The order of convergence of an optimal quadrature formula with\n derivative in the space W₂^(2,1)

2019/08/01 by A.R. Hayotov, Hayotov, Abdullo R., Rashidjon Rasulov +1
Mathematics · #65D30 #65D32 #Differential Equations and Boundary Problems #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1908.00450

openalex publication_date 2019/08/01 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

The present work is devoted to extension of the trapezoidal rule in the space\nW₂^(2,1). The optimal quadrature formula is obtained by minimizing the\nerror of the formula by coefficients at values of the first derivative of a\nintegrand. Using the discrete analog of the operator fracd²dx²-1 the\nexplicit formulas for the coefficients of the optimal quadrature formula are\nobtained. Furthermore, it is proved that the obtained quadrature formula is\nexact for any function of the set\n\F=\span 1,x,e^x,e^-x . Finally, in the space\nW₂^(2,1) the square of the norm of the error functional of the constructed\nquadrature formula is calculated. It is shown that the error of the obtained\noptimal quadrature formula is less than the error of the Euler-Maclaurin\nquadrature formula on the space L₂⁽²⁾.\n

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