Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Extraction of P-and S-wave angle-domain common-image gathers based on first-order velocity-dilatation-rotation equations

Accuracy of angle-domain common-image gathers(ADCIGs)is the key to multiwave AVA inversion and migration velocity analysis,and of which Poynting vectors of pure P-and S-wave are the decisive factors in obtaining multi-component seismic data ADCIGs.A Poynting vector can be obtained from conventional velocity-stress elastic wave equations,but it focused on the propagation direction of mixed P-and S-wave fields,and neither on the propagation direction of the P-wave nor the direction of the S-wave.The Poynting vectors of pure P-or pure S-wave can be calculated from first-order velocity-dilatation-rotation equations.This study presents a method of extracting ADCIGs based on first order velocitydilatation-rotation elastic wave equations reverse-time migration algorithm.The method is as follows:calculating the pure P-wave Poynting vector of source and receiver wavefields by multiplication of P-wave particle-velocity vector and dilatation scalar,calculating the pure S-wave Poynting vector by vector multiplying S-wave particle-velocity vector and rotation vector,selecting the Poynting vector at the time of maximum P-wave energy of source wavefield as the propagation direction of incident P-wave,and obtaining the reflected P-wave(or converted S-wave)propagation direction of the receiver wavefield by the Poynting vector at the time of maximum P-(S-)wave energy in each grid point.Then,the P-wave incident angle is computed by the two propagation directions.Thus,the P-and S-wave ADGICs can obtained Numerical tests show that the proposed method can accurately compute the propagation direction and incident angle of the source and receiver wavefields,thereby achieving high-precision extraction of P-and S-wave ADGICs.

Numerical Analysis of Upwind Difference Schemes for Two-Dimensional First-Order Hyperbolic Equations with Variable Coefficients

In this paper, we consider the initial-boundary value problem of two-dimensional first-order linear hyperbolic equation with variable coefficients. By using the upwind difference method to discretize the spatial derivative term and the forward and backward Euler method to discretize the time derivative term, the explicit and implicit upwind difference schemes are obtained respectively. It is proved that the explicit upwind scheme is conditionally stable and the implicit upwind scheme is unconditionally stable. Then the convergence of the schemes is derived. Numerical examples verify the results of theoretical analysis.

A POSTERIORI ERROR ESTIMATE OF THE DSD METHOD FOR FIRST-ORDER HYPERBOLIC EQUATIONS

A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.

Finite Element Approach for the Solution of First-Order Differential Equations

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION WITH EXTERNAL FORCING IN THE WHOLE SPACE

In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

THE GLOBAL EXISTENCE AND ANALYTICITY OF A MILD SOLUTION TO THE 3D REGULARIZED MHD EQUATIONS

In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small initial data.In addition,we also obtain the Gevrey class regularity and the temporal decay rate of the solution.

THE OPTIMAL LARGE TIME BEHAVIOR OF3D QUASILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DAMPING

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.

EQUATIONS OF MOTION AND BOUNDARY CONDITIONS OF INCREMENTAL RATE TYPE FOR POLAR CONTINUA

The relations between various couple stress tensors and their change rates are derived. The equations of angular momentum and the corresponding boundary conditions of incremental rate type are presented. Thus the equations of motion and the boundary conditions of incremental rate type of Cauchy form, Piola form and Kirchhoff from for polar continua are obtained in combination of these results with those for classical continuum mechanics derived by kuang Zhenbang.

CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.

TIME DECAY RATE OF SOLUTIONS TO THE HYPERBOLIC MHD EQUATIONS IN R3

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R^3. Then we establish that the solutions with initial data belonging to H^m(R^3) ∩ L^1(R^3) have the following time decay rate:║▽~mu(x, t) ║~2+║ ▽~mb(x, t)║~ 2+ ║▽^(m+1)u(x, t)║~ 2+ ║▽^(m+1)b(x, t) ║~2≤ c(1 + t)^(-3/2-m)for large t, where m = 0, 1.

Contributions to Hom-Schunck optical flow equations-part I： Stability and rate of convergence of classical algorithm

Globally exponential stability （which implies convergence and uniqueness） of their classical iterative algorithm is established using methods of heat equations and energy integral after embedding the discrete iteration into a continuous flow. The stability condition depends explicitly on smoothness of the image sequence, size of image domain, value of the regularization parameter, and finally discretization step. Specifically, as the discretization step approaches to zero, stability holds unconditionally. The analysis also clarifies relations among the iterative algorithm, the original variation formulation and the PDE system. The proper regularity of solution and natural images is briefly surveyed and discussed. Experimental results validate the theoretical claims both on convergence and exponential stability.

Quenching Rate for the Porous Medium Equation with a Singular Boundary Condition

We study the porous medium equation ut=(um). 0<x<∞, t>0 with a singular boundary condition (um) (0,t)=u-β(0,t). We prove finite time quenching for the solution at the boundary χ=0. We also establish the quenching rate and asymptotic behavior on the quenching point.

