Numerical Approximation of Fractional Integrals: an Optimal Quadrature Approach Based on Phi-Functions

Background: Riemann–Liouville fractional integrals are non-local operators that model memory effects in applied sciences, and they can rarely be evaluated analytically. Classical quadrature rules of Gregory, Newton–Cotes, Euler, Gauss and Markov type were designed for integer-order integration and cannot, in general, be applied directly to fractional-order operators with singular kernels. Purpose: The aim is to construct a quadrature formula for the left Riemann–Liouville fractional integral that is optimal in the sense of Sard in a Hilbert space of absolutely continuous functions with square-integrable derivative, for an arbitrary fixed set of nodes, and to find its coefficients explicitly. Method: The formula is built by the phi-function method: the error is expressed through a piecewise phi-function, and the integration error is bounded from above through the integral of its square. On each subinterval between neighboring nodes this function solves a first-order linear differential equation with a weakly singular right-hand side. The equation is solved by variation of parameters with a series expansion of the exponential, and the free constants are chosen to minimize the integral of the squared function. The coefficients then follow directly from the values of the function at the nodes. Results: For any fixed set of nodes the optimal formula exists and is unique, and its coefficients are given explicitly by series that depend on the nodes, the upper limit of integration and the order of the fractional integral. The formula is exact for the increasing and the decreasing exponential functions of unit rate, which distinguishes it from classical rules that are exact for polynomials. For equally spaced nodes it reduces to a previously known result. Implication: The paper extends the phi-function method to fractional integrals with arbitrarily placed nodes. The results are analytical and provide a basis for theoretical error bounds and for numerical comparison with existing approximation methods.