Legendre-Galerkin method for nonlinear partial differential equations

In this paper, a Legendre-Galerkin method is proposed and analyzed for the Burgers equation with dirichlet boundary condition. We present in this paper the error estimation concerning Legendre approximations in Sobolev spaces, in which integration is performed with respect to the Legendre weight omega(x) = 1. It is shown that the Legendre-Galerkin approximations are convergent on the interval [-1,1] with spectral accuracy. An efficient and accurate algorithm based on the Legendre-Galerkin approximations to the Burgers equation is developed and implemented. Finally the numerical results which indicate that the high accuracy and effectiveness of this algorithm are presented.