Two-dimensional integrating matrices on rectangular grids

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells

Conservative boundary conditions for 3D gas dynamics problems

A method is described for 3D-gas dynamics computer simulation in regions of complicated shape by means of nonadjusted rectangular grids providing unified treatment of various problems. Some test problem computation results are given

A few remarks on "On certain Vandermonde determinants whose variables separate"

In the recent paper \u201cOn certain Vandermonde determinants whose variables separate\u201d [Linear Algebra and its Applications 449 (2014) pp. 17\u201327], there was established a factorized formula for some bivariate Vandermonde determinants (associated to almost square grids) whose basis functions are formed by Hadamard products of some univariate polynomials. That formula was crucial for proving a conjecture on the Vandermonde determinant associated to Padua-like points. In this note we show that the same formula holds when those polynomials are replaced by arbitrary functions and we extend this formula to general rectangular grids. We also show that the Vandermonde determinants associated to Padua-like points are nonvanishing