In this paper, we investigate a new pseudospectral method for mixed boundary value problems defined on quadrilaterals. We introduce a new Legendre-Gauss type interpolation and establish the basic approximation results, which play important roles in pseudospectral method for partial differential equations defined on quadrilaterals. We propose pseudospectral method for two model problems and prov...

In this paper, we propose a natural collocation method with exact imposition of mixed boundary conditions based on a generalized Gauss-Lobatto-Legendre-Birhoff quadrature rule that builds in the underlying boundary data. We provide a direct construction of the quadrature rule, and show that the collocation method can be implemented as efficiently as the usual collocation scheme for PDEs with Di...

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the θ-methods with 1 2 ≤ θ ≤ 1, the odd stage Gauss-Legendre m...

An iterative algorithm for computing the principal nth root of a positive deenite matrix is presented. The algorithm is based on the Gauss-Legendre approximation of a deenite integral. We p resent a parallelization in which we use as many processors as the order of the approximation. An analysis of the error introduced at each step of the iteration indicates that the algorithm converges more ra...

Abstract The objective of this research is to study numerical integral equation method (NIE) using five quadrature rules namely, composite midpoint rule, trapezoidal Simpson’s Boole’s rule and Gauss-Legendre for approximating the average run length on extended exponentially weighted moving (extended EWMA) control chart autoregressive process with explanatory variables when white noise follows a...

in this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the laplacian equation. themethod is based on the use of the galerkin method with cas wavelets constructed on the unit interval as basis.this approach utilizes the non-uniform gauss-legendre quadrature rule for ...

We now come to the most important result in our course: the law of quadratic reciprocity, or, as Gauss called it, the aureum theorema (“golden theorem”). Many beginning students of number theory have a hard time appreciating this golden theorem. I find this quite understandable, as many first courses do not properly prepare for the result by discussing enough of the earlier work which makes qua...