Interval Criteria for Oscillation of Forced Second-order Half-linear Differential Equations

By employing a generalized Riccati technique and an integral averaging tech-nique, new interval oscillation criteria are established for the forced second-order half-lineardifferential equation [r(t)|x′ (t)|α-1x′ (t)]′ + q(t)|x(t)|α-1x(t) = e(t).

Oscillation Criteria for Second-order Half-linear Impulsive Differential Equations with Damping

By employing the inequality of[8]and a positive continuous function g(t),t∈[to,+∞),oscillation criteria for the second-order half-linear impulsive differential equ-ations with damping are established.Our results generalize and improve some known ones.

New Interval Oscillation Criteria for Forced Second-order Half-linear Differential Equations

这份报纸讨论强迫的秒顺序的一个班一半线性的微分方程。由使用概括 Riccati 技术和平均技术，一些新间隔摆动标准被获得。

Method of Lines for Third Order Partial Differential Equations

The method of lines is applied to the boundary-value problem for third order partial differential equation. Explicit expression and order of convergence for the approximate solution are obtained.

Discrete Arzelà-Ascoli Theorem on the Half Line and Its Application

The compactness criterion in l◇={x（·）∈l^∞（0,∞））｜lim t→∞ x（t）=x（∞）} is given and as an application,we study the existence of positive solutions of second order boundary value problems of difference equation on the half line by the fixedpoint theorem in cones.

COMPUTATION OF STRESS INTENSITY FACTORS BY THE SUB-REGION MIXED FINITE ELEMENT METHOD OF LINES

Based on the sub-region generalized variationM principle, a sub-region mixed version of the newly-developed semi-analytical ＇finite element method of lines＇ （FEMOL） is proposed in this paper for accurate and efficient computation of stress intensity factors （SIFs） of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations （ODEs） and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.

Transmission Lines Distance Protection Using Differential Equation Algorithm and Hilbert-Huang Transform

This paper proposed the scheme of transmission lines distance protection based on differential equation algorithms (DEA) and Hilbert-Huang transform (HHT). The measured impedance based on EDA is affected by various factors, such as the distributed capacitance, the transient response characteristics of current transformer and voltage transformer, etc. In order to overcome this problem, the proposed scheme applies HHT to improve the apparent impedance estimated by DEA. Empirical mode decomposition (EMD) is used to decompose the data set from DEA into the intrinsic mode functions (IMF) and the residue. This residue has monotonic trend and is used to evaluate the impedance of faulty line. Simulation results show that the proposed scheme improves significantly the accuracy of the estimated impedance.

JACOBI SPECTRAL APPROXIMATIONS TO DIFFERENTIALEQUATIONS ON THE HALF LINE

Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Algorithm to prove the stability and convergence of the approximations.

Existence of Solutions for Infinity-Point Nonlinear Fractional Boundary Value Problem at Resonance

A class of the boundary value problem for fractional order nonlinear differential equation with Riemann-Liouville fractional derivative on the half line was studied. By using the coincidence degree theory due to Mawhin and constructing the suitable operators,the existence theorem of at least one solution has been established. An example is given to illustrate our result.

An Accurate Numerical Integrator for the Solution of Black Scholes Financial Model Equation

In this paper the Black Scholes differential equation is transformed into a parabolic heat equation by appropriate change in variables. The transformed equation is semi-discretized by the Method of Lines (MOL). The evolving system of ordinary differential equations (ODEs) is integrated numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10–10.

A Trapezoidal-Like Integrator for the Numerical Solution of One-Dimensional Time Dependent Schrodinger Equation

In this paper, the one-dimensional time dependent Schr?dinger equation is discretized by the method of lines using a second order finite difference approximation to replace the second order spatial derivative. The evolving system of stiff Ordinary Differential Equation (ODE) in time is solved numerically by an L-stable trapezoidal-like integrator. Results show accuracy of relative maximum error of order 10?4 in the interval of consideration. The performance of the method as compared to an existing scheme is considered favorable.

