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More asymptotic expansions for the Harmonic Numbers

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Journal of Mathematical Research with Applications 2020年第3期40卷 297-304页

作者： Hongmei LIU School of Science Dalian Minzu University

In this paper, by exponential complete Bell polynomials, we establish a(general)harmonic number asymptotic expansion, and give the corresponding recurrence of the coefficient sequence in the expansion. By the methods of the generating functions and summation transformations, we also present an explicit expression for the coefficient sequence of the ***, we establish two(general) lacunary harmonic number asymptotic expansions, which contain only even or odd power terms in the logarithmic term.

关键词： asymptotic expansions harmonic numbers Bell polynomials

四阶方程两点边值问题Hermite有限元解的渐近展式与外推

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高等学校计算数学学报 2006年第4期28卷 299-306页

作者：王奇生邓康湘潭大学数学与计算科学学院湖南科技大学数学与计算科学学院

1引言有限元解的渐近展式是提高微分方程数值解精度的重要工具,比如亏量校正和外推就是建立在有限元解的渐近展式的基础之上．许多作者对此进行了大量的研究(见[1]-[4]),特别是文[1],提出了在研究有限元解的渐近展式中十分有用的能量嵌... 详细信息

1引言有限元解的渐近展式是提高微分方程数值解精度的重要工具,比如亏量校正和外推就是建立在有限元解的渐近展式的基础之上．许多作者对此进行了大量的研究(见[1]-[4]),特别是文[1],提出了在研究有限元解的渐近展式中十分有用的能量嵌入技巧．本文利用能量嵌入定理得到了四阶方程两点边值问题Hermite有限元解及其二阶平均导数的渐近展式,进一步我们还讨论了它们的Richardson外推公式．考虑四阶方程两点边值问题

关键词： fourth-order equations Hermite finite element energy embedment method asymptotic expansions Richardson extrapolation.

板振动问题Ciarlet-Raviart混合有限元渐近展开式与Richardson外推

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高等学校计算数学学报 2008年第3期30卷 201-212页

作者：杨一都贵州师范大学数学与计算机科学学院

1引言Richardson外推是数值计算各个方面广泛使用的一种高效算法,它的理论基础是渐近展开式.1983年文对三角形网格首先证明了二阶椭圆问题协调有限元渐近展开式,这以后有限元外推迅速发展成一个国际性研究课题,文[5],[7]对有限元外推作... 详细信息

1引言Richardson外推是数值计算各个方面广泛使用的一种高效算法,它的理论基础是渐近展开式.1983年文对三角形网格首先证明了二阶椭圆问题协调有限元渐近展开式,这以后有限元外推迅速发展成一个国际性研究课题,文[5],[7]对有限元外推作了系统、深

关键词： vibrations of plate eigenvalue Ciarlet-Raviart mixed finite element asymptotic expansions Richardson extrapolation

EXTRAPOLATION AND A-POSTERIORI ERROR ESTIMATORS OF PETROV-GALERKIN METHODS FOR NON-LINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2001年第4期19卷 407-422页

作者： Shu-hua Zhang (Department of Mathematical Sciences, University of Alberta, Edrmonton, Alberta, Canada T6G 2G1) Tao Lin (Department of Mathematics, Virginia Tech, Blacksburg VA 24061) Yan-ping Lin (Department of Mathematical Sciences, University of Alberta University of Alberta 加拿大 T6G 2G1 Virginia Tech 美国 VA 24061 University of Alberta 加拿大 T6G 2G6

In this paper we will show that the Richardson extrapolation can be used to enhance the numerical solution generated by a Petrov-Galerkin finite element method for the initial value problem for a nonlinear Volterra integro-differential equation. As by-products, we will also show that these enhanced approximations can be used to form a class of posteriori estimators for this Petrov-Galerkin finite element method. Numerical examples are supplied to illustrate the theoretical results.

关键词： Volterra integro-differential equations Petrov-Galerkin finite element methods asymptotic expansions Interpolation post-processing A-posteriori error estimators.

The limit of the Boussinesq system of Rayleigh-Bénard convection as the Prandtl number approaches infinity

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Applied Mathematics(A Journal of Chinese Universities) 2011年第4期26卷 493-502页

作者： SHI Jian-guo WU Zhong-lin Department of Mathematics Huanghuai University Zhumadian 463000 China

Benard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods. For ill-prepared initial data, an exact approximating 1 solution with expansions up to any order are given and the convergence rates O（εm＋1/2）and the optimal convergence rates O（εm＋1） are obtained respectively. This improves the result of J.G. SHI.

关键词： Infinite Prandtl number limitl Rayleigh-Benard convection Boussinesq system asymptotic expansions singular perturbation classical energy methods.

Optimal regularity of positive solutions of the Hénon-Hardy equation and related equations

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Science China Mathematics 2024年第10期67卷 2283-2302页

作者： Zongming Guo Fangshu Wan College of Mathematics and Information Science Henan Normal UniversityXinxiang 453007China School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China School of Mathematical Sciences Anhui UniversityHefei 230601China

We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth do... 详细信息

We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\asymptotic expansions)∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈*** is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ωasymptotic expansions) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.

关键词： H´enon-Hardy equation positive solutions optimal regularity singular point asymptotic expansions

Self-adjoint Extensions for the Neumann Laplacian and Applications

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Acta Mathematica Sinica,English Series 2006年第3期22卷 879-906页

作者： S.A.NAZAROV J.SOKOLOWSKI Institute of Mechanical Engineering Problems V.O.Bolshoy pr.61 199178 St Petersburg Russia Institut Elie Cartan Laboratoire de Mathématiques Université Henri Poincaré NancyI B.P.239 54506 Vandoeuvrelès Nancy Cedex France and Systems Research Institute of the Polish Academy of Sciences ul.Newelska601-447Warszawa Poland

A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so-called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self-adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains, The error estimates for proposed approximations of shape functionals are provided.

关键词： shape optimization asymptotic expansions self-adjoint extension weighted spaces with detached asymptotics topological derivatives

HIGHLY OSCILLATORY DIFFUSION-TYPE EQUATIONS

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Journal of Computational Mathematics 2013年第6期31卷 549-572页

作者： Sevda Uskiiplii Altlnbasak Marissa Condon Alfredo Deano Arieh Iserles Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge Cambridge UK School of Electronic Engineering Dublin City University Dublin Ireland Department of Mathematics University Carlos III de Madrid Madrid Spain Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge Cambridge UK

We explore new asymptotic-numeric solvers for partial differential equations with highly oscillatory forcing terms. Such methods represent the solution as an asymptotic series, whose terms can be evaluated by solving non-oscillatory problems and they guarantee high accuracy at a low computational cost. We consider two forms of oscillatory forcing terms, namely when the oscillation is in time or in space： each lends itself to different treatment. Numerical examples highlight the salient features of the new approach.

关键词： Diffusion-type PDEs High oscillation asymptotic expansions Modulated Fourier expansions.

Transient near tip fields in crack dynamics

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Science China(Physics,Mechanics & Astronomy) 2011年第7期54卷 1309-1318页

作者： BRATOV Vladimir PETROV Yuri UTKIN Alexander Institute of Problems of Mechanical Engineering Russian Academy of SciencesSt.-Petersburg 199178Russia St.-Petersburg State University St.-Petersburg 198504Russia

Transient effects of stress-strain fields in the vicinity of a stationary crack tip under high rate loads are *** analytical solutions to near tip stresses are compared to fields prescribed by leading terms(one or several) of Williams asymptotic *** of load application mode,time(or,which is the same,distance from a crack tip) and Poisson's ratio on this discrepancy is extensively *** effects connected with crack tip propagation speed are also *** inconsistencies between real(or received in numerical solutions of state *** element computations) crack tip fields and stress intensity factor(SIF) singular field observed by numerous researchers are *** scope of problems where SIF field can be used for correct prediction of dynamic stress-strain fields in the crack tip region is *** to correctly approximate fields that are not SIF dominated,accounting additional terms of Williams expansion,is studied.

关键词： transient crack tip fields dynamic fracture high-rate loads asymptotic expansions