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A Legendre--Petrov--Galerkin (LPG) method for the third-order differential equation is developed. By choosing appropriate base functions, the method can be implemented efficiently. Also, this new approach enables us to derive an optimal rate of convergence in L2 -norm. The method is applied to some nonlinear problems such as the Korteweg--de Vries (KdV) equation with the Chebyshev collocation treatment for the nonlinear term. It is a Legendre--Petrov--Galerkin and Chebyshev collocation (LPG-CC) method. Numerical experiments are given to confirm the theoretical result.
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Ma et al. (2000) studied this question.
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