few problems lend themselves to closed-form solution, we often need to convert formal definitions into practical numerical methods. One such problem deals with the Principal Value integral, which many students encounter in a course on functions of a complex variable. However, the prospect of evaluating one numerically might seem rather daunting. To the best of my knowledge, the subject remains ...

A general framework is introduced to analyze the approximation properties of mapped Legendre polynomials and of interpolations based on mapped Legendre–Gauss–Lobatto points. Optimal error estimates featuring explicit expressions on the mapping parameters for several popular mappings are derived. These results not only play an important role in numerical analysis of mapped Legendre spectral and ...

This paper gives an efficient numerical method for solving the nonlinear system of Volterra-Fredholm integral equations. A Legendre-spectral method based on the Legendre integration Gauss points and Lagrange interpolation is proposed to convert the nonlinear integral equations to a nonlinear system of equations where the solution leads to the values of unknown functions at collocation points.

This is to certify that I have examined a copy of a technical report by Kazh Brito and found it satisfactory in all respects, and that any and all revisions required by the examining committee have been made. Abstract This is an exploration of Legendre spectral finite-element (LSFE) formulations for Reissner-Mindlin plates. The goal was to compare high-order LSFEs with standard low-order finite...

We introduce a new and efficient Chebyshev-Legendre Galerkin method for elliptic problems. The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation. Hence, it enjoys advantages of both the LegendreGalerkin and Chebyshev-Galerkin methods.

1. Introduction. The estimation of quadrature errors for analytic functions has been considered by Davis and Rabinowitz [1]. An estimate for the error of the Gaussian quadrature formula for analytic functions was obtained by Davis [2]. McNamee [3] has also discussed the estimation of error of the Gauss-Legendre quadrature for analytic functions. Convergence of the Gaussian quadratures was discu...

An asymptotic error expansion for Gauss-Legendre quadrature is derived for an integrand with an endpoint singularity. It permits convergence acceleration by extrapolation.

Smolyak’s sparse grid construction is commonly used in a setting involving quadrature of a function of a multidimensional argument over a product region. However, the method can be applied in a straightforward way to the interpolation problem as well. In this discussion, we outline a procedure that begins with a family of interpolants defined on a family of nested tensor product grids, and demo...