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C1-Conforming Quadrilateral Spectral Element Method for Fourth-order Equations

Communications on Applied Mathematics and Computation 2019年第3期1卷 403-434页

作者： Huiyuan Li Weikun Shan Zhimin Zhang State Key Laboratory of Computer Science/Laboratory of Parallel Computing Institute of SoftwareChinese Academy of SciencesBeijing 100190China School of Science Henan University of TechnologyZhengzhou 450001HenanChina Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for Fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of Fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

关键词： Quadrilateral spectral element method Fourth-order equations Mapped polynomials C1-conforming basis Polynomial inclusion Completeness

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四阶p-Laplacian边值问题正解的存在性（英文）

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Chinese Quarterly Journal of Mathematics 2018年第4期33卷 377-385页

作者：王万鹏 School of Mathematics and Statistics Huazhong University of Science and Technology

This paper investigates the existence of positive solutions for a Fourth-order p-Laplacian nonlinear equation. We show that, under suitable conditions, there exists a positive number λ~*such that the above problem ha... 详细信息

关键词： Fourth-order p-Laplacian Positive solutions Upper and lower solutions method Fixed point theory

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Existence of solutions and positive solutions to a Fourth-order two-point BVP with second derivative

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2004年第3期5卷 104-108页

作者：姚庆六 Department of Applied Mathematics Nanjing University of Finance and Economics

Several existence theorems were established for a nonlinear Fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the height of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends.

关键词： Nonlinear Fourth-order equation Two-point boundary value problem Solution and positive solution Exis

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High Accurate Fourth-order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate

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American Journal of Computational Mathematics 2014年第2期4卷 73-86页

作者： Alemayehu Shiferaw Ramesh Chand Mittal Department of Mathematics Indian Institute of Technology Roorkee India

In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by Fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results.

关键词： Poisson’s Equation Tri-Diagonal Matrix Fourth-order Finite Difference Approximation Hockney’s Method Thomas Algorithm

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Existence and Uniqueness of Positive Solutions for Fourth-order Nonlinear Singular Sturm-Liouville Problems

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Journal of Applied Mathematics and Physics 2014年第9期2卷 875-881页

作者： Ying He School of Mathematics and Statistics Northeast Petroleum University Daqing China

By mixed monotone method, we establish the existence and uniqueness of positive solutions for Fourth-order nonlinear singular Sturm-Liouville problems. The theorems obtained are very general and complement previously ... 详细信息

关键词： Mixed Monotone Operator Fourth-order Boundary Value Problem Singular Uniqueness

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An Efficient Technique for Finding the Eigenvalues of Fourth-order Sturm-Liouville Problems

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Applied Mathematics 2012年第8期3卷 920-925页

作者： Mohamed El-Gamel Mona Sameeh Department of Mathematical Sciences Faculty of Engineering Mansoura University Egypt

In this work, we present a computational method for solving eigenvalue problems of Fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated ... 详细信息

关键词： Chebychev Polynomial Fourth-order Sturm-Liouville Eigenvalue

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Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-order Parabolic Partial Differential Equation

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Journal of Applied Mathematics and Physics 2018年第7期6卷 1476-1480页

作者： Béyi Boukary Justin Loufouilou-Mouyedo Joseph Bonazebi-Yindoula Gabriel Bissanga Universitéde Ouaga 1 Pr JKZ Ouagadougou Burkina Faso Université Marien NGOUABI Brazzaville Republic of Congo

In this paper, the ADM method is used to construct the solution of the singular Fourth-order partial differential equation.

关键词： SBA Method Singular Fourth-order Partial Differential Equation

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Joint frequency, 2D AOA and polarization estimation using Fourth-order cumulants

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Science China(Technological Sciences) 2000年第3期43卷 297-303页

作者：王建英陈天琪 University of Electronic Science and Technology of China Chengdu 610054 China

Based on Fourth-order cumulant and ESPRIT algorithm, a novel joint frequency, two-dimensional angle of arrival (2D AOA) and the polarization estimation method of incoming multiple independent spatial narrow-band non-Gaussian signals in arbitrary Gaussian noise environment are proposed . The array is composed of crossed dipoles parallel to the coordinate axes. The crossed dipole positions are arbitrarily distributed. Computer simulation confirms its feasibility.

关键词： frequency two-dimensional angles of arrival polarization ESPRIT Fourth-order cumulant.

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High-order phase-field model with the local and second-order max-entropy approximants

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Frontiers of Structural and Civil Engineering 2019年第2期13卷 406-416页

作者： Fatemeh AMIRI Department of Mechanical Engineering Isfahan University of TechnologyIsfahan 84156-83111Iran School of Mathematics Institute for Research in Fundamental Sciences (1PM)Tehran 19395-5746Iran

We approximate the fracture surface energy functional based on phase-field method with smooth local maximum entropy (LME) and second-order maximum entropy (SME) approximants. The higher-order continuity of the meshfree methods such as LME and SME approximants allows to directly solve the Fourth-order phase-field equations without splitting the fourth?ordcr differential equation into two second-order differential equations. We will first show that the crack surface functional can be captured more accurately in the Fourth-order model with smooth approximants such as LME, SME and B-spline. Furthermore, smaller length scale parameter is needed for the Fourth-order model to approximate the energy functional. We also study SME approximants and drive the formulations. The proposed meshfree Fourth-order phase-field formulation show more stable results tor SME compared to LME meshfree methods.

关键词： second-order maximum entropy local maximum entropy second-and Fourth-order phase-field models B-spline

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C^0DISCONTINUOUS GALERKIN METHODS FOR A PLATE FRICTIONAL CONTACT PROBLEM

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Journal of Computational Mathematics 2019年第2期37卷 184-200页

作者： Fei Wang Tianyi Zhang Weimin Han School of Mathematics and Statistics Xi 'an Jiaotong University Xi'an 710049 China Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242 USA School of Mathematics and Statistics Xi 'an Jiaotong University Xi'an 710049 China Departinent of Mathematics University of Iowa Iowa City. IA 52242 USA

Numerous C^0 discontinuous Galerkin (DG) schemes for the Kirchhoff plate bending problem are extended to solve a plate frictional contact problem, which is a Fourth-order elliptic variational inequality of the second kind. This variational inequality contains a nondifferentiable term due to the frictional contact. We prove that these C^0 DG methods are consis tent and st able, and derive optimal order error estima tes for the quadratic element. A numerical example is presented to show the performance of the C^0 DG methods;and the numerical convergence orders confirm the theoretical prediction.

关键词： Variational inequality of Fourth-order Discontinuous Galerkin method Plate frictional contact problem Optimal order error estimate

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