一类双调和方程边值问题的四阶和六阶Richardson外推法

首先,对双调和方程建立了具有二阶精度的差分格式,证明了该差分格式的解存在唯一性,并利用极值原理证明了其收敛性和稳定性;利用双重Richardson外推法对该差分格式做了一次外推和二次外推,在内结点处使得计算结果分别具有四阶精度和六阶精度。研究结果表明,外推算法在一定条件下对于提高差分格式解的精度有显著的作用。

Differential transform method for solving Richards' equation

An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.

Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation

By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.

基于Richards方程的薏苡不同部位籽粒灌浆性能和品质的差异及其相关性研究

【目的】探明薏苡不同部位籽粒灌浆结实性能、品质性状差异及其相关性,为薏苡强弱势籽粒产量性能优化与选育优质高产栽培品种提供参考依据。【方法】采用单因素随机区组设计结合Richards方程研究不同薏苡[贵薏1号(GY.1)和师宗黑壳(SZHK)]各部位(上部、中部、下1、下2和下3)籽粒干重、灌浆速率、籽粒充实度和淀粉含量等相关性状指标的差异,灌浆结实特性与灌浆参数及品质性状指标的相关性。【结果】籽粒灌浆方程W=78.76/(1+379259e^(0.27t))^(1/7.91),籽粒最大灌浆速率为1.82 mg/d,平均1.12 mg/d,生长活跃期73.61 d,平均结实率74.55%,平均充实度76.84%,籽粒百粒重和单茎总粒重分别为7.70和13.49 g。籽粒干重随花后天数逐步增加,在第49天时达最大;灌浆速率与花后天数呈先增后降趋势,花后第31天时达最大。籽粒平均灌浆速率,薏苡下1(1.26 mg/d)显著高于其他部位,比最低的下3(1.04 mg/d)高21.10%;最大灌浆速率同样表现为下1(2.03 mg/d)最大,比下3(1.71 mg/d)高18.70%;籽粒百粒重及充实度整体表现为下1>上部>中部>下2>下3,其中,下1(8.37 g)显著高于其他部位,分别比下2(7.41 g)、下3(7.29 g)高12.95%和14.81%;籽粒充实度下1(80.23%)比下3(72.42%)高10.78%;薏苡下1籽粒淀粉含量(38.10%)分别比下2(33.83%)、下3(33.11%)高12.62%和15.07%。相关分析表明,籽粒百粒重与籽粒平均灌浆速率(0.78)、最大灌浆速率(0.89)、淀粉含量(0.95)、淀粉积累速率(0.88)和充实度(0.87)呈极显著正相关,与可溶性糖含量(-0.93)呈极显著负相关。【结论】下1籽粒在粒重、充实度及淀粉含量上均优于其他部位,初步判明下1为薏苡强势籽粒,下2和下3为弱势籽粒。

基于Richards方程的区域连续日蒸散量遥感估算

蒸散发作为自然界水循环的重要组成部分,时空尺度上的蒸散量估算一直是研究热点。遥感手段可以实现区域尺度蒸散量的估算,但是受到卫星过境时间的限制,难以获取连续时间序列的蒸散量。土壤水分作为蒸散发的重要控制因素,结合土壤水分数据改进遥感蒸散发模型,在提高遥感蒸散量估算精度方面也具有重要意义,但是目前大多数遥感方法对土壤水分胁迫性的考虑仍有不足。针对目前蒸散发研究在土壤水分胁迫和连续性方面的不足,以涡度相关法计算的蒸散量作为实际蒸散量,结合联合国粮农组织推荐的单作物系数法,将土壤含水量信息引入Penman-Monteith(P-M)公式计算实际蒸散量,并用Richards方程进行蒸发条件下一维垂向土壤水分运动过程的数值模拟,实现土壤水分胁迫下的连续日蒸散量的估算,并结合遥感数据实现区域尺度的扩展。结果表明:涡度相关法计算的实际日蒸散量与P-M公式计算的潜在日蒸散量具有很强的相关性,相关系数达到0.918;引入土壤含水量信息后的P-M公式,日蒸散量的估算精度显著提升,均方根误差达到0.133 mm/d;基于Richards方程的土壤水分胁迫下连续日蒸散量的估算结果与实测值较为接近,均方根误差为0.288 mm/d;受研究区南北高中间低的地势影响,日蒸散量的高值集中在研究区中部的水域和耕地区域,不同土地利用类型下的平均日蒸散量水域>耕地>林地>草地>未利用土地,且区域扩展的结果与站点的实测结果在时间序列上表现出一致的变化规律。文章可为土壤水分对蒸散发的影响机理研究以及区域蒸散量的估算提供参考。

The Derivation of the Compressible Euler Equation from Quantum Many-Body Dynamics

We study the three-dimensional many-particle quantum dynamics in mean-field set-ting.We forge together the hierarchy method and the modulated energy method.We prove rigorously that the compressible Euler equation is the limit as the particle num-ber tends to infinity and the Planck’s constant tends to zero.We improve the previous sufficient small time hierarchy argument to any finite time via a new iteration scheme and Strichartz bounds first raised by Klainerman and Machedon in this context.We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation.We justify that the macroscopic pressure emerges from the space-time averages of micro-scopic interactions via the Strichartz-type bounds.We have hence found a physical meaning for Strichartz-type bounds.

