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Computer Modeling in Engineering & Sciences 2018年第4期115卷 67-84页

作者： Jiaquan Xie Fuqiang Zhao Zhibin Yao Jun Zhang College of Mechanical Engineering Taiyuan University of Science and TechnologyTaiyuan 030024China Taiyuan Institute of China Coal Technology Engineering Group Taiyuan 030006China Collaborative Innovation Center of Taiyuan Heavy Machinery Equipment Taiyuan 030024China.

In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.

关键词： Three-variable shifted Jacobi polynomials multi-term fractional-order PDEs variable coefficients numerical solution convergence analysis

Fitting objects with implicit polynomials by deep neural network

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Optoelectronics Letters 2023年第1期19卷 60-64页

作者： LIU Jingyi YU Lina SUN Linjun TONG Yuerong WU Min LI Weijun Institute of Semiconductors Chinese Academy of SciencesBeijing100083China Beijing Key Laboratory of Semiconductor Neural Network Intelligent Sensing and Computing Technology Beijing100083China School of Integrated Circuits University of Chinese Academy of SciencesBeijing100049China Shenzhen DAPU Microelectronics Co. Ltd.Shenzhen518116China

Implicit polynomials(IPs)are considered as a powerful tool for object curve fitting tasks due to their simplicity and fewer parameters.The traditional linear methods,such as 3L,Min Var,and Min Max,often achieve good performances in fitting simple objects,but usually work poorly or even fail to obtain closed curves of complex object contours.To handle the complex fitting issues,taking the advantages of deep neural networks,we designed a neural network model continuity-sparsity constrained network(CSC-Net)with encoder and decoder structure to learn the coefficients of IPs.Further,the continuity constraint is added to ensure the obtained curves are closed,and the sparseness constraint is added to reduce the spurious zero sets of the fitted curves.The experimental results show that better performances have been obtained on both simple and complex object fitting tasks.

关键词： continuity constraint polynomials

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre polynomials

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Communications in Theoretical Physics 2014年第9期61卷 331-337页

作者：陈昌远尤源陆法林孙东升董世海 School of Physics and Electronics Yancheng Teachers University Escuela Superior de Físicay Matemáticas Instituto Politécnico Nacional Edificio 9 Unidad Profesional Adolfo López Mateos Mexico D.F. 07738 Mexico Department of Physics and Astronomy Louisiana State University Baton Rouge LA 70803-4001 USA

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.

关键词： double ring-shaped potential super-universal associated Legendre polynomials parity hypergeometric functions Jacobi polynomials

Numerical Study for a Class of Variable Order Fractional Integral-differential Equation in Terms of Bernstein polynomials

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Computer Modeling in Engineering & Sciences 2015年第1期104卷 69-85页

作者： Jinsheng Wang Liqing Liu Yiming Chen Lechun Liu Dayan Liu School of Continuing Education Yanshan UniversityQinhuangdao066004HebeiChina College of Science Yanshan UniversityQinhuangdao066004HebeiChina INSA Centre Val de Loire Universitéd’OrléansPRISME EA 4229Bourges 18022 France

The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials.The fractional derivative is described in the Caputo sense.Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable.By solving the algebraic equations,the numerical solutions are acquired.The method in general is easy to implement and yields good results.Numerical examples are provided to demonstrate the validity and applicability of the method.

关键词： Bernstein polynomials variable order fractional integral-differentialequation operational matrix numerical solution convergence analysis the absolute error

Some Estimates of the Maximum Modulus for polynomials with Gaps

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Analysis in Theory and Applications 2019年第4期35卷 421-426页

作者： Eze R.Nwaeze Department of Mathematics Tuskegee UniversityTuskegeeAL 36088USA

Let p(z)be a polynomial of degree n having some zeros at a point z0 ∈C with |z0|<1 and the rest of the zeros lying on or outside the boundary of a prescribed disk.In this brief note,we consider this class of polynomi... 详细信息

Let p(z)be a polynomial of degree n having some zeros at a point z0 ∈C with |z0|polynomials and obtain some bounds for(max|z|=R|p(z)|)s in terms of (max|z|=1|p(z)|)s for any R≥1 and s∈N.

