• Title, Summary, Keyword: Orthogonal Polynomial Functions

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STRUCTURE RELATIONS OF CLASSICAL MULTIPLE ORTHOGONAL POLYNOMIALS BY A GENERATING FUNCTION

  • Lee, Dong Won
    • Journal of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1067-1082
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    • 2013
  • In this paper, we will find some recurrence relations of classical multiple OPS between the same family with different parameters using the generating functions, which are useful to find structure relations and their connection coefficients. In particular, the differential-difference equations of Jacobi-Pineiro polynomials and multiple Bessel polynomials are given.

A Free Vibration Analysis of Sound-Structure Interaction Plate (구조-음향 연성평판의 자유진동해석)

  • Lee, Dong-Ick;O, Jae-Eung
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.20 no.8
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    • pp.2546-2554
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    • 1996
  • In order to investigate the characteristics of sound-structure interaction problems, we modeled a rectangular cavity and the flexible wall of the cavity. Because the governing equations of motion are coupled through velocity terms, we could redefine them using the velocity potential. We calculated the natural frequencies of plate using orthogonal polynomial functions which satisfy the boundary conditions in the Rayleigh-Ritz Method. As the result, comparisons of theory and experiment show good agreement. and using orthogonal polynomial functions which satisfy the boundary conditions in the Rayleigh-Ritz method show useful method for sound-structure interaction problems too.

SOME FINITE INTEGRALS INVOLVING THE PRODUCT OF BESSEL FUNCTION WITH JACOBI AND LAGUERRE POLYNOMIALS

  • Ghayasuddin, Mohd;Khan, Nabiullah;Khan, Shorab Wali
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.1013-1024
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    • 2018
  • The main object of this paper is to set up two (conceivably) valuable double integrals including the multiplication of Bessel function with Jacobi and Laguerre polynomials, which are given in terms of Srivastava and Daoust functions. By virtue of the most broad nature of the function included therein, our primary findings are equipped for yielding an extensive number of (presumably new) fascinating and helpful results involving orthogonal polynomials, Whittaker functions, sine and cosine functions.

A free vibration analysis of sound-structure interaction plate having a small cut-out (부분적으로 열린 구조-음향 연성평판의 자유진동해석)

  • Oh, Jae-Eung;Rhee, Dong-Ick
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.21 no.10
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    • pp.1666-1673
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    • 1997
  • In order to investigate the characteristics of sound-structure interaction plate having a cut-out, we modeled a rectangular cavity and the flexible plate of the cavity. Because the particle velocity of air is the same as that of plate on the plate, we could easily redefine vibration equation using the velocity potential. We calculated the natural frequencies of plate using orthogonal polynomial functions which satisfy the boundary conditions in the Rayleigh-Ritz method. For the change of vibration characteristics, the effect of sound-structure interaction is more dominant than that of cut-out size.

Dynamic Analysis of the Multi-Span Beam on Elastic Foundation Part one : Natural Frequencies (탄성지반 위에 놓여있는 다지지 보의 동적해석 제1보 : 고유진동수)

  • Y.C. Kim;K.J. Choi
    • Journal of the Society of Naval Architects of Korea
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    • v.28 no.1
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    • pp.83-91
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    • 1991
  • In this paper the dynamic analysis of the multi-span beam on elastic foundation, which include discrete transnational and rotational springs, was performed. Furthermore, the effects of the intermediate supports were investigated. As a solution method, first the orthogonal polynomial functions which satisfy both the geometric and dynamic boundary conditions are obtained by imposing the orthogonality conditions. Then, the Galerkin's method is used to obtain the natural frequencies of the system. From numerical tests for various constraint and boundary conditions, it was found that the higher order orthogonal polynomial functions obtained by the present method can be used to get the accurate solution.

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ERROR BOUNDS FOR GAUSS-RADAU AND GAUSS-LOBATTO RULES OF ANALYTIC FUNCTIONS

  • Ko, Kwan-Pyo
    • Communications of the Korean Mathematical Society
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    • v.12 no.3
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    • pp.797-812
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    • 1997
  • For analytic functions we give an expression for the kernel $K_n$ of the remainder terms for the Gauss-Radau and the Gauss-Lobatto rules with end points of multiplicity r and prove the convergence of the kernel we obtained. The error bound are obtained for the type $$\mid$R_n(f)$\mid$ \leq \frac{1}{\pi}l(\Gamma) max_{z \in \Gamma} $\mid$K_n(z)$\mid$ max_{z \in \Gamma} $\mid$f(z)$\mid$$, where $l(\Gamma)$ denotes the length of contour $\Gamma$.

