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Science China Mathematics 2024年第7期67卷 1641-1664页

作者： Jinye Shen fanhai zeng Martin Stynes School of Mathematics Southwestern University of Finance and EconomicsChengdu611130China School of Mathematics Shandong UniversityJinan250100China Applied and Computational Mathematics Division Beijing Computational Science Research CenterBeijing100193China

Fractional initial-value problems(IVPs) and time-fractional initial-boundary value problems(IBVPs), each with a Caputo temporal derivative of order α ∈(0, 1), are considered. An averaged variant of the well-known L1 scheme is proved to be O(N^(-2)) convergent for IVPs on suitably graded meshes with N points, thereby improving the O(N^(-(2-α))) convergence rate of the standard L1 scheme. The analysis relies on a delicate decomposition of the temporal truncation error that yields a sharp dependence of the order of convergence on the degree of mesh grading used. This averaged L1 scheme can be combined with a finite difference or piecewise linear finite element discretization in space for IBVPs, and under a restriction on the temporal mesh width, one gets again O(N^(-2)) convergence in time, together with O(h^(2)) convergence in space,where h is the spatial mesh width. Numerical experiments support our results.

关键词： time-fractional subdiffusion averaged L1 scheme

Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

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Communications on Applied Mathematics and Computation 2019年第4期1卷 621-637页

作者： Yu bo Yang fanhai zeng Nanhu College Jiaxing UniversityJiaxing 314001ZhejiangChina Department of Mathematics National University of SingaporeSingapore 119076Singapore

In this paper,a new type of the discrete fractional Gronwall inequality is developed,which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdifiFusion *** on the temporal-spatial error splitting argument technique,the discrete fractional Gronwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdififusion equation.

关键词： Time-fractional subdififusion equation Convolution quadrature Fractional linear multistep methods Discrete fractional Gronwall inequality Unconditional stability

A Unified FastMemory-Saving Time-SteppingMethod for Fractional Operators and Its Applications

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Numerical Mathematics(Theory,Methods and Applications) 2022年第3期15卷 679-714页

作者： Yuxiang Huang Qiaoge Li Rongxin Li fanhai zeng Ling Guo Department ofMathematics Shanghai Normal UniversityShanghai 200234China School of Mathematics Shandong UniversityJinan 250100ShandongChina

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time,especially for long-time integration,which taxes computational resources heavily for high-dimensional ***,we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constant-order fractional operators,and identify the current pitfalls of such *** order to overcome the pitfalls,an improved sum-of-exponentials is developed and *** also present several sumof-exponentials for the approximation of the kernel function in variable-order fractional ***,based on the sum-of-exponentials,we propose a unified framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable *** test the fast method based on several benchmark problems,including fractional initial value problems,the time-fractional Allen-Cahn equation in two and three spatial dimensions,and the Schr¨odinger equation with nonreflecting boundary conditions,demonstrating the efficiency and robustness of the proposed *** results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.

关键词： Sum-of-exponentials contour quadrature fractional integral and derivative operators fast time-stepping methods time-fractional Allen-Cahn equation nonreflecting boundary conditions

Bi-Orthogonal fPINN:A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs

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Communications in Computational Physics 2023年第9期34卷 1133-1176页

作者： Lei Ma Rongxin Li fanhai zeng Ling Guo George Em Karniadakis Department of Mathematics Shanghai Normal UniversityShanghai 200234China School of Mathematics Shandong UniversityShandong 250100China Division of Applied Mathematics Brown UniversityProvidenceRI 02906USAand School of EngineeringBrown UniversityProvidenceRI 02906USA

Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal ***,inherent uncertainties arise naturally in real applications due to random forcing or unknown material *** models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and ***,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent ***,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak *** automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network *** weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue ***,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward *** demonstrate the effectiveness of the BO-fPINN method for different benchmark ***,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method *** then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial *** study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN *** results demonstrate the flexibility and efficiency

关键词： Scientific machine learning uncertainty quantification stochastic fractional differential equations PINNs inverse problems

加工时间服从几何分布的流水车间随机排序问题

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辽宁工程技术大学学报（自然科学版） 2009年第3期28卷 502-504页

作者：李扬曾繁慧沈阳化工学院数理系辽宁工程技术大学理学院

为了解决使最长加工时间的数学期望最小、工件的加工时间服从几何分布的两阶段流水车间随机调度问题,采用理论分析的方法,分别研究了两阶段静态随机流水车间和动态随机流水车间工件的最优加工顺序。结果表明:在工件的到达时间均为0的两... 详细信息

为了解决使最长加工时间的数学期望最小、工件的加工时间服从几何分布的两阶段流水车间随机调度问题,采用理论分析的方法,分别研究了两阶段静态随机流水车间和动态随机流水车间工件的最优加工顺序。结果表明:在工件的到达时间均为0的两阶段静态随机流水车间、工件的到达时间不一致的两阶段动态随机流水车间两种情况下,由给出的优先规则的不可中断静态优先策略和不可中断动态优先策略是确定使最长加工时间最小的优先策略,并对算法的最优性进行了证明。该成果对正规目标函数的流水车间随机排序问题的解决具有一定的参考价值和指导意义。

关键词：流水车间随机加工时间几何分布最长加工时间优先策略

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时间分数阶亚扩散方程的高阶差分方法

计算物理 2013年第4期30卷 491-500页

作者：曾凡海李常品上海大学理学院数学系上海200444

提出两差分格式求解时间分数阶亚扩散方程.两个格式都是绝对稳定的,收敛阶均为O(τq+h2),其中q(q=2-β或2)与方程解的光滑性有关,β(0<β<1)是分数阶导数的阶、τ和h分别是时间和空间方向步长.数值实验验证了理论结果的正确性,并与其他... 详细信息

提出两差分格式求解时间分数阶亚扩散方程.两个格式都是绝对稳定的,收敛阶均为O(τq+h2),其中q(q=2-β或2)与方程解的光滑性有关,β(0<β<1)是分数阶导数的阶、τ和h分别是时间和空间方向步长.数值实验验证了理论结果的正确性,并与其他方法进行比较,显示了本文方法的有效性和精确性.

关键词：亚扩散方程分数阶线性多步法高阶方法稳定性收敛性

三峡水库典型支流上游区和回水区营养状态分析

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湖泊科学 2010年第2期22卷 201-207页

作者：张晟宋丹张可曾凡海李斗果重庆市环境科学研究院重庆401147 西南大学资源环境学院重庆400716 重庆市涪陵区环境监测站重庆408000

2006年3、5、8月,对三峡水库成库后5条支流的上游区和回水区水质参数及营养因子分布进行了初步研究.研究结果表明,支流高锰酸盐指数范围在1.00-2.50mg/L、COD范围为6.00-26.5mg/L,上游区与回水区有机物含量差异不大,支流目前未受到有机... 详细信息

2006年3、5、8月,对三峡水库成库后5条支流的上游区和回水区水质参数及营养因子分布进行了初步研究.研究结果表明,支流高锰酸盐指数范围在1.00-2.50mg/L、COD范围为6.00-26.5mg/L,上游区与回水区有机物含量差异不大,支流目前未受到有机物污染影响.NH_4^+-N范围为0.039-0.367mg/L,各支流含量均为丰水期最大.TN范围在0.58-1.67mg/L,TP范围在0.005-0.133mg/L,支流回水区TN和TP含量均远高于水体发生富营养化的最低限制值,水体存在发生富营养化的风险.大多支流N/P比值处于适宜藻类生长范围.Chl.a浓度范围为0.94-28.9mg/m^3,各支流回水区Chl.a浓度均为5月最大河流回水区Chl.a浓度高于上游区,上游区、回水区Chl.a含量有显著差异.选用修正的Carlson营养状态指数(Trophic State Index)TSI_M法,评价了支流水体营养状态,TSI_M指数范围在36.0-64.2,上游区除龙河、澎溪河5月达到富营养水平外,其余支流均为贫-中营养水平.回水区各支流均达到中-富营养水平.支流回水区营养状态指数均高于上游区,但各支流增加幅度不同.支流回水区水体营养状态明显受三峡水库蓄水水体流速减缓的影响.Chl.a与COD(n=15,P<0.01)呈极显著正相关,与其它营养因子无明显相关关系.三峡水库完工后,支流回水区水体流速减缓,富营养化趋势可能加重.

