Fully Discrete Of Spectral Collocation Method For A Partial Integral-differential Equation With A Weakly Singularity Kernel

The integro-differential equation of parabolic type often occurs in applications such as heat conduction in materials with memory, compression of viscoelastic media, nuclear reactor, dynamics, etc.. Many works have been done by Chuan-miao Chen[4], V. Thomee, W. Mclean[10], etc. domastic and overseas. FEM, finite difference scheme, spectral method and spline collocation method are the methods we ofen used to deal with this kind of equations, and we have already get some mature results on FEM[4,10] and finite difference method, but we get few results from spectral method.We consider the spatial and temporal full discrete scheme of a partial integro-differential equation of parabolic type with a weakly singular kernel. The equation is discretized in space by Galerkin spectral method and spectral collocation method, and in time finite differences of first and second oder. Through theoretical analysis, we get the stability and convergence of the solution, and we also present the error of numerical experiments.Primary results as follows:(1)The stability and error bound of Galerkin spectral method spatial semi-discretization and the results of numerical experiments are given.(2)The stability and convergence of spectral collocation method spatial semi-discretization are given.(3)We give full discrete scheme of the first and second oder of this equation, and get the stability and error estimate.