A New Dual-Petrov-Galerkin Method for Third and Higher Odd-Order Differential Equations: Application to the KDV Equation

A new dual-Petrov-Galerkin method is proposed, analyzed and implemented for third and higher odd-order equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the “dual” boundary conditions. The method leads to linear systems which are sparse for problems with constant coefficients and well-conditioned for problems with variable coefficients. Our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient, and most suitable for the study of complex dynamics of higher odd-order equations.