A Legendre Petrov–Galerkin method for fourth-order differential equations

In this paper, we present a Legendre Petrov–Galekin method for one-dimensional linear fourth-order differential equations. A Legendre Petrov–Galerkin and Chebyshev collocation method is developed for the nonlinear Kuramoto–Sivashinsky equation. Numerical results are presented to demonstrate the efficiency of the proposed schemes, and optimal rates of convergence in the L2-norm are rigorously derived.