Composite Fractional Trapezoidal Rule with Romberg Integration

The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator. This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the (n + 1)-composite Trapezoidal rule. In particular, the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule, which has accordingly been used to derive a more accurate fractional-order formula called the (n + 1)-composite Trapezoidal rule. Additionally, in order to increase the accuracy of the proposed approximations by reducing the true errors, we incorporate the so-called Romberg integration, which is an extrapolation formula of the Trapezoidal rule for integration, into our proposed approaches. Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.