非线性波动方程的显式差分法

利用有界延拓法，研究了非线性波动方程周期初边值问题的显式差分解的收敛性与稳定性，避免了较难的先验估计，并放宽了非线性项的条件。     

一类非线性波动方程的显式精确解

本文用直接方法和假设的一种结合求出了一类较广泛的非线性波动方程ｕｔｔ－ａ１ｕｘｘ＋ａ２ｕｔ＋ａ３ｕ＋ａ４ｕＳ＾２＋ａ５ｕ＾３＝０的一些显式精确行波解,贱个有重要的非线性数学物理方程,如φ＾４方程,Ｋｌｅｉｎ－Ｇｏｒｄｏｎ方程,Ｓｉｎｅ－Ｇｏｒｄｏｎ方程,及Ｓｉｎｈ－Ｇｏｒｄｏｎ方程的近似,Ｌａｎｄａｕ－Ｇｉｎｚｂｕｒｇ－Ｈｉｇｇｓ方程,Ｄｕｆｆｉｎｇ方程,非线性电报方程等都可作为该方程的特殊情形得     

一类非线性弹性杆波动方程的求解

对计入横向惯性效应后的非线性弹性杆纵向波动方程进行了分析,得到了一类非线性波动方程,并用完全近似方法求出了该方程的近似解析解．     

一类非线性波动方程的椭圆余弦解

本文研究了一类非线性波动方程讨论了一维波动方程椭圆余弦函数解存在的条件及二维波动方程的简化问题.     

非线性弹性杆波动方程的显式精确解北大核心CSCD

应用sine-cosine方法对非线性弹性杆波动方程进行了求解,得到了该方程的一些新的周期波解和孤波解(材料常数n为不等于1的常数).对部分结果通过数学软件得到了解的图像,获得的结果有助于非线性弹性杆中孤波存在性问题的进一步研究.     

求解2—D波动方程系数反问题的迭代方法及收敛性

本文针对地震勘探中提出一类重要的２－Ｄ波动方程反演问题，通过定义一个新的非线性算子将２－Ｄ波动方程的反演问题归结为一个新的非线性算子方程，详细讨论了非线性算子的性质，给出了求解反问题的迭代方法，并证明了这种迭代方法的收敛性。     

一类具波动算子非线性Schr?dinger方程的精确解

研究一类具波动算子非线性Schr?dinger方程的精确解问题.引入Jacobi椭圆函数组合及双曲函数组合方法,将其应用于求解具有波动算子的非线性Schr?dinger方程中.通过简单代数运算,可以得到具有波动算子非线性Schr?dinger方程的许多新解,并在极限情况下,给出了该方程对应的双曲函数解.同时得出了双曲函数组合解是Jacobi椭圆函数组合解情况下的极限解的结论.该方法可以推广到更多非线性偏微分方程精确解求解问题.     

一类边值问题的有限差分牛顿型方法

本文利用微分方程的非线性差分格式的特殊结构，提出了一种新的牛顿型方法求解非线性差分方程，若新方法每步不队加计算非线性方程组的函数值，那么新自满收敛速度可在室Ｒ－１＋√４／２阶；若新方法每步附加计算一个非线性方程组的向量函数值，那么新算法收敛速度可达到Ｑ－平方阶。     

有限变形弹性杆中的几何非线性波

利用有限变形理论的Lagrange描述,借助非保守系统的Hamilton型变分原理,导出了描述弹性杆中几何非线性波的波动方程．为了使非线性波动方程有稳定的行波解,计及了粘性效应引入的耗散和横向惯性效应导致的几何弥散．运用多重尺度法将非线性波动方程简化为KdV-Bergers方程,这个方程在相平面上对应着异宿鞍-焦轨道,其解为振荡孤波解．如果略去粘性效应或横向惯性,方程将分别退化为KdV方程或Bergers方程,由此得到孤波解或冲击波解,它们在相平面上对应着同宿轨道或异宿轨道．     

非线性波动方程的交替显-隐差分方法

1．引言众所周知，非线性波动方程在自然科学领域有广泛的物理背景，诸如物理、化学反应方程，机械动力学方程，地球物理与大气海洋方程等．差分方法求解非线性波动方程已有研究，如［1］和IZ］就给出了非线性波动方程组的显式和隐式差分格式以及收敛性分析．虽然古典的显式差分格式易于并行计算，但是它的稳定性条件差（条件稳定）；古典的隐式差分格式稳定性条件好（绝对稳定）；但对非线性问题，一般需要线性化，然后求解一个线性代数方程组，并行计算能力差．本文正是在这样一种前题下，给出了一维问题的一种交替分段显一隐差分格式，…     

Nonlinear Implicit One-Step Schemes for Solving Initial Value Problems for Ordinary Differential Equation with Steep Gradients

A general theory for nonlinear implicit one-step schemes for solving initial value problems for ordinary differential equations is presented in this paper. The general expansion of "symmetric" implicit one-step schemes having second-order is derived and stability and convergence are studied. As examples, some geometric schemes are given. Based on previous work of the first author on a generalization of means, a fourth-order nonlinear implicit one-step scheme is presented for solving equations with steep gradients. Also, a hybrid method based on the GMS and a fourth-order linear scheme is discussed. Some numerical results are given.     

非线性Galerkin算法的收敛性和复杂性

本文给出了数值求解非线性发展方程的全离散非线性Ｇａｌｅｒｋｉｎ算法,即将空间离散时的谱非线性Ｇａｌｅｒｋｉｎ算法和时间离散的Ｅｕｌｅｒ差分格式相结合,得到了显式和隐式两种全离散数值格式,相应地也考虑了显式和隐式的Ｇａｌｅｒｋｉｎ全离散格式,并分别分析了上述四种全离散格式的收敛性和复杂性,经过比较得出结论;在某些约束条件下,非线性Ｇａｌｅｒｋｉｎ算法和Ｇａｌｅｒｋｉｎ算法具有相同阶的收敛速度,然而前     

Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H²-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes.

     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis.     

高阶非线性波动方程的有限差分方法

本文研究一类广泛的高阶非线性波动方程组初边值问题的有限差分格式，用离散泛函分析方法和先验估计的技巧得到了有限差分格式的收敛性。     

Space/time convergence analysis of a class of conservative schemes for linear wave equations

This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.     

Strong Convergence of a Fully Discrete Scheme for Multiplicative Noise Driving SPDEs with Non-Globally Lipschitz Continuous Coefficients

This work investigates strong convergence of numerical schemes for nonlinear multiplicative noise driving stochastic partial differential equations under some weaker conditions imposed on the coefficients avoiding the commonly used global Lipschitz assumption in the literature. Space-time fully discrete scheme is proposed, which is performed by the finite element method in space and the implicit Euler method in time. Based on some technical lemmas including regularity properties for the exact solution of the considered problem, strong convergence analysis with sharp convergence rates for the proposed fully discrete scheme is rigorously established.     

Nonlinear scheme with high accuracy for nonlinear coupled parabolic-hyperbolic system

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.     

The Order of Convergence of Difference Schemes for Fractional Equations

In this paper, we continue our research on convergence of difference schemes for fractional differential equations. Using implicit difference scheme and explicit difference scheme, we have a deal with the full discretization of the solutions of fractional differential equations in time variables and get the order of convergence.