An Optimal Quadrature Formula for Numerical Integration of the Right Riemann-Liouville Fractional Integral

In the present article, the problem of construction of the optimal quadrature formula for numerical integration of the right Riemann-Liouville fractional integral in the Hilbert space of real-valued functions is discussed. The error of the quadrature formula is bounded from above by the norm of the error functional. Initially, the norm of the error functional is found using the extremal function of the error functional of the quadrature formula. The norm of the error functional is a multivariate function with respect to the coefficients of the quadrature formula. Then, the Lagrange function is constructed to find the conditional minimum of the norm of the error functional. Thereby, a system of linear equations is obtained for the coefficients of the optimal quadrature formula. The existence and uniqueness of the solution of the obtained system are studied. It is used the discrete analogue of the differential operator d(2)/dx(2) - 1 to solve the obtained system. The analytical forms of the coefficients of the optimal quadrature formula are obtained. The obtained optimal quadrature formula is used in the numerical calculation for the Riemann-Liouville fractional integral of several functions. The errors in the numerical results are analyzed.