The pseudospectral method for third-order differential equations

Abstract

Generalized quadrature rules are derived which assist in the selection of collocation points for the pseudospectral solution of differential equations. In particular, it is shown that for an nth-order differential equation in one space dimension with two-point derivative boundary conditions, an ideal choice of interior collocation points is the set of zeros of a Jacobi polynomial. The pseudospectral solution of a third-order initial-boundary value problem is considered and accuracy is assessed by examining how well the discrete eigenproblem approximates the continuous one. Convergence is established for a special choice of collocation points and numerical results are included to demonstrate the viability of the approach.