Gauss–Jacobi-type quadrature rules for fractional directional integrals

Fractional directional integrals are the extensions of the Riemann–Liouville fractional integrals from one- to multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton–Cotes and Gauss–Legendre rules. It is noted that these kernels after simple transforms can be taken as the Jacobi weight functions which are related to the weight factors of Gauss–Jacobi and Gauss–Jacobi–Lobatto rules. These rules can evaluate the fractional integrals at high accuracy. Comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the Gauss–Jacobi-type rules in handling weakly singular kernels of different strengths. Potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section.