On the optimization of Gegenbauer operational matrix of integration

Elgindy, Kareem T.; Smith‐Miles, Kate

doi:10.1007/s10444-012-9289-5

Cited by 4 publications

(37 citation statements)

References 27 publications

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“…A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by [Elgindy and Smith-Miles (2013a)]. The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme.…”

Section: Discussionmentioning

confidence: 99%

“…One way to achieve such approximations was designed by [Elgindy and Smith-Miles (2013a)] via integrating the orthogonal Gegenbauer interpolant (2.6), and the sought definite integration approximations can be simply expressed in a matrix-vector multiplication; cf. [Elgindy and Smith-Miles (2013a), Theorem 2.1]. [Elgindy and Smith-Miles (2013a)] have further introduced a method for optimally constructing a rectangular GIM by minimizing the magnitude of the quadrature error in some optimality sense; cf.…”

Section: The Barycentric Gim and Quadraturementioning

confidence: 99%

“…[Elgindy and Smith-Miles (2013a), Theorem 2.1]. [Elgindy and Smith-Miles (2013a)] have further introduced a method for optimally constructing a rectangular GIM by minimizing the magnitude of the quadrature error in some optimality sense; cf. [Elgindy and Smith-Miles (2013a), Theorem 2.2].…”

Section: The Barycentric Gim and Quadraturementioning

confidence: 99%

“…In 2013, [Elgindy and Smith-Miles (2013a)] presented some novel numerical quadratures based on the concept of numerical integration matrices. Their unified approach employed the Gegenbauer basis polynomials to achieve rapid convergence rates for small/medium range of spectral expansion terms while using Chebyshev and Legendre bases polynomials for a large-scale number of expansion terms.…”

Section: Introductionmentioning

confidence: 99%

“…This approach provides in turn the luxury of evaluating quadratures for any arbitrary integration points for any desired degree of accuracy; thus increasing the accuracy of collocation schemes using relatively small number of collocation points; cf. [Elgindy and Smith-Miles (2013a), Elgindy and Smith-Miles (2013b), Elgindy and Smith-Miles (2013c), Elgindy (2016a)]. …”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

Elgindy

2017

Applied Numerical Mathematics

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SUMMARYThe work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the wellconditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by [Elgindy and Smith-Miles (2013a)]. The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.

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Section: Discussionmentioning

confidence: 99%

Section: The Barycentric Gim and Quadraturementioning

confidence: 99%

Section: The Barycentric Gim and Quadraturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

Elgindy

2017

Applied Numerical Mathematics

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Gegenbauer Collocation Integration Methods: Advances in Computational Optimal Control Theory

Elgindy¹

2014

Bull. Aust. Math. Soc.

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New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

Abd-Elhameed

2014

The Scientific World Journal

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This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

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On the optimization of Gegenbauer operational matrix of integration

Cited by 4 publications

References 27 publications

High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

Gegenbauer Collocation Integration Methods: Advances in Computational Optimal Control Theory

New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

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