A Class Of Reaction Diffusion Equation The Asymptotic Nature Of The Spectral Method And The Long Error Estimates

In the paper, a class of reaction-diffusion equation is considered.In the first part, first of all, we have a priori estimate for solution of the reaction-diffusion equation with initial condition and Dirichret boundary conditions. Then, we construct a semidiscrete Legendre pseudospectral scheme, and have a priori estimate for approximate solution. We discuss the asymptotic stability of approximate solution. At last, we get a error estimation on large time for solution of the semidiscrete system.In the second part, we construct a completely discrete Legendre pseu-dosepectral scheme. By using the same procedures as the discussion of the semidiscrete system, we get some results which are opposite to the results of semidiscrete system in the completely discrete system.