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Quadrature Formula Study for the Integral with Hilbert Kernel based on Trigonometric Interpolational Polynomial

Affiliations

  • Kazan Federal University, 420008, Kazan, Kremlevskaya str. 18, Russia;

Abstract


Objectives: The article presents the study of the quadrature formula for the integral with Hilbert’s kernel. Methods: The sub integral function is close to interpolation polynomial on such equally spaced nodes that the values of the Weyl fractional integral coincide in these nodes for the function and polynomial. At the derivation of a formula, the known values of the integral are used with Hilbert’s kernel of certain functions, the properties of trigonometric polynomials and the properties of trigonometric functions Results: The obtained quadrature formulas were tested using Wolfram Mathematica system. Calculations performed at different values of node number and the order of integration. The values obtained using the studied quadrature was compared with the values obtained using the previously known formula. Conclusion: The growth of node number improves by the quadrature formula, the dependence of approximation on the values, is observed. At the resemblance to the section ends the difference between integral values calculated by different formulas increases.

Keywords

Fractional Integration, Integral with the Hilbert Kernel, Interpolational Polynomial, Polynomial Operator, Quadrature Formula, Trigonometric Polynomial, Weyl Fractional Integral

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