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Acta Mathematicae Applicatae Sinica 2022年第3期38卷 601-613页

作者： Da-kang CEN Zhi-bo WANG Yan MO School of Mathematics and Statistics Guangdong University of TechnologyGuangzhou 510006China

In this paper,a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed,with O(N^(-(2-α))+M^(-4))accuracy order,where N;M denote the numbers of grids in temporal and spatial direction,α ∈(0,1)is the fractional *** recover the full accuracy based on the regularity requirement of the solution,we adopt the L1 method and the trapezoidal product integration(PI)rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral,respectively,further handle the nonlinear term carefully by the Newton linearized *** on the discrete fractional Gr¨onwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral,the stability and convergence of the proposed scheme are analyzed by the energy *** results are also confirmed by a numerical example.

关键词： nonlinear fractional integro-differential equation graded meshes discrete fractional Gr?nwall inequality compact difference scheme stability and convergence

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Linearly compact difference scheme for the Two-Dimensional Kuramoto-Tsuzuki Equation with the Neumann Boundary Condition

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Annals of Applied Mathematics 2023年第1期39卷 49-78页

作者： Qifeng Zhang Lu Zhang Department of Mathematics Zhejiang Sci-Tech UniversityHangzhouZhejiang 310018China School of Mathematics and Statistics Xuzhou University of TechnologyXuzhouJiangsu 221018China

In this paper,we analyze and test a high-order compact difference scheme numerically for solving a two-dimensional nonlinear Kuramoto-Tsuzuki equation under the Neumann boundary condition.A three-level average technique is utilized,thereby leading to a linearized difference *** main work lies in the pointwise error estimate in H^(2)-*** optimal fourth-order convergence order is proved in combination of induction,the energy method and the embedded ***,we establish the stability of the difference scheme with respect to the initial value under very mild condition,however,does not require any step ratio *** numerical examples with/without exact solutions under diverse cases are implemented to validate the theoretical results.

关键词： Kuramoto-Tsuzuki equation compact difference scheme pointwise error estimate stability numerical simulation

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A compact difference scheme for an Evolution Equation with a Weakly Singular Kernel

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Numerical Mathematics(Theory,Methods and Applications) 2012年第4期5卷 559-572页

作者： Hongbin Chen Da Xu Institute of Mathematics and Physics College of ScienceCentral South University of Forestry and TechnologyChangsha410004HunanChina Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China) College of Mathematics and Computer ScienceHunan Normal UniversityChangsha410081HunanChina

This paper is concerned with a compact difference scheme with the truncation error of order 3/2 for time and order 4 for space to an evolution equation with a weakly singular *** integral term is treated by means of the second order convolution quadrature suggested by *** stability and convergence are proved by the energy method.A numerical experiment is reported to verify the theoretical predictions.

关键词： Evolution equation weakly singular kernel compact difference scheme stability convergence numerical experiment

OPTIMAL POINT-WISE ERROR ESTIMATE OF A compact FINITE difference scheme FOR THE COUPLED NONLINEAR SCHRODINGER EQUATIONS

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Journal of Computational Mathematics 2014年第1期32卷 58-74页

作者： TingchunWang School of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing 2100~ China

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear SchrSdinger equation. The proposed scheme not only conserves the totM mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O（h4 ＋r2） in the discrete L∞ -norm with time step - and mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme.

关键词： Coupled nonlinear SchrSdinger equations compact difference scheme Conservation Point-wise error estimate.

compact Finite difference scheme for the Fourth-Order Fractional Subdiffusion System

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Advances in Applied Mathematics and Mechanics 2014年第4期6卷 419-435页

作者： Seakweng Vong Zhibo Wang Department of Mathematics University of MacaoAv.Padre Tom´as Pereira TaipaMacao

In this paper,we study a high-order compact difference scheme for the fourth-order fractional subdiffusion *** consider the situation in which the unknown function and its first-order derivative are given at the *** s... 详细信息

关键词： Fourth-order fractional subdiffusion equation compact difference scheme energy method stability convergence