Analysis Expression of Rate Equation for Vertical Cavity Lasers

An analysis solution of rate equation is derived for vertical cavity surface-emitting lasers. Based on the enhanced spontaneous emission caused by VCSELs and influence of nonradiative recombination, the relation between output properties and structural parameters of multi-quantum wells (MQWs) is obtained. It was found that the characteristic curve of a“thresholdless”laser is strongly nonradiative depopulation-dependent. When the nonradiative depopulation is no zero, the light-current characteristic is not linearly even for an ideal closed microcavity. The light output is increased by the enhanced well number and by the reduced width. In particular, a lower threshold current density for MQW structure in the short cavity is realized by us, meanwhile the sharpness of the variation depends on spontaneous emission factor.

Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

Accuracy of glomerular filtration rate estimation equations in patients with hematopathy

Renal dysfunction is a common side-effect of chemotherapeutic agents in patients with hematopathy. Although broadly used, glomerular filtration rate(GFR) estimation equations were not fully validated in this specific population. Thus, this study was designed to further assess the accuracy of various GFR equations, including the newly 2012 CKD-EPI equations. Referring to ^(99m)Tc-DTPA clearance method, three Scr-based(MDRD, Peking, and CKD-EPI_(Scr)), three Scys C-based(Steven 1, Steven 2, and CKD-EPI_(Scys C)), and three Scr-Scys C combination based(Ma, Steven 3, and CKD-EPI_(Scr-Scys C)) equations were included. Bias, P_(30), and misclassification rate were applied to compare the applicability of the selected equations. A total of 180 Chinese hematological patients were enrolled.Mean bias, absolute mean bias, P_(30), misclassification rate and Bland-Altman plots of the CKD-EPI_(Scr-Scys C) equation were 7.90 mL/minute/1.73 m^2, 17.77 mL/minute/1.73 m^2, 73.3%, 38% and 79.7 mL/minute/1.73 m^2, respectively.CKD-EPI_(Scr-Scys C) predicted the most precise eGFR both in lymphoma and leukemia subgroups. Additionally, CKDEPI_(Scys C) equation in the rGFR■90 mL/minute/1.73 m^2 subgroup and Steven 2 equation in the rGFR<90 mL/minute/1.73 m^2 subgroup provided more accurate estimates in each subgroup. The CKD-EPI_(Scr-Scys C) equation could be recommended to monitor kidney function in hematopathy patients. The accuracy of GFR equations may be closely related with GFR level and kidney function markers, but not the primary cause of hematopathy.

KINETIC MODELING OF ESTERIFICATION OF EPOXY RESIN IN THE PRESENCE OF TRIPHENYLPHOSPHINE FOR PRODUCING VINYL ESTER RESIN: MECHANISTIC RATE EQUATION

Due to its mechanical properties and ease of use, vinyl ester resin is enjoying increasing consideration. This resin normally is produced by reaction between epoxy resin and unsaturated carboxylic acid. In the present study, bis-phenol A based epoxy resin and methacrylic acid was used to produce vinyl ester resin. The reaction was conducted under both stoichiometric and non-stoichiometric conditions in the presence of triphenylphosphine as catalyst. The stoichiometric and non-stoichiometric experiments were conducted at 95, 100, 105 and 110℃ and at 90 and 95℃, respectively. The first order rate equation and mechanism based rate equation were examined. Parameters are evaluated by least square method. A comparison of mechanism based rate equation and experimental data show an excellent agreement. Finally, Arrhenius equation and activation energy were presented.

OPTIMAL CONVERGENCE RATE OF THE LANDAU EQUATION WITH FRICTIONAL FORCE

The Cauchy problem of the Landau equation with frictional force is investigated. Based on Fourier analysis and nonlinear energy estimates, the optimal convergence rate to the steady state is obtained under some conditions on initial data.

Rate Equation of Small Particle-Air Bubble Attachment in Turbulent Flow of Flotation Cells

A rate equation of small particle-air bubble attachment in the turbulent now of flotation cells has beenderived. The equation, integrating both the collision probability and adhesion probability together, represents theprobability of attachment between particle and bubble in the turbulent flow. 'Capture efficiency' f(a) is introducedinto the rate equation to reflect the influence of energy hairier on the attachment rate. Three typical situations of particle-bubble interaction in flotation process have been discussed. For a completely hydrophobic particle-bubble system,f(a) = 1. This means that all collision leads to attachment. Whereas for hydrophilic particle-bubble systems, .f(a) =0. Thus no adhesion of particle on bubble occurs at all. In real notation circumstances, however, there always existsa certain energy barrier between the particle and the bubble. Therefore, f(a) = 0～1. In such cases, not all collisionsresult in particle-bubble attachment.