In mathematical modeling and computational mathematics, special attention is paid to the creation of various optimal calculation methods. This article is devoted to the construction of optimal quadrature formulas with derivative in the space of differentiable functions using the Sobolev method. This quadrature formula consists of a linear combination of the values of the interval 0, 1 up to the second derivative of the function at all nodes. The error of the quadrature formulas is estimated by the norm of the error function. We obtain the optimal quadrature formula by minimizing the norm of the error functional by the coefficients of the quadrature formula with derivative. The resulting optimal quadrature formulas are exact for all functions which polinomials degree of m-1. In addition, some methods for the numerical solution of Fredholm integral equations of the second kind are given. These methods are optimal quadrature formulas and Simpson’s method. Numerical examples are provided to demonstrate the effectiveness and accuracy of the work presented.
Shadimetov et al. (Wed,) studied this question.