Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain

The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, the integrated Green’s function method has been adopted to solve the 3D Poisson equation subject to open boundary conditions. In this paper, we report on the efficient implementation of this method, which can save more than a factor of 50 computing time compared with the direct brute force implementation and its improvement under certain extreme conditions.

A Novel Accurate Method forMulti-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.

Detection of decline in estimated glomerular filtration rate in patients with type 2 diabetes by cystatin C-based equations

BACKGROUND Aging population is a significant issue in Viet Nam and across the globe.Elderly individuals are at higher risk of chronic kidney disease(CKD),especially those with diabetes.Several studies found that the estimated glomerular filtration rate(eGFR)determined using creatinine-based equations was not as accurate as that determined using cystatin C-based equations.Cystatin C-based equations may be beneficial in elderly patients with an age-associated decline in kidney function.Early determination of eGFR decline and associated factors would aid in appropriate interventions to improve kidney function in elderly patients with diabetes.AIM To determine the utility of cystatin C-based equations in early detection of eGFR decline and to explore factors associated with eGFR decline in elderly patients with diabetes.METHODS This cross-sectional study included 93 participants aged≥60 years evaluated in Can Tho University of Medicine and Pharmacy Hospital between October 2022 and July 2023,including 47 and 46 participants with and without diabetes respectively,according to the American Diabetes Association criteria for diabetes.The kappa coefficient,Student’s t,Mann-Whitney,χ2,Pearson’s correlation,multivariate logistic regression,and multiple linear regression analyses were employed.RESULTS The eGFRs were lower with the cystatin C-based equations than with the creatinine-based equations.Good agreement was found between the Modification of Diet in Renal Disease(MDRD)and CKD Epidemiology Collaboration(CKD-EPI)2021 creatinine-cystatin C equations(kappa=0.66).In the diabetes group,30%of the participants had low eGFR.Both plasma glucose and glycated hemoglobin were associated with an increased risk of eGFR decline(P<0.05)and negatively correlated with eGFR(P=0.001).By multivariate logistic regression,total cholesterol,and exercise were independently associated with low eGFR.By multiple linear regression,higher plasma glucose levels were correlated with lower eGFR(P=0.026,r=-0.366).CONCLUSION Cystatin C-based equations were superior in the early detection of a decline in eGFR,and the MDRD equation may be considered as an alternative to the CKD-EPI 2021 creatinine-cystatin C equation.Exercise,plasma glucose,and total cholesterol were independently associated with eGFR in patients with diabetes.

Enhancing Proficiency in Quadratic Equations and Functions Through the MILAPlus Strategy

This study investigates the efficacy of the Mathematics Independent Learning Activity Practice and Play Unite Scheme(MILAPlus)as an instructional strategy to improve the proficiency levels of Grade 9 students in quadratic equations and functions through a study carried out at Quezon National High School.The research involved 116 Grade 9 students and utilized a quantitative approach,incorporating both pre-assessment and post-assessment measures.The research utilizes a quasi-experimental design,examining the academic performance of students before and after the introduction of MILAPlus.The pre-assessment establishes a baseline,and the subsequent post-assessment measures the impact of the instructional strategy.Statistical analyses,including t-tests,assess the significance of differences in mean scores and mean percentage scores,providing quantitative insights into the effectiveness of MILAPlus.Findings from the study revealed a statistically significant improvement in both mean scores and mean percentage scores after the utilization of MILAPlus,indicating enhanced proficiency in quadratic equations and functions.The Mean Proficiency Scores(MPS)also showed a substantial increase,demonstrating a marked improvement in overall proficiency levels among Grade 9 students.In light of the results,recommendations were given including the continued utilization of MILAPlus as an instructional strategy and aligning its development with prescribed learning competencies.Emphasizing the consistent adherence to policies and guidelines for MILAPlus implementation is suggested for sustaining positive effects on students’long-term performance in mathematics.This research contributes valuable insights into the practical application and effectiveness of MILAPlus within the context of Grade 9 mathematics education at Quezon National High School.

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.

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.

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.

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.

Modified Scherrer Equation to Estimate More Accurately Nano-Crystallite Size Using XRD

Scherrer Equation, L=Kλ/β.cosθ, was developed in 1918, to calculate the nano crystallite size (L) by XRD radiation of wavelength λ (nm) from measuring full width at half maximum of peaks (β) in radian located at any 2θ in the pattern. Shape factor of K can be 0.62 - 2.08 and is usually taken as about 0.89. But, if all of the peaks of a pattern are going to give a similar value of L, then β.cosθ must be identical. This means that for a typical 5nm crystallite size and λ Cukα1 = 0.15405 nm the peak at 2θ = 170° must be more than ten times wide with respect to the peak at 2θ = 10°, which is never observed. The purpose of modified Scherrer equation given in this paper is to provide a new approach to the kind of using Scherrer equation, so that a least squares technique can be applied to minimize the sources of errors. Modified Scherrer equation plots lnβ against ln(1/cosθ) and obtains the intercept of a least squares line regression, ln=Kλ/L, from which a single value of L is obtained through all of the available peaks. This novel technique is used for a natural Hydroxyapatite (HA) of bovine bone fired at 600°C, 700°C, 900°C and 1100°C from which nano crystallite sizes of 22.8, 35.5, 37.3 and 38.1 nm were respectively obtained and 900°C was selected for biomaterials purposes. These results show that modified Scherrer equation method is promising in nano materials applications and can distinguish between 37.3 and 38.1 nm by using the data from all of the available peaks.

EXISTENCE OF WEAK SOLUTIONS FOR A DEGENERATE GENERALIZED BURGERS EQUATION WITH LARGE INITIAL DATA

It is obtained the existence of the weak solution for a degenerate generalized Burgers equation under the restriction u0 ∈ L∞. The main method is to add viscosity perturbation and obtain some estimates in L1 norm. Meanwhile it is obtained the solution is exponential decay when the initial data has compact support.

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.

EQUATION OF STATE BASED HYDRATE MODEL FOR NATURAL GAS SYSTEMS CONTAINING BRINE AND POLAR INHIBITOR

A new thermodynamic model for gas hydrates was established by combining the modified Patel-Teja equation of state proposed for aqueous electrolyte systems and the simplified Holder -John multi -shell hydrate model. The new hydrate model is capable of predicting the hydrate formation/dissociation conditions of natural gas systems containing pure water/formation water (brine) and polar inhibitor without using activity coefficient model. Extensive test results indicate very encouraging results.

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.

A REMARK OF THE HLDER ESTIMATE OFSOLUTIONS OF SOME DEGENERATE PARABOLICEQUATIONS

In this paper, the Cauchy problem of the degenerate parabolic equationsis studied for some cases, and the explicit Holder estimates of the solution u with respectto x is given.

A GENERALIZED PROJECTION-SUCCESSIVE LINEAR EQUATIONS ALGORITHM FOR NONLINEARLY EQUALITY AND INEQUALITY CONSTRAINED OPTIMIZATION AND ITS RATE OF CONVERGENCE

In this paper, the topological pressure is preserved under some semi conjugates, and a formula of computing topological pressure by use of periodic points for positively expansive continuous map with specification is given.

Solving mKdV-sinh-Gordon equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.