A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

多波段Richards模型及其数据拟合分析

目的 在传染病传播的过程中,往往会经过反复感染,出现传播多次高峰。基于这个特点,提出了一种多波段Richards模型,对传染病的感染人数进行数值模拟。方法 利用多波段Richards模型结合最小二乘法对西太平洋区域的累计感染病例进行拟合,得到感染率、拐点、最终感染人数等参数估计值并判断其是否收敛,从而确定传染病传播的波段数以及每个波段的拐点日期。结果 结果表明:西太平洋区域在2020-01-04—2020-09-13暴发的传染病分为3个波段,其中03-15和07-04为分割3个波段的两个拐点,由第一波段求得的基本再生数约为8.53,说明这次传染病的传播能力很强。结论 借助多波段Richards模型可以有效拟合多波段传染病的感染情况,能够确定传染病在多阶段暴发中的关键转折点,以及预测未来一段时间的感染规模,为制定传染病防控措施提供参考。

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

Matrix Riccati Equations in Optimal Control

In this paper, the matrix Riccati equation is considered. There is no general way for solving the matrix Riccati equation despite the many fields to which it applies. While scalar Riccati equation has been studied thoroughly, matrix Riccati equation of which scalar Riccati equations is a particular case, is much less investigated. This article proposes a change of variable that allows to find explicit solution of the Matrix Riccati equation. We then apply this solution to Optimal Control.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R^(2),due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T×R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperatureθinto the horizontal average(ū,θ)and the corresponding oscillation(ū,θ),we can derive the global stability in H~2 and the exponential decay of(ū,θ)to zero in H^(1).Moreover,we also obtain that(ū_(2),θ)decays exponentially to zero in H^(1),and thatū_(1)decays exponentially toū_(1)(∞)in H^(1)as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H^(3)for the 3D case R^(3).

The Maxwell-Heaviside Equations Explained by the Theory of Informatons

In the articles “Newtons Law of Universal Gravitation Explained by the Theory of Informatons” and “The Gravitational Interaction between Moving Mass Particles Explained by the Theory of Informatons” the gravitational interaction has been explained by the hypothesis that information carried by informatons is the substance of gravitational fields, i.e. the medium that the interaction in question makes possible. From the idea that “information carried by informatons” is its substance, it has been deduced that—on the macroscopic level—a gravitational field manifests itself as a dual entity, always having a field- and an induction component (Egand Bg) simultaneously created by their common sources. In this article we will mathematically deduce the Maxwell-Heaviside equations from the kinematics of the informatons. These relations describe on the macroscopic level how a gravitational field (Eg, Bg) is generated by whether or not moving masses and how spatial and temporal changes of Egand Bgare related. We show that there is no causal link between Egand Bg.

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL

In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.

Prediction of ILI following the COVID-19 pandemic in China by using a partial differential equation

The COVID-19 outbreak has significantly disrupted the lives of individuals worldwide.Following the lifting of COVID-19 interventions,there is a heightened risk of future outbreaks from other circulating respiratory infections,such as influenza-like illness(ILI).Accurate prediction models for ILI cases are crucial in enabling governments to implement necessary measures and persuade individuals to adopt personal precautions against the disease.This paper aims to provide a forecasting model for ILI cases with actual cases.We propose a specific model utilizing the partial differential equation(PDE)that will be developed and validated using real-world data obtained from the Chinese National Influenza Center.Our model combines the effects of transboundary spread among regions in China mainland and human activities’impact on ILI transmission dynamics.The simulated results demonstrate that our model achieves excellent predictive performance.Additionally,relevant factors influencing the dissemination are further examined in our analysis.Furthermore,we investigate the effectiveness of travel restrictions on ILI cases.Results can be used to utilize to mitigate the spread of disease.

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].

Equivalence between the internal observability and equation

This paper is concerned with a third order in time linear Moore-Gibson-Thompson equation which describes the acoustic velocity potential in ultrasound wave program.Influenced by the work of Kaltenbacher,Lasiecka and Marchand(Control Cybernet.2011,40:971-988),we establish an observability inequality of the conservative problem,and then discuss the equivalence between the exponential stabilization of a dissipative system and the internal observational inequality of the corresponding conservative system.