reexpressed numerically using the aforementioned formulas,and by dividing the considered mesh into multiple nodes,a system is generated and algebraically solved with the aid of MATLAB.This would allow us to obtain the desired approximate solution for the problem at hand.

Nontrivial solutions for a class of fractional difference boundary value problems and fixed-point problems

In this work,we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:{T△^(v)_(t-1)(t△^(v)_(v-1)x(t))=f(x(t+v-1)),t∈[0,T]N_(0),x(v-2)=[tΔ^(v)_(v-1)x(t)]_(t=T=0.The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.

SOME PROBLEMS OF FRACTIONAL PARTIAL DIFFERENCE DIFFUSION EQUATIONS

This paper mainly discusses the problems of fractional variational problems and fractional diffusion problems using fractional difference and summation. Through the Euler finite difference method we get a variational formulation of the variation problem and the discrete solution to the time-fractional and space-fractional difference equation using separating variables method and two-side Z-transform method.

A Mixed-finite Volume Element Coupled with the Method of Characteristic Fractional Step Difference for Simulating Transient Behavior of Semiconductor Device of Heat Conductor And Its Numerical Analysis

The mathematical system is formulated by four partial differential equations combined with initial- boundary value conditions to describe transient behavior of three-dimensional semiconductor device with heat conduction. The first equation of an elliptic type is defined with respect to the electric potential, the successive two equations of convection dominated diffusion type are given to define the electron concentration and the hole concentration, and the fourth equation of heat conductor is for the temperature. The electric potential appears in the equations of electron concentration, hole concentration and the temperature in the formation of the intensity. A mass conservative numerical approximation of the electric potential is presented by using the mixed finite volume element, and the accuracy of computation of the electric intensity is improved one order. The method of characteristic fractional step difference is applied to discretize the other three equations, where the hyperbolic terms are approximated by a difference quotient in the characteristics and the diffusion terms are discretized by the method of fractional step difference. The computation of three-dimensional problem works efficiently by dividing it into three one-dimensional subproblems and every subproblem is solved by the method of speedup in parallel. Using a pair of different grids （coarse partition and refined partition）, piecewise threefold quadratic interpolation, variation theory, multiplicative commutation rule of differential operators, mathematical induction and priori estimates theory and special technique of differential equations, we derive an optimal second order estimate in L2-norm. This numerical method is valuable in the simulation of semiconductor device theoretically and actually, and gives a powerful tool to solve the international problem presented by J. Douglas, Jr.

EXISTENCE OF MULTIPLE SOLUTIONS TO A FRACTIONAL DIFFERENCE BOUNDARY VALUE PROBLEM WITH PARAMETER

By establishing the corresponding variational framework, and using critical point theory, we give the existence of multiple solutions to a fractional difference boundary value problem with parameter. Under some suitable assumptions we obtain some results which ensure the existence of well precise interval of parameter for which the problem admits multiple solutions.

THE UPWIND FINITE DIFFERENCE FRACTIONAL STEPS METHOD FOR NONLINEAR COUPLED SYSTEM OF DYNAMICS OF FLUIDS IN POROUS MEDIA

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, trod two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.