Optimal Methods for Approximating the Riemann–Liouville Fractional Integral in the Sobolev Space

Abstract

We construct and analyze optimal quadrature formulas for the right-sided Riemann–Liouville fractional integral in the Sobolev space L2(m)(t,1), using φ,φ′,…,φ(m−1) at N+1 equally spaced nodes. The analysis is carried out through the Peano kernel of the error functional, for which we give an exact closed form; this yields the norm of the error functional, existence and uniqueness of the optimal coefficients by an orthogonal-projection argument, and computable error bounds. Our main result is structural: the functions attached to the coefficients form a basis of the space of discontinuous piecewise polynomials of degree m−1 on the mesh, so that the optimality problem is an L2 projection that decouples panel by panel. Consequently, the globally optimal (Sard) coefficients—not only the sequentially optimal ones—are available in closed form for every m, at a cost of O(m3N) operations with no global linear system and with a condition number independent of N, h, α and t. The globally optimal rule is exact on polynomials of degree 2m−1 and converges as O(h2m) for smooth integrands, whereas the sequential rule is exact on degree m; we quantify the gap between them. We also prove sharp asymptotics for the error norm, ∥R∥2∼κm(1−t)2α−1(2α−1)−1h2m for α>12 with κm=(−1)m−1B2m/(2m)!, with a ln(1/h) factor exactly at α=12 and a loss of half an order for α<12. An extensive numerical study over five fractional orders, three evaluation points and eight integrands of prescribed Sobolev regularity confirms every theoretical statement, and the formulas are compared with product, spline and Gauss–Jacobi quadratures and applied to an Abel integral equation.