• Title/Summary/Keyword: three term recurrence relation

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FINITE ORTHOGONAL POLYNOMIALS SATISFYING A SECOND ORDER DIFFERENTIAL EQUATION

  • Yoo, Byeong-Hoon;Lee, Dong-Won
    • Communications of the Korean Mathematical Society
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    • v.20 no.4
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    • pp.765-774
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    • 2005
  • The orthogonality of polynomials plays an important role in many areas and in many cases only finite orthogonalities are used. Concerning this fact we find characterizations of a finite orthogonal polynomial system satisfying a second order differential equation and then give several examples.

CHARACTERIZATIONS OF SOME POLYNOMIAL VARIANCE FUNCTIONS BY d-PSEUDO-ORTHOGONALITY

  • KOKONENDJI CELESTIN C.
    • Journal of applied mathematics & informatics
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    • v.19 no.1_2
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    • pp.427-438
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    • 2005
  • From a notion of d-pseudo-orthogonality for a sequence of poly-nomials ($d\;\in\;{2,3,\cdots}$), this paper introduces three different characterizations of natural exponential families (NEF's) with polynomial variance functions of exact degree 2d-1. These results provide extended versions of the Meixner (1934), Shanbhag (1972, 1979) and Feinsilver (1986) characterization results of quadratic NEF's based on classical orthogonal polynomials. Some news sets of polynomials with (2d-1)-term recurrence relation are then pointed out and we completely illustrate the cases associated to the families of positive stable distributions.

A NON-ASYMPTOTIC METHOD FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS

  • File, Gemechis;Reddy, Y.N.
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.39-53
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    • 2014
  • In this paper, a non-asymptotic method is presented for solving singularly perturbed delay differential equations whose solution exhibits a boundary layer behavior. The second order singularly perturbed delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. Then, Simpson's integration formula and linear interpolation are employed to get three term recurrence relation which is solved easily by Discrete Invariant Imbedding Algorithm. Some numerical examples are given to validate the computational efficiency of the proposed numerical scheme for various values of the delay and perturbation parameters.

NUMERICAL METHOD FOR SINGULAR PERTURBATION PROBLEMS ARISING IN CHEMICAL REACTOR THEORY

  • Andargie, Awoke
    • Journal of applied mathematics & informatics
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    • v.28 no.1_2
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    • pp.411-423
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    • 2010
  • In this paper, a numerical method for singular perturbation problems arising in chemical reactor theory for general singularly perturbed two point boundary value problems with boundary layer at one end(left or right) of the underlying interval is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

NUMERICAL INTEGRATION METHOD FOR SINGULAR PERTURBATION PROBLEMS WITH MIXED BOUNDARY CONDITIONS

  • Andargie, Awoke;Reddy, Y.N.
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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    • pp.1273-1287
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    • 2008
  • In this paper, the numerical integration method for general singularly perturbed two point boundary value problems with mixed boundary conditions of both left and right end boundary layer is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

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q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0 FOR POSITIVE INTEGERS N

  • Moreno, Samuel G.;Garcia-Caballe, Esther M.
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.913-926
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    • 2011
  • The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.