A Well-Conditioned Spectral Integration Method for High-Order Differential Equations with Variable Coefficients

A well-conditioned spectral integration (SI) method is introduced, developed and applied to nth-order differential equations with variable coefficients and general boundary conditions. The approach is based on integral reformulation techniques which lead to almost banded linear matrices, and the main system to be solved is further banded by utilizing a Schur complement approach. Numerical experiments indicate the spectral integration method can solve high order equations efficiently, oscillatory problems accurately and is adaptable to large systems. Applications in Korteweg-de Vries (KdV) type and Kawahara equations are carried out to illustrate the proposed method is effective to complicated mathematical models.