求解Burgers方程的两层隐式紧致差分格式

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高等学校计算数学学报 2017年第4期39卷 340-354页

作者：杨晓佳魏剑英葛永斌宁夏大学数学统计学院

1引言Burgers方程是流体力学中扩散波最简单的非线性模型方程,它出现在许多物理问题当中,包括气体动力学问题、交通流问题和流体力学问题等.同时Burgers方程也可以作为流体动力学Navier-Stokes方程的简化模型方程.近年来,求解一维Burger... 详细信息

1引言Burgers方程是流体力学中扩散波最简单的非线性模型方程,它出现在许多物理问题当中,包括气体动力学问题、交通流问题和流体力学问题等.同时Burgers方程也可以作为流体动力学Navier-Stokes方程的简化模型方程.近年来,求解一维Burgers方程的计算方法受到科研工作者的广泛关注,有关的文献报道已有很多,如文献[1-5].这些方法在空间

关键词： Burgers equation compact difference scheme Implicit scheme High accuracy Error remainder correction method

compact FOURTH-ORDER FINITE difference schemeS FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS

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Journal of Computational Mathematics 2008年第1期26卷 98-111页

作者： Yiping Fu School of Mathematical Sciences South China University of Technology Guangzhou 510640 China

In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O（h^4） term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.

关键词： Helmholtz equation compact difference scheme FFT algorithm Convergence.

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一类具有非线性边界条件的四阶方程的紧差分迭代解法

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高等学校计算数学学报 2016年第4期38卷 289-300页

作者：庄清渠任全伟华侨大学数学科学学院上海师范大学数理学院

1引言科学和工程计算中许多问题的数学模型都可用四阶微分方程来表述.如刻画火焰传播的Kuramoto-Sivashinsky方程以及描述固体复杂的相位分离以及粗化现象的CahnHilliard方程[1,2,3].而在应用惩罚法于带有边界约束的四阶变分不等式时则... 详细信息

1引言科学和工程计算中许多问题的数学模型都可用四阶微分方程来表述.如刻画火焰传播的Kuramoto-Sivashinsky方程以及描述固体复杂的相位分离以及粗化现象的CahnHilliard方程[1,2,3].而在应用惩罚法于带有边界约束的四阶变分不等式时则会产生带非线性边界条件的四阶方程[4].本文的目的是考虑如下四阶非线性边值问题

关键词： Fourth-order differential equation Nonlinear boundary condition compact difference scheme Iterative algorithm

compact difference Method for Time-Fractional Neutral Delay Nonlinear Fourth-Order Equation

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Engineering（科研） 2022年第12期14卷 544-566页

作者： Huan Wang Qing Yang School of Mathematics and Statistics Shandong Normal University Jinan China

In this paper, we present a compact finite difference method for a class of fourth-order nonlinear neutral delay sub-diffusion equations in two-dimensional space. The fourth-order problem is first transformed into a second-order system by a reduced-order method. Next by using compact operator to approximate the second order space derivatives and L2-1σ formula to approximate the time fractional derivative, the difference scheme which is fourth order in space and second order in time is obtained. Then, the existence and uniqueness of solution, the convergence rate of and the stability of the scheme are proved. Finally, numerical results are given to verify the accuracy and validity of the scheme.

关键词： Two-Dimensional Nonlinear Sub-Diffusion Equations Neutral Delay compact difference scheme Convergence Stability

Estimation of 1-dimensional nonlinear stochastic differential equations based on higher-order partial differential equation numerical scheme and its application

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Frontiers of Mathematics in China 2017年第6期12卷 1441-1455页

作者： Peiyan LI Wei GU School of Mathematics and Statistics Central China Normal University Wuhan 430079 China School of Statistics and Mathematics Zhongnan University of Economics and Law Wuhan 430073 China

A method based on higher-order partial differential equation （PDE） numerical scheme are proposed to obtain the transition cumulative distribution function （CDF） of the diffusion process （numerical differentiation of the transition CDF follows the transition probability density function （PDF））, where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numericM tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.

关键词： Kolmogorov partial differentiM equations transition probability density function transition cumulative distribution function compact difference scheme