The Global Behavior Of Spectral Collocation Methods Full Discretization For A Partial Integro-differential Equation With A Weakly Singular Kernel

The integro-differential equation of parabolic type often occurs in applications such as heat conduction in material with memory, compression of poro-viscoelastic media, population dynamics, nuclear reactor dynamics, etc.. There are lots of documents of V. Thomee [1 , 5 , 7, 16 , 17 , 18, 19 , 20 , 21, 22 , 23 , 24 , 31], Stig. Larsson [19]. W. M(?)l(?)an [5 , 17 , 20,24], Ch. Lubich[18], J. C. Lopez-Marcos [14], J. M. Sanz-Serna [6], G. Fairweather[3 , 15], L. Wahlbin [1 , 17 , 19], I. H. Sloan [7 , 18 , 22 , 23], Yanping Lin [31] in overseas and Chuan-miao Chen [1 , 35], Yun-qing Huang [2], Da Xu [8 , 9 , 10 , 11 , 12 , 13], Tao Tang [33], Qiya Hu [34], Tie Zhang [45] in home. A lot of them use FEM;Spline collocation methods;finite difference methods;Spectral collocation methods. But a few of them make the global behavior of full discretization by spectral methods.We study a partial integro-defferential equations of parabolic type with a weakly singular kernel, using spectral collocation derived stabilities and error estimated respectively.Main results follows:(1)Given the stability, error estimate of spatial semi-discretization Legendre-Galerkin spectral methods for the linear equation;(2)We discuss the stability and estimate of spatial semi-discretization based on the spectral collocation methods;(3)Given the stability, error estimate of full discretization for the linear equation.(4)Numerical experiments.