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Special Function Inverse Series Pairs

  • Alsardary, Salar Yaseen;Gould, Henry Wadsworth
    • Kyungpook Mathematical Journal
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    • v.50 no.2
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    • pp.177-193
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    • 2010
  • Working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the type $F_n(x)=\sum\limits_{k=0}^{n}A^n_kG_k(x)$ and $ G_n(x)=\sum\limits_{k=0}^{n}B_k^nF_k(x)$, where 0, 1, 2,$\cdots$. Here $F_n(x)$, $G_n(x)$ denote special polynomial functions, and $A_k^n$, $B_k^n$ denote coefficients found by use of the orthogonal properties of $F_n(x)$ and $G_n(x)$, or by skillful series manipulations. Typically $G_n(x)=x^n$ and $F_n(x)=P_n(x)$, the n-th Legendre polynomial. We give a collection of inverse series pairs of the type $f(n)=\sum\limits_{k=0}^{n}A_k^ng(k)$ if and only if $g(n)=\sum\limits_{k=0}^{n}B_k^nf(k)$, each pair being based on some reasonably well-known special function. We also state and prove an interesting generalization of a theorem of Rainville in this form.

Reconstruction of missing response data for identification of higher modes

  • Shrikhande, Manish
    • Earthquakes and Structures
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    • v.2 no.4
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    • pp.323-336
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    • 2011
  • The problem of reconstruction of complete building response from a limited number of response measurements is considered. The response at the intermediate degrees of freedom is reconstructed by using piecewise cubic Hermite polynomial interpolation in time domain. The piecewise cubic Hermite polynomial interpolation is preferred over the spline interpolation due to its trend preserving character. It has been shown that factorization of response data in variable separable form via singular value decomposition can be used to derive the complete set of normal modes of the structural system. The time domain principal components can be used to derive empirical transfer functions from which the natural frequencies of the structural system can be identified by peak-picking technique. A reduced-rank approximation for the system flexibility matrix can be readily constructed from the identified mass-orthonormal mode shapes and natural frequencies.

TD-CFIE Formulation for Transient Electromagnetic Scattering from 3-D Dielectric Objects

  • Lee, Young-Hwan;Jung, Baek-Ho;Sarkar, Tapan K.;Yuan, Mengtao;Ji, Zhong;Park, Seong-Ook
    • ETRI Journal
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    • v.29 no.1
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    • pp.8-17
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    • 2007
  • In this paper, we present a time domain combined field integral equation formulation (TD-CFIE) to analyze the transient electromagnetic response from dielectric objects. The solution method is based on the method of moments which involves separate spatial and temporal testing procedures. A set of the RWG functions is used for spatial expansion of the equivalent electric and magnetic current densities, and a combination of RWG and its orthogonal component is used for spatial testing. The time domain unknowns are approximated by a set of orthonormal basis functions derived from the Laguerre polynomials. These basis functions are also used for temporal testing. Use of this temporal expansion function characterizing the time variable makes it possible to handle the time derivative terms in the integral equation and decouples the space-time continuum in an analytic fashion. Numerical results computed by the proposed formulation are compared with the solutions of the frequency domain combined field integral equation.

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Free Vibration Analysis of Stiffened Plates Using Polynomials Having the Property of Timoshenko Beam Functions (Timoshenko 보함수 성질을 갖는 다항식을 이용한 보강판의 교유진동 해석)

  • 김병희;김진형;조대승
    • Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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    • pp.623-628
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    • 2004
  • In this study, the assumed-mode method using characteristic polynomials of Timoshenko beam is applied for the free vibration analysis of rectangular stiffened plates. The polynomial is derived considering the rotational constraint along the boundary edges of plate and the orthogonal relation of Timoshenko beam functions, which enables to simplify the free vibration analysis of plate structure having various boundary conditions. To verify the validity and effectiveness of the adopted method, numerical analysis for cross-stiffened plates were carried out and its results were compared with those obtained by the general purpose FEA software.

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