Approximate Calculation of the Caputo Fractional-Order Derivative

Background: The Caputo fractional-order derivative of order ???? ∈ (0, 1) is represented by a weakly singular integral of the classical first-order derivative against the kernel (????−????)−????, and its accurate numerical evaluation is central to the modeling of processes with memory. Existing methods (the ????1 scheme, Grünwald–Letnikov-type sums, product-integration schemes) fix an interpolation rule for the derivative first and only afterwards estimate the resulting error, so their coefficients are not chosen to directly minimize that error. Purpose: The purpose of this work is to construct, for an arbitrary partition of the interval of integration, a quadrature formula for the Caputo derivative whose coefficients are optimal from the outset, in the sense of Sard, in a suitable Sobolev-type Hilbert space. Method: We pose the approximation problem directly in the space ????(1)2 (????, ????) and use the ????-function method: on each subinterval of the partition we construct an explicit function ???????? related to the singular kernel, determine its free constant by minimizing the corresponding quadratic error functional, and thereby obtain the coefficients of the quadrature formula in closed form. Results: Explicit closed-form expressions are obtained for the coefficients ????0, ????1, . . . , ???????? of the optimal quadrature formula for an arbitrary node partition, and these are specialized to the case of equally spaced nodes. As an independent check, the constructed formula is shown to reduce, in a particular case, to a known optimal quadrature formula obtained earlier by a different method for the generalized Abel integral equation. Implication: The result extends the ????-function construction of Sard-optimal quadrature formulas, previously used mainly for regular and oscillatory integrals, to the weakly singular kernels that arise in fractional calculus, and it provides explicit, ready-to-use coefficients for the accurate numerical evaluation of the Caputo derivative on arbitrary meshes.