A universal Birkhoff pseudospectral method for solving boundary value problems

Inspired by the well-conditioned spectral collocation method of Wang et al., a new Birkhoff pseudospectral method for solving boundary value problems is proposed. More specifically, we develop new Birkhoff interpolants that jointly satisfy boundary conditions on both the derivatives and values of the dependent variable. This key design criterion obviates the need to generate different Birkhoff matrices for different boundary conditions while main-taining the O(1) condition number with respect to N, where, N 2 is the size of the Birkhoff matrix. In addressing the resulting theoretical and numerical problems, we generate com-putationally efficient and stable formulas for producing the new Birkhoff matrices. Fur-thermore, by transforming a generic mth order differential equation to its canonical form, we show that the same first-order Birkhoff matrix can be repeatedly used to solve a wide variety of boundary value problems. Illustrative numerical examples demonstrate the gen-erality, simplicity and utility of the new approach. Published by Elsevier Inc.