The asymptotic representation at a point of the derivative of orthonormal polynomials

1976 ◽  
Vol 19 (5) ◽  
pp. 397-404 ◽  
Author(s):  
B. L. Golinskii
Keyword(s):  
Asymptotic Representation ◽  
Orthonormal Polynomials

The Sector Problem

1957 ◽  
Vol 24 (4) ◽  
pp. 574-581
Author(s):  
G. Horvay ◽  
K. L. Hanson
Keyword(s):  
Variational Method ◽  
Orthogonal Polynomials ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Circular Sector ◽  
Orthonormal Polynomials ◽  
Circular Boundary ◽  
Stress Functions ◽  
Complete Set ◽  
Derivatives Of

Abstract On the basis of the variational method, approximate solutions f k ( r ) h k ( θ ) , f k ( r ) g k ( θ ) , F k ( θ ) H k ( r ) , F k ( θ ) G k ( r ) of the biharmonic equation are established for the circular sector with the following properties: The stress functions fkhk create shear tractions on the radial boundaries; the stress functions fkgk create normal tractions on the radial boundaries; the stress functions FkHk create both shear and normal tractions on the circular boundary, and the stress functions FkGk create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors fk(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors Fk(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions Fk″ + Fk constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.

New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials

2016 ◽  
Vol 174 (1) ◽  
pp. 295-320 ◽  
Author(s):  
Samer S. Ezz-Eldien ◽  
Ramy M. Hafez ◽  
Ali H. Bhrawy ◽  
Dumitru Baleanu ◽  
Ahmed A. El-Kalaawy
Keyword(s):  
Variational Problems ◽  
Numerical Approach ◽  
Orthonormal Polynomials ◽  
Fractional Variational Problems

scholarly journals A New Asymptotic Treatment of g-modes of a Star

1995 ◽  
Vol 155 ◽  
pp. 285-286
Author(s):  
P. Smeyers ◽  
T. Van Hoolst ◽  
I. De Boeck ◽  
L. Decock
Keyword(s):  
Singular Perturbation ◽  
Fourth Order ◽  
Asymptotic Expansions ◽  
Equilibrium Model ◽  
Asymptotic Representation ◽  
Radial Displacement ◽  
Low Frequency ◽  
Order System ◽  
Oscillation Modes ◽  
Perturbation Problems

An asymptotic representation of low-frequency, linear, isentropic g-modes of a star is developed without the usual neglect of the Eulerian perturbation of the gravitational potential. Our asymptotic representation is based on the use of asymptotic expansions adequate for solutions of singular perturbation problems (see, e.g., Kevorkian & Cole 1981).Linear, isentropic oscillation modes with frequency different from zero are governed by a fourth-order system of linear, homogeneous differential equations in the radial parts of the radial displacement ξ(r) and the divergence α(r). These equations take the formThe symbols have their usual meaning. N2 is the square of the frequency of Brunt-Väisälä. The functions K1 (r), K2 (r), K3 (r), K4 (r), depend on the equilibrium model, e.g.,We introduce the small expansion parameterand assume, for the sake of simplification, N2 to be positive everywhere in the star so that the star is everywhere convectively stable.

scholarly journals Azarin limiting sets of functions and asymptotic representation of integrals

2019 ◽  
Vol 11 (2) ◽  
pp. 97-113
Author(s):  
Konstantin Gennadyevich Malyutin ◽  
Taisiya Ivanovna Malyutina ◽  
Tat'yana Vasil'evna Shevtsova
Keyword(s):  
Asymptotic Representation
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