Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

ABJM理论的精确孤立子解

对出现在规范场中的Aharony-Bergman-Jafferis-Maldacena(简记为ABJM)理论,首先在阿贝尔ansatz之下,导出二阶Euler-Lagrange方程组;其次,利用BPS技巧导出一阶BPS方程组,并证明了二阶Euler-Lagrange方程组与一阶BPS方程组的等价性;最后,通过求解一阶BPS方程组得到了ABJM理论的所有有限能量的(正则)精确孤立子解.     

一类正整数是否是孤立数的讨论

对于任意正整数n,令σ(n)表示为n的所有正因数的和函数.对于正整数n,若存在正整数m满足关系式σ(n)=σ(m)=n+m,则称正整数数对(n,m)为一对亲和数;若不存在正整数m满足关系式σ(n)=σ(m)=n+m,则称n为孤立数.亲和数与孤立数是数论中的两类重要的整数.利用初等方法结合计算机python语言,证明了整数E(33,t)=1/2(33^(2^(t))+1)是孤立数.     

一类多维非线性波动方程组的有限元分析

一、引 言 在文献[1,2]中研究了一类三维非线性波动方程组的三维孤立子问题,证明了三维孤立子的存在和它的稳定性。文[3]考察了如下一类更广泛的多维非线性波动方程组     

非线性KdV-mKdV方程的新的复合函数解

利用孤立子方程KdV-mKdV的朗斯基解的形式和结构,我们提出了朗斯基形式展开法,运用这一方法获得了KdV-mKdV方程的丰富的新的复合函数解,并且朗斯基行列式中的元素不满足任何线性偏微分方程组.所得到的复合函数解是使用其它的方法得不到的.     

复Schrdinger场和实Boussinesq场耦合作用下一类方程组的有限元分析

复Schrodinger场和实Boussinesq场相互作用下的孤立子问题,已经引起广泛的研究.这类问题孤立子解的最重要特征在于孤立子之间的相互作用是非弹性的,见[1].在[1]中讨论了复Schrodinger场和实Boussinesq场耦合作用下一类方程组:     

一类KdV非线性Schrdinger组合微分方程组周期初值问题和柯西问题整体解的存在性唯一性

<正> 在[1]、[2]、[3]中研究了组合微分方程组——低频电场扰动密度满足具有质动力项的KdV方程和电场满足的Schrodinger类方程——的偶合孤立子问题.在[1]中用数值解方法研究了Langmuir波和离子声波偶合的C孤立子结构,分析了它和非线性Schrodinger孤立子、Langmuir孤立子以及离子声波孤立子的相互作用问题.为了更好地研究这类方程组及其孤立子的性质,有必要对它的整体解的存在性、唯一性加以论证. 本文考虑如下一类KdV非线性Schrodinger组合微分方程组     

非线性波方程的精确孤立波解

立了一种求解非线性波方程精确孤立波解的双曲函数方法,并在计算机代数系统上加以实现,推导出了一大批非线性波方程的精确孤立波解.方法的基本原理是利用非线性波方程孤立波解的局部性特点,将方程的孤立波解表示为双曲函数的多项式,从而将非线性波方程的求解问题转化为非线性代数方程组的求解问题.利用吴消元法或Gröbner基方法在计算机代数系统上求解非线性代数方程组, 最终获得非线性波方程的精确孤立波解,其中有很多新的精确孤立波解.     

换元Newton型方法的计算效能和最佳换元周期

考虑非线性方程组: F(x)=0, (1.1)其中F:R~n→R~n是二次连续可微函数.一般地说,解方程组(1.1)的拟Newton法较Newton法更为有效.我们可以将拟Newton法解释为逐次在R~n的子空间上构造F′(x)的近似(割线近似)得到的算法.按照这种思想,如果将子空间依次循环取成F′(x)的例     

常微分方程组的整体解及两点边值问题——Ⅰ.整体解方法

众所周知,n维向量函数u(x)的一阶常微分方程组,如在某点上只给出n_1相似文献   

A time-fractional competition ecological model with cross-diffusion

This paper is concerned with some mathematical and numerical aspects of a Lotka-Volterra competition time-fractional reaction-diffusion system with cross-diffusion effects. First, we study the existence of weak solutions of the model following the well-known Faedo-Galerkin approximation method and convergence arguments. We demonstrate the convergence of approximate solutions to actual solutions using the energy estimates. Next, the Galerkin finite element scheme is proposed for the considered model. Further, various numerical simulations are performed to show that the fractional-order derivative plays a significant role on the morphological changes of the considered competition model.     

Energy Error Estimates for the Projection-Difference Method with the Crank–Nicolson Scheme for Parabolic Equations

We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001     

Strong-Norm Error Estimates for the Projective-Difference Method for Parabolic Equations with Modified Crank--Nicolson Scheme

A parabolic problem in a separable Hilbert space is solved approximately by the projective-difference method. The problem is discretized with respect to space by the Galerkin method and with respect to time by the modified Cranck--Nicolson scheme. In this paper, we establish efficient (in time and space) strong-norm error estimates for approximate solutions. These estimates allow us to obtain the rate of convergence with respect to time of the error to zero up to the second order. In addition, the error estimates take into account the approximation properties of projective subspaces, which is illustrated for subspaces of finite element type.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

The dynamical behavior of the discontinuous Galerkin method and related difference schemes

We study the dynamical behavior of the discontinuous Galerkin finite element method for initial value problems in ordinary differential equations. We make two different assumptions which guarantee that the continuous problem defines a dissipative dynamical system. We show that, under certain conditions, the discontinuous Galerkin approximation also defines a dissipative dynamical system and we study the approximation properties of the associated discrete dynamical system. We also study the behavior of difference schemes obtained by applying a quadrature formula to the integrals defining the discontinuous Galerkin approximation and construct two kinds of discrete finite element approximations that share the dissipativity properties of the original method.

     

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori $L^2(0, T ;L^2(Ω))$ error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.     

Navier-Stokes方程的非奇异解分支的谱Galerkin逼近

No error estimate of the spectral Galerkin approximation for the steady-state Navier-Stokes equations was given without assuming that the data of the externalforce field and the boundary conditions are small enough. In this paper, under the condition that the solutions of the Navier-Stokes equations are nonsingular,we proved the existence and convergence of the spectral Galerkin approximation solutions and gave the error estimate. At last, this approximation method wasapplied to simulate the spherical Couette flow.     

A Modified Weak Galerkin Finite Element Method for Sobolev Equation

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete $H^1$ and $L^2$ norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.