New Kamenev-type Oscillation Criteria for Half-linear Partial Differential Equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping div（A（x）｜｜ u｜｜^p-2 u）＋〈b（x）,｜｜ u｜｜^p-2 u〉＋c（x）｜u｜^p-2u=0（E）under quite general conditions. These results are extensions of the recent results developed by Sun [Y.C. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 （2004） 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.

具阻尼项的二阶非线性中立型微分方程的振动准则

建立了二阶非线性中立型阻尼微分方程[a(t)|z′(t)|^(α-1 )z′(t)]′+b(t)|z′(t)|^(α-1) z′(t)+q(t)|x(σ(t))|^(β-1) x(σ(t))=0的若干振动准则,其中z(t)=x(t)+p(t)x(τ(t))。改进、推广和统一了已有文献的相关结果,并通过实例说明了所得准则的广泛应用效果。

求非线性方程组所有根的Newton场线法

为求解非线性方程组F(x)=0,提出Newton场线微分方程xt(t)=-(DF(x))-1F(x),x(0)=x0.在m重根x*的中心场域中任取初始点x0,证明了用前向Eu ler格式得到的解序列xn一定收敛到此根,故场线法大范围收敛.由此提出求非线性方程组所有根的场线算法,其有效性为数值试验所证实.

OSCILLATIONS OF SECOND-ORDER HALF-LINEAR DIFFERENTIAL EQUATIONS WITH IMPULSES

A kind of second-order half-linear differential equations with impulses is studied in this paper. Our results improve and generalize some known ones.

融合局部感知Transformer模型的微积分方程求解

针对局部信息在数学表达式的符号计算中非常重要,而Transformer模型通常存在局部缺失从而忽略字符间的语义信息的问题,提出一种将一维卷积(Conv1d)和Transformer模型相结合的符号计算模型Convld_Transformer。该模型通过在嵌入层中引入卷积网络提取局部特征信息,可有效增强局部感知。此外,还提出了一种生成一类偏微分方程数据集的算法,该算法结合特征线法和常微分方程变换,可实现对常系数一阶线性(pde1_cc)、变系数一阶线性(pde1_vc)以及满足一定条件的二阶抛物线偏微分方程(parapde2_cc)的求解。实验结果表明:Conv1d_Transformer模型在函数积分任务中相比Transformer模型精度更高,在解决pde1_cc、pde1_vc和parapde2_cc问题时准确率分别达到了96.00%、77.18%和86.18%,其性能要优于Mathematica和SymPy数学求解器。

用Jacobi拟谱方法求解半直线上的非线性Klei-Gordon方程(英文)

研究半直线上的非线性Klein-Gordon方程的数值求解方法。通过适当的变换,将此问题变成有限区间上的某类奇异问题。然后利用Jacobi拟谱方法来求解。证明了拟谱格式的稳定性和收敛性。数值结果也说明了该方法的有效性。

OSCILLATION OF A SECOND-ORDER HALF-LINEAR NEUTRAL DAMPED DIFFERENTIAL EQUATION WITH TIME-DELAY

In this paper,the oscillation for a class of second-order half-linear neutral damped differential equation with time-delay is studied.By means of Yang-inequality,the generalized Riccati transformation and a certain function,some new sufficient conditions for the oscillation are given for all solutions to the equation.

Simulation of electrically driven jet using Chebyshev collocation method

The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations(PDEs).Following the method of lines,the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations(DAEs).By differentiating constrains in DAEs twice,the system is transformed into a set of ordinary differential equations(ODEs) with invariants.Then the implicit differential equations solver 'ddaskr' is used to solve the ODEs and post-stabilization is executed at the end of each step.Results show the distributions of radius,linear charge density,stretching ratio and also the horizontal velocity at a time point.Meanwhile,the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.

Introduction to an Invariant Quantity Method

It is impossible,mathematically, to use a time series which is regarded as a set of observational facts of a dynamicsystem to reconstruct the particular system.Physically, however, with a few assumptions put, a dynamic system canbe rebuilt approximately by means of observational facts.This is the goal of the so called invariant quantity method(IQM),whose research and experiment are of potential significance to atmospheric sciences.