Numerical Solutions of Three-Dimensional Coupled Burgers’ Equations by Using Some Numerical Methods

In this paper, we found the numerical solution of three-dimensional coupled Burgers’ Equations by using more efficient methods: Laplace Adomian decomposition method, Laplace transform homotopy perturbation method, variational iteration method, variational iteration decomposition method and variational iteration homotopy perturbation method. Example is examined to validate the efficiency and accuracy of these methods and they reduce the size of computation without the restrictive assumption to handle nonlinear terms and it gives the solutions rapidly.

Application of OHAM-DJ to the System of Burgers’ Equations

In this paper, the system of Burgers’ equations is solved by the optimal homotopy asymptotic method with Daftardar-Jafari polynomials OHAM-DJ. Two numerical examples are illustrated the efficient of this methods for solving the system of Burgers’ equations.

Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations

In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.

Higher-Order Numerical Solution of Two-Dimensional Coupled Burgers’ Equations

We proposed a higher-order accurate explicit finite-difference scheme for solving the two-dimensional heat equation. It has a fourth-order approximation in the space variables, and a second-order approximation in the time variable. As an application, we developed the proposed numerical scheme for solving a numerical solution of the two-dimensional coupled Burgers’ equations. The main advantages of our scheme are higher accurate accuracy and facility to implement. The good accuracy of the proposed numerical scheme is tested by comparing the approximate numerical and the exact solutions for several two-dimensional coupled Burgers’ equations.

一种风向监督双流神经网络--以一维Burgers方程求解为例

针对一维Burgers方程下单一建模方式难以充分考虑不同阶段风向对系数的影响比重,无法有效获得各节点间的关联信息的问题,本文提出了一种风向监督双流神经网络分别预测上下风向的有限差分系数。同时设计了一种风向判断模块,实现了对预测得到有限差分系数的权重融合。通过风向监督双流神经网络,并结合先验知识对学得的系数分配一定的权重,以突出上下风向对预测结果的不同影响,可以有效实现对不同风向上的点分别进行预测,使得空间结构特征信息挖掘更加充分,从而提高差分系数预测的精度。在比传统数值求解方法网格分辨率粗4~8倍的同时,提高了谷歌团队工作的精度,以此提高了计算的速度。

定常Burgers方程的周期解

本文首先对Burgers方程的周期解关于粘性系数ε作幂级数展开,得到逼近函数之间的递推关系。然后,通过对逼近函数作Fourier级数展开,给出了逼近函数Fourier展开的通项形式,并证明了每个逼近函数的Fourier级数是一致且绝对收敛的。

基于有限差分-谱方法的分数阶Burgers流体的非稳态驻点流动

研究了分数阶Burgers流体通过拉伸平板的非稳态驻点流动问题。将分数阶导数引入Burgers流体模型可以更好地模拟流动过程,但也增加了模型的复杂性和求解难度。首次运用有限差分-谱方法求解分数阶Burgers流体模型,离散格式构造简单有效。采用谱方法对控制方程中的空间项进行离散,利用有限差分方法分别结合L-1和L-2算法离散控制方程中的时间项,给出了两种离散格式,并且通过构造数值算例证明了离散格式的收敛性。结果表明,在靠近平板处,速度随着分数阶导数的增加而减小,而无穷远处的流体速度呈现出相反的趋势,体现了分数阶导数的记忆特性。此外,雷诺数越小,流体的粘度越大,导致流体速度越大。由于松弛时间参数的松弛特性,靠近平板处松弛时间参数对速度分布有抑制作用,远离平板处松弛时间促进流体流动。

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.

一类非线性Burgers型问题的预测校正紧差分方法

针对一类非线性Burgers型方程,提出一种预测-校正紧差分方法。首先,对时间一阶导数采用一阶Euler格式,时间积分项运用一阶卷积求积公式进行离散,并以MacCormack方法的两步预测-校正方法处理非线性项;然后采用四阶紧差分离散空间的一阶和二阶导数,构造了Euler预测-校正紧差分全离散格式。最后通过案例验证了所提出算法的有效性。

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.

BBM-Burgers方程的非协调有限元方法的超收敛分析

研究Benjamin-Bona-Mahony-Burgers(BBM-Burgers)方程的非协调EQ_(1)^(rot)元的线性化BDF格式下的超收敛性质。通过使用数学归纳法来处理非线性项,并利用该单元已有的高精度结果及插值后处理技术,得到了在对空间剖分尺度和时间步长无网格比约束的前提下,关于离散H^(1)-模意义下具有O(h^(2)+τ^(2))阶的超逼近和超收敛结果。最后,通过给出数值算例验证了理论分析的正确性。

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.

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].

Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.

THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THERMOMECHANICAL CUCKER-SMALE-STOKES EQUATIONS IN THE WHOLE DOMAIN

We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.

THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS

We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.

基于PINN的Burgers方程求解模型

传统的数值求解方法面临维数灾难和效率与精度平衡问题,而基于数据驱动的神经网络求解方法又存在训练量冗余和不可解释性问题。针对此问题,物理信息神经网络(Physical Information Neural Networks,PINNs)关注了训练数据中隐含的物理先验知识,融合了神经网络拟合复杂变量的能力,赋予了传统神经网络所缺乏的物理可解释性。应用该算法模型,提出了一种基于PINN的Burgers方程求解模型,该算法模型在训练中施加物理信息约束,因此能用少量的训练样本学习预测到分布在时空域上的偏微分方程模型。实验结果表明,在1+1维Burgers方程算例下,所提方法相比于经典的机器学习算法能有效捕抓到方程的变化并进行精确模拟,相比于有限差分法,可以大幅度缩短模拟时间。通过对不同的网络参数进行比较实验,所提方法在10%的噪声破坏下能产生合理的识别准确度,网络逼近方程的待定系数误差在0.001以内。

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.

Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.