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 ...

In this article we study the implementation of the Nonlinear Galerkin method as a multiresolution method when a two-level Chebyshev-collocation discretization is used. A fine grid containing an even number of Gauss-Lobatto points is considered. The grid is decomposed into two coarse grids based on half as many Gauss-Radau points. This splitting suggests a decomposition of the unknowns in low mo...

In this paper, we mainly study the superconvergence properties in approximation of second- and third-order BVPs by spectral collocation methods. The theoretical analyses identify points interpolation function. Ample numerical experiments are carried out which perfectly match results. addition, results show that also hold solving PDEs.

High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed collocating at Lobatto quadrature points. Re...

The main purpose of our work is to provide Fekete and Lebesgue points on three basic bidimensional compact sets, the simplex, the square, and the disk, by solving numerically the corresponding large-scale nonlinear optimization problems up to degree n = 18. Our results reach and often improve those previously known [1], [3]. In the case of the simplex, due to their relevance in developing spect...

We present a family of four-point quadrature rule, a generalization of Gauss-two point, Simpson’s 3/8, and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover, it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different cla...

A new adaptive quadrature algorithm that places a greater emphasis on cost reduction while still maintaining an acceptable accuracy is demonstrated. The different needs of science and engineering applications are highlighted as the existing algorithms are shown to be inadequate. The performance of the new algorithm is compared with the well known adaptive Simpson, Gauss-Lobatto and GaussKronrod...

Kronrod extensions to two classes of Gauss and Lobatto integration rules for the evaluation of Cauchy principal value integrals are derived. Since in one frequently occurring case, the Kronrod extension involves evaluating the derivative of the integrand, a new extension is introduced using n + 2 points which requires only values of the integrand. However, this new rule does not exist for all n...

We propose and analyze the spectral collocation approximation for the partial integrodifferential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomia...