QUADRATURE FORMULAS FOR APPROXIMATING FRACTIONAL RIEMANN-LIOUVILLE INTEGRALS

Abstract

Fractional integrals play a fundamental role in fractional calculus and are widely used in the modeling of complex physical and engineering processes with memory and hereditary properties. In particular, the Riemann–Liouville fractional integral is characterized by a weakly singular kernel, which makes its exact analytical evaluation difficult in most practical cases. Therefore, the development of efficient and accurate numerical methods for approximating fractional integrals remains an important research topic. In this paper, several quadrature formulas for approximating Riemann–Liouville fractional integrals are investigated. The proposed approaches are based on polynomial interpolation, including fractional left and right rectangular formulas as well as the fractional trapezoidal (L1) method. Exact analytical solutions for power-type functions are derived and used as benchmark solutions. Numerical approximations are constructed and compared with exact solutions to assess accuracy. The absolute error is analyzed to evaluate the performance of each method. The results demonstrate that polynomial interpolation–based methods provide reliable approximations of fractional integrals. In particular, the fractional trapezoidal (L1) method shows improved accuracy compared to rectangular formulas while maintaining computational simplicity. These findings confirm the effectiveness of the proposed quadrature formulas for practical applications in fractional calculus.