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In this paper, we prove that the Legendre tau method has the optimal rate of convergence in L2-norm, H1-norm and H2-norm for one-dimensional second-order steady differential equations with three kinds of boundary conditions and in C([0,T];L2(I))-norm for the corresponding evolution equation with the Dirichlet boundary condition. For the generalized Burgers equation, we develop a Legendre tau-Chebyshev...
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