Fractional directional integrals are the extensions of the Riemann-Liouville fractional integrals from oneto 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 ...

A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are i) non-repeating interior point operators, ii) nonuniform nodal distribution in the computational domain, iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditio...

Gauss quadrature rules associated with a nonnegative measure support on (part of) the real axis find many applications in Scientific Computing. It is important to be able estimate error when replacing an integral by ℓ-node rule order choose suitable number of nodes. A classical approach this evaluate (2ℓ+1)-node Gauss–Kronrod rule. However, 2ℓ+1 nodes might not exist. The generalized averaged f...

We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the “half-point rule”, indicating that optimal rules involve a number of points roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order o...