An Optimal Quadrature Formula With Derivative for Weakly Singular Integrals

ABSTRACT This article presents the derivation and analysis of an optimal quadrature formula for the numerical integration of fractional integrals in the Hilbert space . In this space, functions satisfy certain smoothness conditions. The proposed quadrature formula is expressed as a linear combination of function values and its first derivative at equidistant nodes over the interval . The coefficients of the formula are derived by minimizing the norm of the error functional in the dual space . This error functional represents the difference between the exact integral and its quadrature approximation. We provide explicit formulas for the coefficients and for the norm of the error functional. The optimization problem is formulated and solved, leading to a system of linear equations for the coefficients. Analytical solutions of the system are obtained, providing explicit expressions for the optimal coefficients. Fractional integrals of several functions are numerically calculated with the constructed optimal quadrature formula, and the convergence with the exact value of the integral is analyzed in numerical experiments.