A WEIGHTED OPTIMAL QUADRATURE FORMULA WITH DERIVATIVE

This article focuses on the derivation and analysis of a weighted optimal quadra- ture formula in the Hilbert space W (2,1) 2 (0, 1). The formula is expressed as a linear combination of function values and its first-order derivatives at equidistant nodes in the interval [0, 1]. The coefficients are determined by minimizing the norm of the error functional in the dual space W (2,1)∗ 2 (0, 1). The article presents explicit expressions for the coefficients and the norm of the error func- tional. These results are obtained by formulating and solving an optimization problem, which leads to a system of linear equations for the coefficients. Analytical solutions of the system are obtained using the Sobolev method, providing an explicit expression for the optimal coefficients.

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