A combined integrating-and differentiating-matrix formulation for boundary-value problems on rectangular domains (Q1081390)

Integrating and differentiating matrices allow the numerical integration and differentiation of functions whose values are known at points of a discrete grid. Previous derivations of these matrices have been restricted to one-dimensional grids or to rectangular grids with uniform spacing in at least one direction. The present work develops integrating and differentiating matrices for rectangular grids with non-uniform spacing in both directions. The use of these matrices as operators to reformulate boundary-value problems on rectangular domains as matrix problems for a finite-dimensional solution vector is considered. The method requires non-uniform grids which include ''near-boundary'' points. An eigenvalue problem for the transverse vibrations of a simply-supported rectangular plate is solved to illustrate the method.