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legendre polynomials-Based Numerical Differentiation: A Convergence Analysis in a Weighted L2 Space

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Journal of Mathematical Research with Applications 2016年第2期36卷 247-252页

作者： Qin FANG Haojie LI Min XU College of Information and Engineering Dalian University School of Mathematical Sciences Dalian University of Technology

We consider the problem of estimating the derivative of a function f from its noisy version fδby using the derivatives of the partial sums of Fourier-legendre series of f. Instead of the observation Lspace, we perform the reconstruction of the derivative in a weighted Lspace. This takes full advantage of the properties of legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.

关键词： legendre polynomials numerical differentiation Jacobi polynomials weighted L2 space

Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and legendre polynomials

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Optics and Photonics Journal 2016年第6期6卷 113-123页

作者： Igor Di Varano Leibniz Institut für Astrophysik Potsdam (AIP) Potsdam Germany

For every astronomical instrument, the operating conditions are undoubtedly different from those defined in a setup experiment. Besides environmental conditions, the drives, the electronic cabinets containing heaters and fans introduce disturbances that must be taken into account already in the preliminary design phase. Such disturbances can be identified as being mostly of two types: heat sources/sinks or cooling systems responsible for heat transfer via conduction, radiation, free and forced convection on one side and random and periodic vibrations on the other. For this reason, a key role already from the very beginning of the design process is played by integrated model merging the outcomes based on a Finite Element Model from thermo-structural and modal analysis into the optical model to estimate the aberrations. The current paper presents the status of such model, capable of analyzing the deformed surfaces deriving from both thermo-structural and vibrational analyses and measuring their effect in terms of optical aberrations by fitting them by Zernike and legendre polynomial fitting respectively for circular and rectangular apertures. The independent contribution of each aberration is satisfied by the orthogonality of the polynomials and mesh uniformity.

关键词： FEM Wavefront Error Zernike legendre polynomials Astronomical Instrumentation

Duality between Bessel Functions and Chebyshev polynomials in Expansions of Functions

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Advances in Pure Mathematics 2023年第8期13卷 504-536页

作者： Alfred Wünsche Humboldt-Universitä t Berlin Institut für Physik Nichtklassische Strahlung (MPG) Berlin Germany

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

关键词： Spherical Bessel Functions Chebyshev polynomials legendre polynomials Hermite polynomials Derivatives of Delta Functions Normally and Anti-Normally Ordered Operators

A one-dimensional polynomial chaos method in CFD–Based uncertainty quantification for ship hydrodynamic performance

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Journal of Hydrodynamics 2013年第5期25卷 655-662页

作者：贺伟 DIEZ Matteo CAMPANA Emilio Fortunato STERN Frederick 邹早建 School of Naval Architecture Ocean and Civil Engineering Shanghai Jiao Tong University IIHR-Hydroscience and Engineering The University of Iowa IIHR-Hydroscience and Engineering The University of Iowa CNR-INSEAN National Research Council-Maritime Research Centre CNR-INSEAN National Research CouncilMaritime Research Centre School of Naval Architecture Ocean and Civil Engineering and State Key Laboratory of Ocean Engineering Shanghai Jiao Tong University

A one-dimensional non-intrusive Polynomial Chaos （PC） method is applied in Uncertainty Quantification （UQ） studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value （EV） and Standard Deviation （SD） are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the legendre polynomials is applied to：（1） stochastic input domain and （2） Cumulative Distribution Function （CDF） image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization （RDO） and Reliability-Based Design Optimization （RBDO） of ship hydrodynamic performances.

关键词： Uncertainty Quantification （UQ） Polynomial Chaos （PC） method legendre polynomials ship design

A Unified Petrov-Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

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Communications on Applied Mathematics and Computation 2021年第1期3卷 61-90页

作者： Mehdi Samiee Ehsan Kharazmi Mark M.Meerschaert Mohsen Zayernouri Department of Mechanical Engineering Michigan State UniversityEast LansingMI 48824USA Department of Computational Mathematics Science and EngineeringMichigan State UniversityEast LansingMI 48824USA Department of Probability and Statistics Michigan State UniversityEast LansingMI 48824USA

Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings.In complex multi-fractal anomalous transport phenomena,distributed-order partial differential equations appear as tractable mathematical models,where the underlying derivative orders are distributed over a range of values,hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics.We develop a unified,fast,and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and legendre polynomials as temporal and spatial basis/test functions,respectively.By defining the proper underlying distributed Sobolev spaces and their equivalent norms,we rigorously prove the well-posedness of the weak formulation,and thereby,we carry out the corresponding stability and error analysis.We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

关键词： Distributed Sobolev space Well-posedness analysis Discrete inf-sup condition Spectral convergence Jacobi poly-fractonomials legendre polynomials

On Polynomial Maximum EntropyMethod for ClassicalMoment Problem

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Advances in Applied Mathematics and Mechanics 2016年第1期8卷 117-127页

作者： Jiu Ding Noah H.Rhee Chenhua Zhang Department of Mathematics The University of Southern MississippiHattiesburgMS 39406-5045USA Department ofMathematics and Statistics University ofMissouri–Kansas CityKansas CityMO 64110-2499USA

The maximum entropy method for the Hausdorff moment problem suffers from ill conditioning as it uses monomial basis{1,x,x2,···,xn}.Themaximum entropy method for the Chebyshev moment probelm was studied to overcome this drawback in[4].In this paper we review and modify the maximum entropy method for the Hausdorff and Chebyshev moment problems studied in[4]and present the maximum entropy method for the legendre moment problem.We also give the algorithms of converting the Hausdorff moments into the Chebyshev and Lengendre moments,respectively,and utilizing the corresponding maximum entropy method.

