A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations

A Legendre Petrov Galerkin ( LPG) method for the third-order differential e quation is developed. By choosing appropriate base functions, the method ca n be implemented efficiently. Also, this new approach enables us to derive an optimal rate of convergence in L-2-norm. The method is applied to some n onlinear problems such as the Korteweg de Vries ( KdV) equation with the Ch ebyshev collocation treatment for the nonlinear term. It is a Legendre Petr ov Galerkin and Chebyshev collocation ( LPG-CC) method. Numerical experimen ts are given to con rm the theoretical result.