关键词： polynomials maximum modulus zeros prescribed disk

Accurate and fast simulation of Kolmogorov phase screen by combining spectral method with Zernike polynomials method

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Chinese Optics Letters 2010年第10期8卷 969-971页

作者：张宝东秦石乔王省书 College of Photoelectric Science and Engineering National University of Defense Technology

A new method is proposed to simulate the Kolmogorov phase screen. A hybrid phase screen is achieved by linearly combining two phase screens simulated by both spectral and Zernike polynomials methods. Unlike the imperfection of phase structure function existing at some spatial frequencies in each of the two methods, the mean structure function of the hybrid phase screens coincides remarkably well with theory in the whole spatial frequency domain. Apart from advantage on accuracy, the proposed method also saves considerable computational time. For a phase screen with certain accuracy, the new method is ten times and seven times faster compared with the spectral method and Zernike polynomials method, respectively.

关键词： Computational complexity polynomials Spectroscopy

Certain Results for the 2-Variable Peters Mixed Type and Related polynomials

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Analysis in Theory and Applications 2018年第2期34卷 147-164页

作者： Ghazala Yasmin Department of Applied Mathematics Faculty of Engineering Aligarh Muslim University

In this article, the 2-variable general polynomials are taken as base with Peters polynomials to introduce a family of 2-variable Peters mixed type polynomials.These polynomials are framed within the context of monomiality principle and their properties are established. Certain summation formulae for these polynomials are also derived. Examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored.

关键词： 2-Variable general polynomials Peters polynomials 2-variable truncated exponential polynomials Sheffer sequences monomiality principle

Inequalities Concerning The Maximum Modulus of polynomials

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Analysis in Theory and Applications 2018年第2期34卷 175-186页

作者： B.A.Zargar A.W.Manzoor Shaista Bashir Department of Mathematics University of Kashmir

Let P(z) be a polynomial of degree n having all its zeros in |z|≤k, k ≤1, then for every real or complex number β, with |β|≤ 1 and R ≥ 1, it was shown by A.Zireh et al. [7] that for |z|=1,min|z|=1|P(Rz)+β((R+k)/(1+k))~nP(z)|≥k^(-n)|R^n+β((R+k)/(1+k))~n|min|z|=k|P(z)|.In this paper, we shall present a refinement of the above inequality. Besides, we shall also generalize some well-known results.

关键词： Growth of polynomials minimum modulus of polynomials inequalities

Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems

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Control Theory and Technology 2019年第3期17卷 297-306页

作者： Sheng-Guo WANG Shu LIANG Liang MA Kaixiang PENG College of Engineering and College of Computing and Informatics University of North Carolina at Charlotte NC 28223-0001 U.S.A. School of Automation and Electrical Engineering University of Science and Technology Beijing Beijing 100083 China

A Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems is presented via an auxiliary integer degree polynomial. The presented Routh test is a classical Routh table test on the auxiliary integer degree polynomial derived from and for the commensurate fractional degree polynomial. The theoretical proof of this proposed approach is provided by utilizing Argument principle and Cauchy index. Illustrative examples show efficiency of the presented approach for stability test of commensurate fractional degree polynomials and commensurate fractional order systems. So far, only one Routh-type test approach [1] is available for the commensurate fractional degree polynomials in the literature. Thus, this classical Routh-type test approach and the one in [1] both can be applied to stability analysis and design for the fractional order systems, while the one presented in this paper is easy for peoples, who are familiar with the classical Routh table test, to use.

关键词： Fractional order systems stability commensurate fractional degree polynomials Routh table test

Jacobi-Sobolev Orthogonal polynomialsand Spectral Methods for Elliptic Boundary Value Problems

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Communications on Applied Mathematics and Computation 2019年第2期1卷 283-308页

作者： Xuhong Yu Zhongqing Wang Huiyuan Li University of Shanghai for Science and Technology Shanghai 200093China State Key Laboratory of Computer Science/Laboratory of Parallel Computing Institute of SoftwareChinese Academy of SciencesBeijing 100190China

Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

关键词： Generalized Jacobi polynomials Spectral method - Jacobi-Sobolev orthogonal basis functions Elliptic boundary value problems Error analysis