关键词： Classical moment problem monomials Chebyshev polynomials legendre polynomials

On the solution of the Human Immunodeficiency Virus(HIV)infection model using spectral collocation method

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International Journal of Biomathematics 2021年第2期14卷 15-40页

作者： Sagithya Thirumalai Rajeswari Seshadri Suayip Yuzbası Department of Mathematics Pondicherry UniversityKalapetPuducherry 605014India Department of Mathematics Akdeniz University TH-07058 AntalyaTurkey

In this research work,we study the Human Immunodeficiency Virus(HIV)infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations.The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time.The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev,legendre and Jacobi polynomials.Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts.Detailed error analysis is done to compare our results with the existing methods.It is shown that the spectral collocation method is a very reliable,efficient and robust method of solution compared to many other solution procedures available in the literature.All these results are presented in the form of tables and figures.

关键词： HIV infection model spectral collocation method Chebyshev polynomials legendre polynomials Jacobi polynomials error analysis

An analysis of the characteristics of crustal magnetic anomaly in China based on CHAMP satellite data

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Geodesy and Geodynamics 2018年第4期9卷 328-333页

作者： Jianguo Zhang Xiaodong Yang Jungang Yan Xiaoping Wu Handan Central Seismic Station of Hebei Earthquake Agency Handan 056001 China School of Earth and Space Sciences University of Science and Technology of China Hefei 230026 China

Based on the observation data of CHAMP satellite from 2006 to 2009, a 2D crustal magnetic anomaly model in China is established to study the distribution characteristics of crustal magnetic anomaly. In this paper, the 2D anomaly model is derived from the legendre polynomial expansion of harmonic term N =6-50. The result shows that many elaborate structures reflected in magnetic anomaly map well correspond to the geologic structures in China and its adjacent area. The magnetic anomaly at low satellite height behaves complexly, which is mainly caused by the magnetic disturbance of shallow rocks.In contrast, the magnetic field isolines at high satellite height are relatively sparse and only magnetic anomalies of deep crust are reflected. This fact implies that the 2D model of crustal magnetic anomaly provides an important method of the space prolongation of geomagnetic field, and is of theoretical and practice importance in geologic structure analysis and geophysical prospecting.

关键词： CHAMP satellite legendre polynomials Crustal magnetic anomaly Magnetic anomaly model

SOME RESULTS ON THE COEFFICIENTS OF INTEGRATED EXPANSIONS OF ULTRASPHERICAL polynomials AND THEIR INTEGRALS

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Analysis in Theory and Applications 2001年第3期17卷 69-84页

作者： E.H.Doha S.I.EL-Soubhy Department of Mathematics Faculty of Science Cairo UniversityGiza-Egypt Department of Mathematics Girl's College of EducationJeddahSaudi Arabia

The formula of expressing the coefficients of an expansion ofultraspherical polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is stated in a more compact form and proved in a simpler way than the formula of Phillips and Karageorghis (1990). A new formula is proved for the q times integration of ultraspherical polynomials, of which the Chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.CLC Number：O17 Document ID：AAuthor Resume：E. H. Doha，e-mail: eiddoha@frcu, eun. eg References：[1]Canuto,C. ,Spectral Methods in Fluid Dynamics,Springer,Belrin,1988.[2]Doha,E.H.,An Accurate Solution of Parabolic Equations by Expansion in Ultraspherical polynomials,Comput. Math. Appl. ,19(1990),75-88.[3]Doha,E. H. ,The Coefficients of Differentiated Expansions and Derivatives of Ultraspherical polynomials,Comput. Math. Appl.,21(1991),115-122.[4]Doha,E.H. ,The Chebyshev Coefficients of General order Derivatives of an Infinitely Differen-tiable Function in Two or Three Variables,Ann. Univ. Sci. Budapest. Sect. Comput. ,13(1992),83-91.[5]Doha,E. H.,On the Cefficients of Differentiable Expansions of Double and Triple legendre polynomials,Ann Univ. Sci. Budapest. Sect. Comput. ,15(1995),25-35.[6]Doha,E.H. ,The Ultraspherical Coefficients of the Moments of a General-Order Derivatives of an Infinitely Differentiable Function,J. Comput. Math. ,89(1998),53-72.[7]Doha,E.H. ,The Coefficients of Differentiated Expansions of Double and Triple Ultraspherical polynomials,Annales Univ. Sci. Budapest.,Sect. Comp.,19(200),57-73.[8]Doha,E.H. and Al-Kholi,F. M. R. ,An Efficient Double Legerdre Spectral Method for Parabolic and Elliptic Partial Differential Equations,Intern. J. Computer. Math. (toAppear).[9]Fox,L. and Parker,I.B. ,Chebyshev polynomials in Numerical Analysis,Clarendon Press,Oxford,1972.[10]Doha,E.H. an

关键词： differential equations Partial Differential Equations Spectral Methods Chebyshev polynomials legendre polynomials Numerical Analysis Boundary Value Problems Spectral Approximation Parabolic Equations Applied Mathematics Accurate Solution Fluid Dynamics Heat Transfer new formula Methods for