Hybrid Dual Quadrature Rules Combining Open and Closed Quadrature Rules Enhanced by Kronrod Extension or Richardson's Extrapolation for Numerical Integration.

• Haniyah Saed Ben Hamdin

  Mathematics Department, Faculty of SciencesSirte University

  Author

• Faoziya S. M. Musbah

  Mathematics Department, Faculty of Education, University of Bani Waleed, Bani Waleed, Libya

  Author

Numerical integration is a powerful way to integrate certain categories of integrals, such as those whose closed-form anti-derivative is missing, improper integrals, and tabular data where a function is absent. In this paper, open and closed dual hybrid quadrature rules have been designed for the numerical integration of real definite integrals with either a singular integrand or a non-elementary anti-derivative, respectively.  Such quadrature rules couple a Gauss-type rule with a Newton-Cotes-type rule such that both rules are of the same degree of precision, say p to achieve a hybrid rule of a degree of precision greater than or equal to p+2.  The open/closed-type hybrid quadrature rule has been constructed as a linear combination between the two-point Gauss-Legendre quadrature enhanced by Kronrod extension and a derivative-based open/closed Newton-Cotes formula, yielding a hybrid rule of degree of precision equal to nine. Furthermore, a hybrid quadrature rule was created by merging the numerically enhanced Lobatto-Gauss rule and Bool's rule, which was enhanced by Richardson extrapolation. An error analysis analytically confirms that the proposed rules perform better than their ingredients' quadrature rules. The effectiveness of the suggested hybrid rules has been demonstrated with some integral examples that exhibit good agreement with the precise outcomes. An adaptive algorithm has been implemented to enhance the accuracy of the results obtained.

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