关键词：三峡水库支流回水区营养状态

雪峰造山带北西缘永顺-大明逆冲推覆构造的厘定及其地质意义

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矿物岩石 2017年第4期37卷 62-73页

作者：李泽泓曾广乾凌跃新熊延旺曾凡海吴开彬湖南省地质调查院湖南长沙410016 贵州省地质调查院贵州贵阳550005

通过1∶5×10~4区域地质调查及对重点区段的解剖,首次对湘西永顺-大明逆冲推覆构造的几何样式与变形特征进行详细研究。推覆构造总体呈北东—南西向展布,在逆冲推覆构造的前锋断层沿线,寒武系及奥陶系向北西逆冲推覆于奥陶系及志留系之... 详细信息

通过1∶5×10~4区域地质调查及对重点区段的解剖,首次对湘西永顺-大明逆冲推覆构造的几何样式与变形特征进行详细研究。推覆构造总体呈北东—南西向展布,在逆冲推覆构造的前锋断层沿线,寒武系及奥陶系向北西逆冲推覆于奥陶系及志留系之上,形成规模不等的串珠状排列的飞来峰与构造窗。推覆主断层具有典型的台阶式结构特征,即推覆断层在寒武系上统和奥陶系以灰岩、白云岩为主的强硬岩层中形成高角度切层断坡,进入下志留统下部以泥岩、页岩为主的软弱岩层后断层倾角变缓,转为断坪。推覆构造沿走向变形强度以及结构特征存在明显变化,由北东往南西各区段的水平位移总体上呈现递减趋势,最大推覆距离约4km。永顺-大明逆冲推覆构造的厘定确证雪峰造山带北西缘推覆构造的存在,但诸多证据表明该区域并不存在大规模推覆作用,推覆体为准原地型。分析认为推覆构造是印支—燕山期在北西—南东向挤压应力场下递进变形的产物。

关键词：逆冲推覆构造几何学与运动学飞来峰与构造窗慈利-保靖断裂带雪峰造山带

变系数Burgers方程的一种预处理谱方法

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应用数学与计算数学学报 2010年第1期24卷 99-106页

作者：邓芳芳马和平曾凡海上海大学数学系上海200444

本文用预处理Legendre-Galerkin-Chebyshev配置方法来求解二阶变系数Burgers方程,该方法首先对方程进行预处理,然后在空间方向应用Legendre-Galerkin-Chebyshev配置方法,对时间方向采用leap-frog/Crank-Nicolson格式进行离散,这样可使得变系数项和非线性项能够显式计算.我们对该方法进行了误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.

关键词： Legendre-Galerkin-Chebyshev配置方法预处理变系数

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烟草特有亚硝胺(TSNA)的研究进展

中国农学通报 2010年第10期26卷 82-86页

作者：曾凡海李卫周冀衡张一扬胡志明湖南农业大学烟草科学与工程技术中心长沙410128

综述了烟草特有亚硝胺(TSNA)的发现、形成过程、危害、检测方法及国内外研究成果,并对进一步开展降低TSNA的研究进行了展望,以期为生产优质低害的烟叶提供参考依据。

关键词：烟草烟草特有亚硝胺(TSNA) 进展